Program Studi
Sains Data
Universitas
INSTITUT TEKNOLOGI SAINS BANDUNG
Dosen Pengampu
Bakti Siregar, M Sc., CSD
STUDY CASES
Confidence Interval - Week 13
Case Study 1
Confidence Interval for Mean, \(\sigma\) Known: An
e-commerce platform wants to estimate the
average number of daily transactions per user after
launching a new feature. Based on large-scale historical data, the
population standard deviation is known.
\[
\begin{eqnarray*}
\sigma &=& 3.2 \quad \text{(population standard deviation)} \\
n &=& 100 \quad \text{(sample size)} \\
\bar{x} &=& 12.6 \quad \text{(sample mean)}
\end{eqnarray*}
\]
Identify the
appropriate statistical test and justify your choice.
Karena tujuan analisisnya adalah mengestimasi mean populasi
\(\mu\) dan nilai standar deviasi
populasi \(\sigma\) diketahui
(\(\sigma = 3.2\)), maka metode yang
sesuai adalah confidence interval berbasis Z untuk
mean:
\[
\bar{x} \pm z_{\alpha/2} \left(\frac{\sigma}{\sqrt{n}}\right)
\]
Compute the
Confidence Intervals for:
- \(90\%\)
- \(95\%\)
- \(99\%\)
Data:
\[
\sigma = 3.2, \quad n = 100, \quad \bar{x} = 12.6
\]
- Hitung Standard Error (SE):
\[
SE = \frac{\sigma}{\sqrt{n}}
= \frac{3.2}{\sqrt{100}}
= \frac{3.2}{10}
= 0.32
\]
- Rumus CI (karena \(\sigma\)
known → Z):
\[
CI = \bar{x} \pm z_{\alpha/2} \cdot SE
\]
1. Confidence Interval 90%
Nilai kritis: \[
z_{0.95} = 1.645
\]
Margin of Error: \[
ME = 1.645 \times 0.32 = 0.526
\]
CI 90%: \[
12.6 \pm 0.526 \;\Rightarrow\; (12.074,\; 13.126)
\]
2. Confidence Interval 95%
Nilai kritis: \[
z_{0.975} = 1.960
\]
Margin of Error: \[
ME = 1.960 \times 0.32 = 0.627
\]
CI 95%: \[
12.6 \pm 0.627 \;\Rightarrow\; (11.973,\; 13.227)
\]
3. Confidence Interval 99%
Nilai kritis: \[
z_{0.995} = 2.576
\]
Margin of Error: \[
ME = 2.576 \times 0.32 \approx 0.824
\]
CI 99%: \[
12.6 \pm 0.824 \;\Rightarrow\; (11.776,\; 13.424)
\]
comparison
visualization of the three confidence intervals.

Interpret the
results in a business analytics context.
Visualisasi ini menunjukkan perkiraan rata-rata transaksi
harian per user berada di sekitar \(\bar{x} = 12.6\) (garis vertikal/titik
tengah).lalu tiga confidence interval (garis horizontal) memberi rentang
nilai mean populasi yang masuk akal. yaitu: - CL 90% =
(12.074, 13.126) paling sempit. - CL 95%
= (11.973, 13.227) lebih lebar. - CL
99% = (11.776, 13.424) paling lebar.
Intinya, semakin tinggi tingkat
kepercayaan, rentang makin melebar karena kita
ingin “lebih yakin” mean sebenarnya ada di dalam rentang
itu.
jadi, toleransi ketidakpastian
diperbesar. akibatnya 90% paling presisi tapi kurang
konservatif, sedangkan 99% paling aman/konservatif tapi
paling tidak presisi.
Case Study 2
Confidence Interval for Mean, \(\sigma\) Unknown: A UX
Research team analyzes task completion time (in
minutes) for a new mobile application. The data are collected
from 12 users:
\[
8.4,\; 7.9,\; 9.1,\; 8.7,\; 8.2,\; 9.0,\;
7.8,\; 8.5,\; 8.9,\; 8.1,\; 8.6,\; 8.3
\]
Identify the
appropriate statistical test
Metode yang tepat adalah confidence interval berbasis
distribusi t (one-sample t-interval) untuk mean, karena yang
ingin diestimasi adalah rata-rata populasi waktu
penyelesaian tugas, sementara standar deviasi populasi (σ) tidak
diketahui sehingga harus diganti dengan standar deviasi
sampel (s), dan ukuran sampel juga kecil (n =
12) sehingga pendekatan yang paling sesuai menggunakan
t-Student dengan derajat bebas df = 11.
Compute the
Confidence Intervals for:
- \(90\%\)
- \(95\%\)
- \(99\%\)
Data:
\[
n = 12,\quad df = n - 1 = 11
\]
\[
x = (8.4,\; 7.9,\; 9.1,\; 8.7,\; 8.2,\; 9.0,\; 7.8,\; 8.5,\; 8.9,\;
8.1,\; 8.6,\; 8.3)
\]
- Hitung rata-rata sampel (\(\bar{x}\)):
\[
\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} = 8.4583
\]
- Hitung standar deviasi sampel (\(s\)):
\[
s = 0.4209
\]
- Hitung Standard Error (SE):
\[
SE = \frac{s}{\sqrt{n}}
= \frac{0.4209}{\sqrt{12}}
= 0.1215
\]
- Rumus CI (karena \(\sigma\)
unknown \(\rightarrow\) pakai
distribusi t):
\[
CI = \bar{x} \pm t_{\alpha/2,\,df} \cdot SE
\]
1. Confidence Interval 90%
Nilai kritis: \[
t_{0.95,\,11} = 1.796
\]
Margin of Error: \[
ME = 1.796 \times 0.1215 = 0.2182
\]
CI 90%: \[
8.4583 \pm 0.2182 \;\Rightarrow\; (8.2401,\; 8.6766)
\]
2. Confidence Interval 95%
Nilai kritis: \[
t_{0.975,\,11} = 2.201
\]
Margin of Error: \[
ME = 2.201 \times 0.1215 = 0.2675
\]
CI 95%: \[
8.4583 \pm 0.2675 \;\Rightarrow\; (8.1909,\; 8.7258)
\]
3.Confidence Interval 99%
Nilai kritis: \[
t_{0.995,\,11} = 3.106
\]
Margin of Error: \[
ME = 3.106 \times 0.1215 = 0.3774
\]
CI 99%: \[
8.4583 \pm 0.3774 \;\Rightarrow\; (8.0809,\; 8.8357)
\]
Visualize the
three intervals on a single plot.

Explain
sample size and confidence level influence the interval
width.
Lebar confidence interval untuk mean (\(\sigma\) unknown) ditentukan oleh:
\[
CI = \bar{x} \pm t_{\alpha/2,\,df}\left(\frac{s}{\sqrt{n}}\right)
\]
Jadi “lebar” CI bergantung pada dua hal utama:
(1) Sample size
Kalau \(n\) makin
besar, maka \(\dfrac{s}{\sqrt{n}}\) (standard
error) makin kecil. \(\rightarrow\) interval makin sempit (lebih
presisi).
Kalau \(n\)
kecil (kayak di kasus ini \(n =
12\)), standard error masih relatif besar \(\rightarrow\) interval lebih
lebar.
(2) Confidence level (90% vs 95% vs 99%)
Semakin tinggi confidence level, nilai kritis
\(t_{\alpha/2,\,df}\) makin besar \(\rightarrow\) interval makin
lebar.
Makanya urutannya selalu: \[
\text{CI 90\% (paling sempit)}
< \text{ CI 95\% }
< \text{ CI 99\% (paling lebar)}
\]
Intinya: \(n\)
besar bikin CI menyempit, sedangkan confidence level tinggi bikin CI
melebar (lebih aman tapi kurang presisi).
Case Study 3
Confidence Interval for a Proportion, A/B Testing: A
data science team runs an A/B test on a new
Call-To-Action (CTA) button design. The experiment yields:
\[
\begin{eqnarray*}
n &=& 400 \quad \text{(total users)} \\
x &=& 156 \quad \text{(users who clicked the CTA)}
\end{eqnarray*}
\]
Compute the
sample proportion \(\hat{p}\).
Diketahui:
\[
n = 400 \quad \text{(total users)}
\] \[
x = 156 \quad \text{(users yang klik CTA)}
\]
Proporsi sampel:
\[
\hat{p} = \frac{x}{n} = \frac{156}{400} = 0.39
\]
Jadi sample proportion / CTR sampel = 0.39 (atau 39%).
- Compute Confidence Intervals for the proportion at:
- \(90\%\)
- \(95\%\)
- \(99\%\)
Data:
\[
n = 400,\quad x = 156,\quad \hat{p} = \frac{156}{400} = 0.39
\]
- Hitung Standard Error (SE) untuk proporsi:
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
= \sqrt{\frac{0.39(1 - 0.39)}{400}}
= \sqrt{\frac{0.39 \cdot 0.61}{400}}
= \sqrt{\frac{0.2379}{400}}
= \sqrt{0.00059475}
\approx 0.0244
\]
\[
CI = \hat{p} \pm z_{\alpha/2} \cdot SE
\]
1. Confidence Interval 90%
Nilai kritis: \[
z_{0.95} = 1.645
\]
Margin of Error: \[
ME = 1.645 \times 0.0244 = 0.0401
\]
CI 90%: \[
0.39 \pm 0.0401 \;\Rightarrow\; (0.3499,\; 0.4301)
\]
2. Confidence Interval 95%
Nilai kritis: \[
z_{0.975} = 1.960
\]
Margin of Error: \[
ME = 1.960 \times 0.0244 = 0.0478
\]
CI 95%: \[
0.39 \pm 0.0478 \;\Rightarrow\; (0.3422,\; 0.4378)
\]
3. Confidence Interval 99%
Nilai kritis: \[
z_{0.995} = 2.576
\]
Margin of Error: \[
ME = 2.576 \times 0.0244 = 0.0628
\]
CI 99%: \[
0.39 \pm 0.0628 \;\Rightarrow\; (0.3272,\; 0.4528)
\]
Visualize and
compare the three intervals.

Explain
confidence level affects decision-making in product
experiments.
Dalam eksperimen produk (misalnya A/B testing CTA),
confidence level menentukan seberapa “ketat” kita saat
menyimpulkan CTR sebenarnya. Dari visualisasi, terlihat bahwa
CI 90% paling sempit, CI 95% lebih
lebar, dan CI 99% paling lebar. artinya
semakin tinggi confidence level, semakin lebar rentang CTR yang
dianggap masuk akal, karena kita ingin keyakinan yang
lebih tinggi.
Rekomendasi untuk pengambilan keputusan:
Confidence level tinggi (99%) → interval lebih
lebar → keputusan lebih konservatif/aman (risiko salah klaim lebih
kecil), tetapi bisa lebih sering “belum cukup bukti” untuk
mengambil keputusan cepat.
Confidence level lebih rendah (90%) → interval
lebih sempit → keputusan lebih cepat dan responsif, tetapi
risikonya lebih besar dibanding 95%/99%.
Rekomendasi:
Untuk keputusan high-risk (rollout besar, dampak revenue
besar, biaya tinggi) → lebih cocok pakai 95%–99%.
Untuk keputusan low-risk (iterasi UI kecil, eksplorasi
awal) → 90%–95% biasanya sudah cukup.
Case Study 4
Precision Comparison (Z-Test vs t-Test): Two data
teams measure API latency (in milliseconds) under
different conditions.
\[\begin{eqnarray*}
\text{Team A:} \\
n &=& 36 \quad \text{(sample size)} \\
\bar{x} &=& 210 \quad \text{(sample mean)} \\
\sigma &=& 24 \quad \text{(known population standard deviation)}
\\[6pt]
\text{Team B:} \\
n &=& 36 \quad \text{(sample size)} \\
\bar{x} &=& 210 \quad \text{(sample mean)} \\
s &=& 24 \quad \text{(sample standard deviation)}
\end{eqnarray*}\]
Identify the
statistical test used by each team.
Team A menggunakan Z confidence
interval untuk mean karena standar deviasi populasi
diketahui (\(\sigma =
24\)).
Sedangkan Team B menggunakan t confidence
interval untuk mean karena standar deviasi populasi
tidak diketahui dan digantikan oleh standar deviasi sampel
(\(s = 24\)) dengan derajat bebas \(df = n - 1 = 35\).
Compute
Confidence Intervals for 90%, 95%, and 99%.
Data
Team A: \(n = 36,\
\bar{x} = 210,\ \sigma = 24\) (\(\sigma\) diketahui)
Team B: \(n = 36,\
\bar{x} = 210,\ s = 24\) (\(\sigma\) tidak diketahui)
Standard error (keduanya sama):
\[
SE = \frac{24}{\sqrt{36}} = \frac{24}{6} = 4
\]
Team A (Z-interval, \(\sigma\) known)
\[
CI = \bar{x} \pm z_{\alpha/2} \cdot SE
\]
90%: \[
210 \pm 1.645(4)
\;\Rightarrow\;
(203.421,\; 216.579)
\]
95%: \[
210 \pm 1.960(4)
\;\Rightarrow\;
(202.160,\; 217.840)
\]
99%: \[
210 \pm 2.576(4)
\;\Rightarrow\;
(199.696,\; 220.304)
\]
Team B (t-interval, \(\sigma\) unknown, \(df = 35\))
\[
CI = \bar{x} \pm t_{\alpha/2,35} \cdot SE
\]
90%: \[
210 \pm 1.690(4)
\;\Rightarrow\;
(203.242,\; 216.758)
\]
95%: \[
210 \pm 2.030(4)
\;\Rightarrow\;
(201.880,\; 218.120)
\]
99%: \[
210 \pm 2.724(4)
\;\Rightarrow\;
(199.105,\; 220.895)
\]
Visualization
comparing all intervals.

Explain why
the interval widths differ, even with similar data.
Meskipun Team A dan Team B punya ringkasan data yang terlihat
sama (\(n = 36\), \(\bar{x} = 210\), dan ukuran sebaran 24),
lebar intervalnya tetap berbeda karena sumber
ketidakpastiannya beda. Team A memakai Z-interval karena \(\sigma\) populasi benar-benar
diketahui, sehingga ketidakpastian hanya berasal dari proses
sampling.
Sementara itu Team B memakai t-interval karena \(\sigma\) tidak diketahui dan harus
diperkirakan dari sampel (pakai \(s\)),
sehingga ada ketidakpastian tambahan.
Akibatnya, nilai kritis distribusi t untuk level
kepercayaan yang sama sedikit lebih besar daripada nilai
z, sehingga margin of error menjadi lebih besar dan
interval Team B jadi lebih lebar.
Jadi, perbedaan lebar interval terutama karena
t-interval lebih konservatif saat \(\sigma\) tidak diketahui.
Case Study 5
One-Sided Confidence Interval: A Software as
a Service (SaaS) company wants to ensure that at least
70% of weekly active users utilize a premium feature.
From the experiment:
\[
\begin{eqnarray*}
n &=& 250 \quad \text{(total users)} \\
x &=& 185 \quad \text{(active premium users)}
\end{eqnarray*}
\]
Management is only interested in the lower bound of
the estimate.
Identify the type of
Confidence Interval and the appropriate test.
Case Study 5 menggunakan confidence interval satu sisi
(one-sided) dengan fokus pada batas bawah itu sebagai proporsi
pengguna aktif yang memakai fitur premium.
Metode yang tepat adalah , karena parameter yang diestimasi
adalah proporsi \(p\) dan ukuran sampel
cukup besar (\(n = 250\))
sehingga pendekatan normal dapat digunakan.
Karena perusahaan ingin memastikan “minimal 70%”,
analisis ini juga sejalan dengan pengujian satu arah untuk target, yaitu
\[
H_0 : p \le 0.70
\quad \text{versus} \quad
H_1 : p > 0.70.
\]
Compute the
one-sided lower Confidence Interval
- \(90\%\)
- \(95\%\)
- \(99\%\)
Data:
\[
n = 250,\quad x = 185,\quad \hat{p} = \frac{x}{n} = \frac{185}{250} =
0.74
\]
- Hitung Standard Error (SE) proporsi:
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
= \sqrt{\frac{0.74(1 - 0.74)}{250}}
= \sqrt{\frac{0.74 \cdot 0.26}{250}}
= \sqrt{0.0007696}
\approx 0.02774
\]
- Rumus one-sided lower CI (Z):
Pada confidence interval dua sisi, kita biasanya menghitung dua
batas:
\[
CI = \hat{p} \pm z_{\alpha/2} \cdot SE
\]
Namun pada kasus ini manajemen hanya tertarik pada
batas bawah (“minimal proporsinya berapa”).
Karena itu digunakan , sehingga kita hanya menghitung batas
bawah yang dinotasikan dengan \(L\), di mana \(L\) adalah singkatan dari (batas
bawah).
Untuk confidence level \(C\), batas
bawah one-sided ditulis:
\[
L = \hat{p} - z_C \cdot SE
\]
dan interval one-sided dapat dituliskan sebagai:
\[
(L,\, 1)
\]
karena nilai proporsi maksimum adalah 1.
1. One-sided lower CI 90%
Nilai kritis: \[
z_{0.90} = 1.282
\]
Margin of Error: \[
ME = 1.282 \times 0.02774 = 0.03555
\]
Lower bound 90%: \[
L = 0.74 - 0.03555 = 0.70445
\]
Jadi one-sided lower CI 90%: \[
(0.70445,\; 1)
\]
2. One-sided lower CI 95%
Nilai kritis: \[
z_{0.95} = 1.645
\]
Margin of Error: \[
ME = 1.645 \times 0.02774 = 0.04563
\]
Lower bound 95%: \[
L = 0.74 - 0.04563 = 0.69437
\]
Jadi one-sided lower CI 95%: \[
(0.69437,\; 1)
\]
Visualize the
lower bounds for all confidence levels.

Determine whether the
70% target is statistically satisfied.
Untuk menentukan apakah target 70% “terpenuhi secara
statistik”, kita bandingkan batas bawah (lower bound) one-sided
CI dengan 0.70. Jika \(L \ge
0.70\), maka kita bisa yakin (sesuai tingkat confidence)
bahwa proporsi sebenarnya setidaknya 70%.
Dari hasil yang sudah dihitung:
Lower bound 90%: \[
L_{90} = 0.70445
\] lebih besar dari 0.70 \(\Rightarrow\) target pada 90%
confidence.
Lower bound 95%: \[
L_{95} = 0.69437
\] lebih kecil dari 0.70 \(\Rightarrow\) target pada 95%
confidence.
Lower bound 99%: \[
L_{99} = 0.67546
\] lebih kecil dari 0.70 \(\Rightarrow\) target pada 99%
confidence.
Kesimpulannya, target 70% hanya dapat dinyatakan
terpenuhi secara statistik pada tingkat kepercayaan 90%, tetapi tidak
terpenuhi jika perusahaan mensyaratkan keyakinan yang lebih ketat
seperti 95% atau 99%.
---
title: "Study Cases"            
subtitle: "Confidence Interval~ Week 13"  
author: "Nurul Iffah"      
date:  "`r format(Sys.Date(), '%B %d, %Y')`" 
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  --icon-moon-light: url("data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHZpZXdCb3g9IjAgMCAyNCAyNCIgZmlsbD0iI0Y4RjlCQSIgc3Ryb2tlPSIjNTU1NTU1IiBzdHJva2Utd2lkdGg9IjIiIHN0cm9rZS1saW5lY2FwPSJyb3VuZCIgc3Ryb2tlLWxpbmVqb2luPSJyb3VuZCI+PHBhdGggZD0iTTIxIDEyLjc5QTkgOSAwIDEgMSAxMS4yMSAzYTcgNyAwIDAgMCA5Ljc5IDkuNzl6Ii8+PC9zdmc+");
  --icon-sun-dark: url("data:image/svg+xml;base64,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");
  --icon-moon-dark: url("data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHZpZXdCb3g9IjAgMCAyNCAyNCIgZmlsbD0iI0Q2RENGRiIgc3Ryb2tlPSIjRjZGOEZGIiBzdHJva2Utd2lkdGg9IjIiIHN0cm9rZS1saW5lY2FwPSJyb3VuZCIgc3Ryb2tlLWxpbmVqb2luPSJyb3VuZCI+PHBhdGggZD0iTTIxIDEyLjc5QTkgOSAwIDEgMSAxMS4yMSAzYTcgNyAwIDAgMCA5Ljc5IDkuNzl6Ii8+PC9zdmc+");
}

*,
*::before,
*::after {
  box-sizing: border-box;
}

html,
body {
  height: 100%;
  margin: 0;
  padding: 0;
}

body {
  font-family: "Quicksand", "Trebuchet MS", sans-serif;
  font-size: 16px;
  line-height: 1.7;
  color: var(--text-body);
  background: linear-gradient(135deg, var(--bg-left) 0%, var(--bg-right) 100%);
}

h1.title,
h2.title,
h3.title,
h1.subtitle,
h2.subtitle,
h3.subtitle {
  display: none !important;
}

#main p {
  margin: 0.35rem 0 0.9rem 0;
}

#main ul,
#main ol {
  margin: 0.2rem 0 0.9rem 0;
  padding-left: 1.4rem;
}

#main li {
  margin: 0.15rem 0;
}

#main ul li::marker {
  content: "🐰 ";
  color: var(--text-title);
  font-size: 0.9rem;
}

a {
  color: var(--text-body);
  text-decoration: none;
  border-bottom: 1px dashed var(--link-underline);
  transition: all 0.18s ease;
}

a:hover {
  color: var(--text-body);
  background: var(--link-hover-bg);
  border-bottom-style: solid;
}

hr {
  border: none;
  height: 2px;
  max-width: 72%;
  margin: 1.4rem auto;
  border-radius: 999px;
  background: linear-gradient(90deg, transparent, var(--ear-pink), var(--snow-fur), transparent);
}

h1,
h2,
h3,
h4,
h5 {
  font-family: "Fredoka", "Quicksand", sans-serif;
  color: var(--text-title);
  letter-spacing: 0.02em;
}

body #content {
  max-width: 1200px !important;
  margin: 0 auto !important;
  padding: 2.4rem 2.4rem 3rem !important;
  transition: margin-left 0.25s ease, max-width 0.25s ease;
}

body #main {
  max-width: 1100px !important;
  width: 100% !important;
  margin: 2rem auto 3rem !important;
  padding: 2rem 2.4rem 2.6rem !important;
  background: var(--card-bg);
  border-radius: var(--radius-lg) !important;
  box-shadow: var(--shadow-soft) !important;
  border: 1px solid var(--label-gray);
}

#sidebar {
  background: var(--sidebar-bg);
  color: var(--text-title);
  padding-top: 1.2rem;
  width: 280px;
  border-right: 1px solid var(--sidebar-border);
  box-shadow: var(--sidebar-shadow);
}

#sidebar h2 {
  background: linear-gradient(135deg, var(--snow-fur), var(--ear-pink));
  color: var(--text-title);
  font-family: "Fredoka", "Quicksand", sans-serif;
  font-size: 1rem;
  letter-spacing: 0.06em;
  text-align: center;
  padding: 0.7rem 0.6rem;
  border-radius: 18px;
  box-shadow: var(--sidebar-h2-shadow);
  margin: 0 0.9rem 1rem 0.9rem;
}

#sidebar .tocify,
#sidebar ul {
  font-size: 0.95rem;
}

#sidebar a {
  display: block;
  padding: 4px 10px;
  border-radius: 999px;
  margin: 1px 0;
  color: var(--text-title);
}

#sidebar a:hover,
#sidebar .tocify-focus > a {
  background: var(--snow-fur) !important;
  color: var(--text-title) !important;
  font-weight: 600;
}

#postamble {
  background: linear-gradient(135deg, var(--bg-left), var(--bg-right));
  border-top: 2px solid var(--label-gray);
  color: var(--text-title);
  padding: 0.7rem 0.9rem 0.8rem;
  font-size: 0.85rem;
  text-align: center;
}

#postamble p {
  margin: 0.15rem 0;
}

#postamble span,
#postamble a {
  color: var(--text-title) !important;
}

#postamble .date {
  color: var(--text-body) !important;
}

#postamble .date .glyphicon {
  color: inherit;
}

#postamble .author .glyphicon.glyphicon-user {
  font-size: 0;
  position: relative;
}

#postamble .author .glyphicon.glyphicon-user::before {
  content: "🧕";
  font-size: 1rem;
  margin-right: 0.35rem;
  position: relative;
  top: 1px;
}

#postamble .date .glyphicon.glyphicon-calendar {
  margin-right: 0.35rem;
}

body:not(.sidebar-collapsed) #content {
  margin-left: 300px !important;
  max-width: calc(100% - 300px) !important;
}

@media (max-width: 900px) {
  body:not(.sidebar-collapsed) #content {
    margin-left: 0 !important;
    max-width: 100% !important;
  }

  #sidebar {
    width: 80vw;
    max-width: 260px;
    position: fixed;
    top: 0;
    left: 0;
    height: 100%;
    overflow-y: auto;
    z-index: 9990;
    padding-bottom: 3rem;
  }

  body:not(.sidebar-collapsed) #sidebar {
    box-shadow: var(--sidebar-shadow-open);
  }
}

#main .section.level2 {
  max-width: 980px;
  margin: 1.6rem auto;
  padding: 1.3rem 1.5rem 1.6rem;
  border-radius: 18px;
  border: 1px solid var(--label-gray);
  border-left: 6px solid var(--ear-pink);
  box-shadow: var(--shadow-soft);
  background: var(--card-bg);
}

#main .section.level2 > h2 {
  margin-top: 0;
  display: inline-block;
  padding: 0.45rem 0.9rem;
  border-radius: 14px;
  background: var(--snow-fur);
  color: var(--text-title);
  box-shadow: var(--section-title-shadow);
}

#main .section.level3 > h3 {
  display: inline-block;
  padding: 0.35rem 0.75rem;
  border-radius: 12px;
  background: var(--label-gray);
  color: var(--text-title);
}

.logo-wrapper {
  display: flex;
  flex-direction: column;
  align-items: center;
  margin: 1.4rem 0 1rem;
}

.logo-frame {
  position: relative;
  width: 180px;
  height: 180px;
  aspect-ratio: 1 / 1;
  border-radius: 50%;
  padding: 8px;
  background: conic-gradient(from 210deg, var(--snow-fur), var(--ear-pink), var(--label-gray), var(--bg-right), var(--accent-star), var(--snow-fur));
  box-shadow: var(--logo-frame-shadow);
}

.logo-inner {
  width: 100%;
  height: 100%;
  aspect-ratio: 1 / 1;
  border-radius: 50%;
  overflow: hidden;
  background: radial-gradient(circle at 30% 0, #ffffff 0, var(--card-bg) 60%, var(--accent-star) 100%);
  display: flex;
  align-items: center;
  justify-content: center;
}

#main .logo-inner img {
  width: 100% !important;
  height: 100% !important;
  max-width: none !important;
  max-height: none !important;
  object-fit: cover;
  object-position: center;
  display: block;
  margin: 0 !important;
  padding: 0 !important;
  border: 0 !important;
  border-radius: 50% !important;
  box-shadow: none !important;
  pointer-events: none;
  user-select: none;
}

.logo-frame::before {
  content: "*";
  position: absolute;
  top: -4px;
  right: 18px;
  font-size: 1.1rem;
  color: var(--accent-star);
  text-shadow: var(--logo-star-shadow);
}

.logo-frame::after {
  content: "o";
  position: absolute;
  bottom: 6px;
  left: 18px;
  font-size: 0.9rem;
  color: var(--ear-pink);
  text-shadow: var(--logo-dot-shadow);
}

.logo-caption {
  margin-top: 0.35rem;
  font-size: 0.9rem;
  font-weight: 700;
  letter-spacing: 0.12em;
  color: var(--text-title);
}

.info-badges {
  display: flex;
  flex-wrap: wrap;
  justify-content: center;
  gap: 0.8rem;
  margin: 1rem 0 0.6rem;
}

.badge-card {
  min-width: 170px;
  padding: 0.7rem 1.2rem;
  border-radius: 16px;
  text-align: center;
  box-shadow: var(--shadow-card);
  border: 1px solid var(--label-gray);
}

.badge-primary {
  background: var(--snow-fur);
}

.badge-secondary {
  background: var(--ear-pink);
}

.badge-tertiary {
  background: var(--label-gray);
}

.badge-label {
  display: block;
  text-transform: uppercase;
  letter-spacing: 0.08em;
  font-size: 0.7rem;
  color: var(--text-body);
}

.badge-value {
  display: block;
  font-size: 0.95rem;
  font-weight: 700;
  color: var(--text-title);
}

.hero-title {
  text-align: center;
  font-family: "Fredoka", "Quicksand", sans-serif;
  font-size: clamp(1.7rem, 3vw, 2.5rem);
  letter-spacing: 0.12em;
  margin: 0.6rem 0 0.1rem;
  color: var(--text-title);
}

.hero-subtitle {
  text-align: center;
  color: var(--text-body);
  font-weight: 600;
  margin-bottom: 0.8rem;
}

.logo-wrapper,
.info-badges,
.hero-title,
.hero-subtitle {
  animation: floatIn 0.65s ease both;
}

.logo-wrapper {
  animation-delay: 0.05s;
}

.info-badges {
  animation-delay: 0.12s;
}

.hero-title {
  animation-delay: 0.18s;
}

.hero-subtitle {
  animation-delay: 0.22s;
}

@keyframes floatIn {
  from {
    opacity: 0;
    transform: translateY(12px);
  }
  to {
    opacity: 1;
    transform: translateY(0);
  }
}

#main img {
  display: block;
  margin: 1.2rem auto 1.4rem;
  border-radius: 12px;
  box-shadow: var(--shadow-soft);
  width: 100% !important;
  max-width: 100% !important;
  height: auto !important;
}

#main .figure,
#main .figure img {
  width: 100% !important;
  max-width: 100% !important;
}

#main figure,
#main figure img {
  width: 100% !important;
  max-width: 100% !important;
}

#main table {
  width: 100%;
  border-collapse: collapse;
  margin: 1.2rem 0 1.6rem;
  background: var(--card-bg);
  border-radius: 14px;
  overflow: hidden;
  box-shadow: var(--shadow-soft);
}

#main th,
#main td {
  padding: 0.6rem 0.8rem;
  font-size: 0.9rem;
  text-align: left;
}

#main th {
  color: var(--text-title);
  font-weight: 700;
}

#main td {
  color: var(--text-body);
}

#main thead tr {
  background: linear-gradient(to right, var(--snow-fur), var(--ear-pink));
}

#main tbody tr:nth-child(even) {
  background: var(--table-even-bg);
}

#main tbody tr:hover {
  background: var(--table-hover-bg);
}

#main blockquote {
  margin: 1rem 0;
  padding: 0.9rem 1rem;
  border-left: 5px solid var(--ear-pink);
  background: var(--card-bg);
  border-radius: 12px;
  font-size: 0.95rem;
  color: var(--text-body);
  box-shadow: var(--shadow-soft);
}

pre,
code {
  font-family: "Fira Code", "JetBrains Mono", Consolas, "Courier New", monospace;
}

#main pre {
  background: var(--code-block-bg);
  border-radius: 12px;
  padding: 0.8rem 1rem;
  border: 1px solid var(--label-gray);
  box-shadow: var(--shadow-soft);
  overflow-x: auto;
  position: relative;
}

#main pre::before {
  content: "code";
  position: absolute;
  top: 6px;
  right: 12px;
  font-size: 0.7rem;
  padding: 2px 7px;
  border-radius: 999px;
  background: var(--ear-pink);
  color: var(--text-title);
}

#main p > code,
#main li > code {
  background: var(--code-inline-bg);
  padding: 2px 6px;
  border-radius: 6px;
  border: 1px solid var(--label-gray);
  font-size: 0.9em;
}

body.dark-mode mjx-container,
body.dark-mode .mjx-container,
body.dark-mode .MathJax_Display,
body.dark-mode .MathJax,
body.dark-mode span[id*='MathJax-Span'],
body.dark-mode span[id*='MathJax-Span'] * {
  color: var(--text-body) !important;
}

body.dark-mode mjx-container svg,
body.dark-mode .mjx-container svg,
body.dark-mode .MathJax_SVG svg,
body.dark-mode .MathJax_SVG svg * {
  fill: var(--text-body) !important;
  stroke: var(--text-body) !important;
  filter: none;
}

.caption,
.footnotes,
.footnote {
  font-size: 0.82rem;
  color: var(--text-muted);
}

details {
  background: var(--details-bg);
  border-radius: 12px;
  border: 1px dashed var(--label-gray);
  padding: 0.4rem 0.8rem;
  margin: 0.8rem 0;
}

summary {
  cursor: pointer;
  font-weight: 600;
}

#sidebar-toggle {
  position: fixed;
  top: 16px;
  left: 22px;
  width: 52px;
  height: 36px;
  border-radius: 14px;
  background: var(--snow-fur);
  box-shadow: var(--toggle-shadow);
  display: flex;
  align-items: center;
  justify-content: center;
  cursor: pointer;
  z-index: 9999;
}

#sidebar-toggle span,
#sidebar-toggle span::before,
#sidebar-toggle span::after {
  content: "";
  display: block;
  width: 26px;
  height: 3px;
  border-radius: 999px;
  background: var(--text-body);
  position: relative;
}

#sidebar-toggle span::before {
  position: absolute;
  top: -6px;
}

#sidebar-toggle span::after {
  position: absolute;
  top: 6px;
}

body:not(.sidebar-collapsed) #sidebar-toggle {
  left: 310px !important;
}

.btn-group.pull-right > .btn.btn-default.btn-xs.dropdown-toggle {
  background: var(--btn-explore);
  color: var(--btn-text);
  border: none;
  border-radius: 999px;
  padding: 6px 16px;
  font-size: 0.78rem;
  text-transform: uppercase;
  letter-spacing: 0.05em;
  box-shadow: var(--shadow-soft);
}

.btn-group.pull-right > .btn.btn-default.btn-xs.dropdown-toggle .caret {
  border-top-color: var(--btn-text);
}

.btn-group.pull-right .dropdown-menu {
  border-radius: 14px;
  padding: 0.25rem 0;
  border: 1px solid var(--label-gray);
  box-shadow: var(--shadow-soft);
}

.btn-explore,
.btn-relax {
  display: inline-flex;
  align-items: center;
  justify-content: center;
  padding: 0.55rem 1.2rem;
  border-radius: 999px;
  border: none;
  font-weight: 600;
  letter-spacing: 0.02em;
  color: var(--btn-text);
  text-decoration: none;
  box-shadow: var(--shadow-soft);
}

.btn-explore {
  background: var(--btn-explore);
}

.btn-relax {
  background: var(--btn-relax);
}

#mode-toggle {
  position: fixed;
  top: 16px;
  right: 26px;
  width: 38px;
  height: 38px;
  border-radius: 50%;
  background: var(--card-bg);
  border: 1px solid var(--label-gray);
  box-shadow: var(--shadow-soft);
  display: flex;
  align-items: center;
  justify-content: center;
  gap: 0;
  padding: 0;
  cursor: pointer;
  line-height: 0;
  z-index: 9999;
}

#mode-toggle .sun,
#mode-toggle .moon {
  flex: 0 0 auto;
  width: 18px;
  height: 18px;
  display: inline-flex;
  align-items: center;
  justify-content: center;
  margin: 0;
  background-repeat: no-repeat;
  background-position: center;
  background-size: contain;
  transition: transform 0.2s ease;
  transform: translateY(var(--toggle-icon-shift));
}

#mode-toggle .sun {
  background-image: var(--icon-sun-light);
}

#mode-toggle .moon {
  background-image: var(--icon-moon-light);
}

body:not(.dark-mode) #mode-toggle .sun {
  display: none;
}

body:not(.dark-mode) #mode-toggle .moon {
  display: inline-flex;
}

body.dark-mode #mode-toggle {
  background: var(--card-bg);
  border: 1px solid var(--label-gray);
  box-shadow: var(--shadow-soft);
}

body.dark-mode #mode-toggle .sun {
  background-image: var(--icon-sun-dark);
}

body.dark-mode #mode-toggle .moon {
  background-image: var(--icon-moon-dark);
}

body.dark-mode #mode-toggle .sun {
  display: inline-flex;
}

body.dark-mode #mode-toggle .moon {
  display: none;
}

body.dark-mode {
  color: var(--text-body);
  --bg-left: #121826;
  --bg-right: #24304a;
  --card-bg: #1c2338;
  --text-title: #ffffff;
  --text-body: #ffffff;
  --text-muted: #ffffff;
  --label-gray: #3a465f;
  --snow-fur: #8aa0ff;
  --ear-pink: #ff8fc7;
  --btn-explore: #5fb2d6;
  --btn-relax: #c98ac7;
  --accent-star: #2a334d;
  --shadow-soft: 0 18px 35px rgba(6, 10, 20, 0.45);
  --shadow-card: 0 8px 18px rgba(6, 10, 20, 0.55);
  --link-underline: rgba(95, 178, 214, 0.7);
  --link-hover-bg: rgba(242, 182, 212, 0.2);
  --sidebar-bg: rgba(24, 30, 49, 0.96);
  --sidebar-border: rgba(58, 70, 95, 0.9);
  --sidebar-shadow: 10px 0 22px rgba(6, 10, 20, 0.55);
  --sidebar-shadow-open: 14px 0 26px rgba(6, 10, 20, 0.6);
  --sidebar-h2-shadow: 0 6px 14px rgba(6, 10, 20, 0.55);
  --section-title-shadow: 0 6px 14px rgba(6, 10, 20, 0.5);
  --logo-frame-shadow: 0 12px 26px rgba(6, 10, 20, 0.6), 0 0 0 3px rgba(255, 255, 255, 0.12);
  --logo-star-shadow: 0 0 8px rgba(242, 182, 212, 0.55);
  --logo-dot-shadow: 0 0 6px rgba(214, 220, 255, 0.55);
  --table-even-bg: rgba(42, 52, 79, 0.6);
  --table-hover-bg: rgba(242, 182, 212, 0.2);
  --code-inline-bg: rgba(214, 220, 255, 0.2);
  --code-block-bg: #151b2c;
  --details-bg: rgba(23, 29, 45, 0.85);
  --toggle-shadow: 0 4px 10px rgba(6, 10, 20, 0.55);
}

body.dark-mode a {
  color: var(--text-body);
  border-bottom-color: rgba(255, 255, 255, 0.45);
}

body.dark-mode a:hover {
  color: var(--text-body);
  background: rgba(255, 255, 255, 0.12);
}

@media (prefers-reduced-motion: reduce) {
  *,
  *::before,
  *::after {
    animation: none !important;
    transition: none !important;
  }
}

@media (max-width: 1100px) {
  body #content {
    padding: 1.8rem 1.3rem 2.4rem !important;
  }

  body #main {
    margin: 1.6rem auto 2.3rem !important;
    padding: 1.6rem 1.2rem 2rem !important;
    border-radius: 18px !important;
  }
}

@media (max-width: 800px) {
  body #content {
    padding: 1.4rem 0.8rem 2rem !important;
  }

  body #main {
    margin: 1.4rem auto 2.1rem !important;
    padding: 1.3rem 0.9rem 1.8rem !important;
    border-radius: 14px !important;
  }

  #sidebar-toggle {
    top: 10px !important;
    left: 12px !important;
    width: 40px !important;
    height: 30px !important;
  }

  body:not(.sidebar-collapsed) #sidebar-toggle {
    left: 12px !important;
  }

  #mode-toggle {
    top: 10px !important;
    right: 14px !important;
    transform: scale(0.9);
  }

  .btn-group.pull-right {
    margin-top: 52px;
    margin-right: 12px;
  }

  .info-badges {
    flex-direction: column;
    align-items: center;
  }

  .badge-card {
    min-width: 0;
    width: 90%;
  }

  .logo-frame {
    width: 140px;
    height: 140px;
  }

  .logo-caption {
    font-size: 0.8rem;
    letter-spacing: 0.08em;
  }
}

@media (max-width: 700px) {
  #main table {
    display: block;
    overflow-x: auto;
  }

  #main th,
  #main td {
    white-space: nowrap;
  }

  #main img {
    max-width: 100%;
    height: auto;
  }

  #main .MathJax_Display,
  #main .mjx-container {
    overflow-x: auto;
    overflow-y: hidden;
  }
}

@media (max-width: 600px) {
  body {
    --content-top: 4.2rem;
  }

  body #content {
    padding: 1rem 0.6rem 1.6rem !important;
  }

  body #main {
    margin: 1.1rem auto 1.6rem !important;
    padding: 1.1rem 0.75rem 1.4rem !important;
    border-radius: 12px !important;
  }

  #sidebar {
    width: 85vw;
    max-width: 240px;
  }

  .hero-title {
    font-size: clamp(1.4rem, 6vw, 2rem);
    letter-spacing: 0.08em;
  }

  .hero-subtitle {
    font-size: 0.9rem;
  }

  .info-badges {
    gap: 0.6rem;
  }

  .badge-card {
    width: 100%;
    padding: 0.6rem 0.9rem;
  }

  .logo-frame {
    width: 120px;
    height: 120px;
  }

  .logo-caption {
    font-size: 0.75rem;
    letter-spacing: 0.06em;
  }

  #mode-toggle {
    width: 32px;
    height: 32px;
    padding: 0;
    --toggle-icon-shift: 0px;
  }

  #mode-toggle .sun,
  #mode-toggle .moon {
    width: 16px;
    height: 16px;
  }

  #sidebar-toggle {
    width: 38px !important;
    height: 28px !important;
  }

  #sidebar-toggle span,
  #sidebar-toggle span::before,
  #sidebar-toggle span::after {
    width: 22px;
    height: 2px;
  }

  #sidebar-toggle span::before {
    top: -5px;
  }

  #sidebar-toggle span::after {
    top: 5px;
  }
}

body #content {
  padding-top: var(--content-top) !important;
}

body.sidebar-collapsed #sidebar {
  display: none !important;
}

body.sidebar-collapsed #content {
  margin-left: 0 !important;
  max-width: 100% !important;
}

.logo-wrapper pre,
.logo-wrapper code,
.logo-wrapper + pre,
.logo-wrapper + div pre,
.logo-wrapper + div code {
  display: none !important;
}
/* Foto kamu sebagai lingkaran di dalam */
.logo-frame::after {
  content: "";
  position: absolute;
  top: 8px;
  left: 8px;
  right: 8px;
  bottom: 8px;
  border-radius: 50%;
  background:
    url("C:/Users/Nurul Iffah/Downloads/fotoku.jpg") center top / cover no-repeat;
  background-color: #f7f4ff;
}
/* ===============================
   BACKGROUND FULL + SMOOTH
   (anti putus-putus / tiling)
   =============================== */
html, body{
  height: 100%;
}


/* kalau masih ada bagian yang keliatan “kepotong putih”
   (container bawaan theme) */
.main-container,
.container-fluid,
.toc-content,
.page-wrapper{
  background: transparent !important;
}
/* =========================
   SIDEBAR DEFAULT: CLOSED
   (fix kiri kosong)
   ========================= */

/* sidebar ditutup (geser keluar layar, bukan cuma display none) */
body.sidebar-closed #TOC,
body.sidebar-closed .tocify{
  transform: translateX(-120%) !important;
  opacity: 0 !important;
  visibility: hidden !important;
  pointer-events: none !important;
  transition: .25s ease !important;
}

/* konten balik full ke kiri */
body.sidebar-closed .toc-content{
  margin-left: 0 !important;
  padding-left: 0 !important;
  width: 100% !important;
  max-width: 100% !important;
}

/* kadang wrapper ikut ngasih space */
body.sidebar-closed .main-container,
body.sidebar-closed .container-fluid{
  padding-left: 0 !important;
}

/* optional: biar saat open animasinya halus */
#TOC, .tocify, .toc-content{
  transition: .25s ease;
}
/* ===== SIDEBAR CLOSED (default) ===== */
body.sidebar-collapsed #TOC,
body.sidebar-collapsed .tocify,
body.sidebar-collapsed #sidebar{
  display: none !important;
}

/* konten full lebar, no sisa space kiri */
body.sidebar-collapsed .toc-content{
  margin-left: 0 !important;
  padding-left: 0 !important;
  width: 100% !important;
  max-width: 100% !important;
}

/* kadang kolom bootstrap masih ngunci lebar */
body.sidebar-collapsed .col-sm-9,
body.sidebar-collapsed .col-md-9,
body.sidebar-collapsed .col-lg-9{
  width: 100% !important;
  max-width: 100% !important;
  flex: 0 0 100% !important;
}
/* ===============================
   SNOWBALL BACKGROUND (FULL + MIRIP FOTO)
   =============================== */

/* 1) Tambahin variabel ini di :root (biar var(--bg1) dst bener) */
:root{
  --bg1: #F6F5F5;   /* putih soft */
  --bg2: #F7F1F0;   /* warm mist */
  --bg3: #E5CCCC;   /* pink bawah */
  --bg4: #DEBEBD;   /* pink lebih tegas (pojok kanan bawah) */
}

/* 2) Background dipasang ke SEMUA root layer (biar ga kepotong) */
html,
body,
#page,
.container-fluid.main-container,
body > .container-fluid{
  min-height: 100vh !important;
  margin: 0 !important;

  background:
    radial-gradient(circle at 18% 0%,
      #FFFFFF 0%,
      var(--bg2) 38%,
      rgba(247,241,240,0) 72%
    ),
    radial-gradient(circle at 85% 20%,
      rgba(255,221,226,0.55) 0%,   /* ear-pink lembut */
      rgba(255,221,226,0) 60%
    ),
    radial-gradient(circle at 80% 88%,
      rgba(222,190,189,0.85) 0%,   /* rose bawah */
      rgba(222,190,189,0) 62%
    ),
    linear-gradient(135deg,
      var(--bg1) 0%,
      var(--bg2) 45%,
      var(--bg3) 78%,
      var(--bg4) 100%
    ) fixed !important;

  background-repeat: no-repeat !important;
  background-size: cover !important;
  background-attachment: fixed !important;
}

/* 3) Wrapper dalamnya wajib transparan (biar ga “ketutup putih”) */
.main-container,
.container-fluid,
.toc-content,
.page-wrapper,
.page-inner,
.row-fluid{
  background: transparent !important;
}

/* ==========================================
   FIX: BAGIAN ATAS PUTIH (WRAPPER NUTUPIN BG)
   BACKGROUND FULL 100% VIEWPORT (ANTI PUTUS)
   ========================================== */

/* 1) Matikan background putih bawaan wrapper */
#page,
#content,
.toc-content,
.container-fluid,
.container-fluid.main-container,
body > .container-fluid,
.row-fluid,
.page-wrapper,
.page-inner{
  background: transparent !important;
}

/* 2) Paksa 1 layer background nempel ke layar */
html, body{
  min-height: 100vh !important;
  background: transparent !important;
}

body::before{
  content:"";
  position: fixed;
  inset: 0;
  z-index: -9999;
  pointer-events: none;

  /* Background mirip foto Snowball (soft putih -> pink bawah) */
  background:
    radial-gradient(circle at 18% 0%,
      #FFFFFF 0%,
      #F6F3F3 40%,
      rgba(246,243,243,0) 72%
    ),
    radial-gradient(circle at 85% 20%,
      rgba(255,221,226,0.55) 0%,   /* blush */
      rgba(255,221,226,0) 60%
    ),
    radial-gradient(circle at 80% 88%,
      rgba(215,182,181,0.85) 0%,   /* pink_d */
      rgba(215,182,181,0) 62%
    ),
    linear-gradient(135deg,
      #FFFFFF 0%,
      #F7F1F0 45%,
      #E5CCCC 78%,
      #D7B6B5 100%
    );
  background-size: cover;
  background-repeat: no-repeat;
}

/* =========================================================
   SOFT BLUR BACKGROUND + GLASS CARDS (SNOWBALL VIBE)
   Tempel PALING BAWAH CSS kamu
   ========================================================= */

/* 1) Background blur ala screenshot */
html, body { min-height: 100vh; }

body{
  background: transparent !important; /* biar yang kerja body::before */
}

body::before{
  content:"";
  position: fixed;
  inset: -40px;               /* biar blur ga kepotong pinggir */
  z-index: -9999;
  pointer-events: none;

  /* blob-blur pakai warna CSS kamu */
  background:
    radial-gradient(circle at 18% 8%,
      rgba(255,255,255,0.95) 0%,
      rgba(255,255,255,0.70) 35%,
      rgba(255,255,255,0) 70%
    ),
    radial-gradient(circle at 88% 12%,
      rgba(255,221,226,0.65) 0%,
      rgba(255,221,226,0) 62%
    ),
    radial-gradient(circle at 78% 88%,
      rgba(214,193,193,0.70) 0%,
      rgba(214,193,193,0) 60%
    ),
    radial-gradient(circle at 12% 92%,
      rgba(248,249,186,0.55) 0%,
      rgba(248,249,186,0) 62%
    ),
    linear-gradient(135deg,
      rgba(255,255,255,1) 0%,
      rgba(247,241,240,1) 45%,
      rgba(229,204,204,1) 78%,
      rgba(214,193,193,1) 100%
    );

  filter: blur(22px) saturate(115%);
  opacity: 0.95;
  transform: scale(1.06);
}

/* 2) Pastikan wrapper nggak nutupin background */
#page, #content, .toc-content,
.container-fluid, .container-fluid.main-container,
body > .container-fluid, .row-fluid,
.page-wrapper, .page-inner{
  background: transparent !important;
}

/* 3) Card/section jadi “glassy” (nggak ngeblok) */
#main .section.level2{
  position: relative;
  overflow: hidden;

  /* glass */
  background: rgba(255,255,255,0.62) !important;
  backdrop-filter: blur(14px);
  -webkit-backdrop-filter: blur(14px);

  border: 1px solid rgba(226,229,233,0.75) !important;
  box-shadow: 0 18px 35px rgba(139,114,114,0.12) !important;

  /* default accent kalau belum di-set per case */
  --case-accent: rgba(255,221,226,0.85);
  --case-blob: rgba(255,221,226,0.28);

  border-left: 6px solid var(--case-accent) !important;
}

/* blob blur kecil di dalam card (bikin soft kaya SS) */
#main .section.level2::before{
  content:"";
  position:absolute;
  inset: -60px -80px auto auto;
  width: 260px;
  height: 260px;
  border-radius: 999px;
  background: radial-gradient(circle,
    var(--case-blob) 0%,
    rgba(255,255,255,0) 65%
  );
  filter: blur(18px);
  opacity: 0.9;
}

/* 4) Warna beda tiap CASE (tetep soft & blur) */
/* Case 1 = butter */
#case1.section.level2, #case-study-1.section.level2{
  --case-accent: rgba(248,249,186,0.95);
  --case-blob: rgba(248,249,186,0.30);
}
/* Case 2 = blush */
#case2.section.level2, #case-study-2.section.level2{
  --case-accent: rgba(255,221,226,0.95);
  --case-blob: rgba(255,221,226,0.30);
}
/* Case 3 = mist */
#case3.section.level2, #case-study-3.section.level2{
  --case-accent: rgba(226,229,233,0.95);
  --case-blob: rgba(226,229,233,0.35);
}
/* Case 4 = mint/teal */
#case4.section.level2, #case-study-4.section.level2{
  --case-accent: rgba(143,186,177,0.95);
  --case-blob: rgba(143,186,177,0.26);
}
/* Case 5 = dusty pink */
#case5.section.level2, #case-study-5.section.level2{
  --case-accent: rgba(214,169,170,0.95);
  --case-blob: rgba(214,169,170,0.26);
}

/* 5) Dark mode: tetap blur tapi lebih “malem” */
body.dark-mode::before{
  opacity: 0.75;
  filter: blur(24px) saturate(120%);
}
body.dark-mode #main .section.level2{
  background: rgba(28,35,56,0.62) !important;  /* glass gelap */
  border-color: rgba(58,70,95,0.85) !important;
}
/* ===== OVERRIDE: lebih BLUR + lebih sedikit putih ===== */
body::before{
  content:"";
  position: fixed;
  inset: -80px;                 /* biar blur ga kepotong */
  z-index: -9999;
  pointer-events: none;

  background:
    /* kurangi putih: ganti jadi “soft mist” dan bikin lebih tipis */
    radial-gradient(circle at 18% 8%,
      rgba(247,241,240,0.70) 0%,
      rgba(247,241,240,0.30) 28%,
      rgba(247,241,240,0) 55%
    ),

    /* blush kanan atas (dibuat lebih kuat) */
    radial-gradient(circle at 88% 12%,
      rgba(255,221,226,0.95) 0%,
      rgba(255,221,226,0) 58%
    ),

    /* pink bawah kanan (lebih tegas) */
    radial-gradient(circle at 78% 88%,
      rgba(214,193,193,0.95) 0%,
      rgba(214,193,193,0) 58%
    ),

    /* butter bawah kiri (lebih kelihatan) */
    radial-gradient(circle at 12% 92%,
      rgba(248,249,186,0.78) 0%,
      rgba(248,249,186,0) 60%
    ),

    /* base gradient (lebih pink, kurang putih) */
    linear-gradient(135deg,
      #F7F1F0 0%,
      #F3E8E8 38%,
      #E5CCCC 75%,
      #D6C1C1 100%
    );

  /* BLUR NAIK + warna lebih “nempel” */
  filter: blur(40px) saturate(135%) contrast(102%);
  opacity: 1;
  transform: scale(1.10);
}
/* ==============================
   CASE CARDS — BLUR PASTEL (NO BLOK)
   tempel PALING BAWAH
   ============================== */

#main .section.level2{
  position: relative;
  overflow: hidden;
  isolation: isolate;

  /* “kaca” putih tipis biar lembut, bukan blok */
  background: rgba(255,255,255,0.58) !important;
  border: 1px solid rgba(226,229,233,0.75) !important;
  border-left: 6px solid var(--ear-pink) !important;

  /* efek glass */
  backdrop-filter: blur(10px);
  -webkit-backdrop-filter: blur(10px);
}

#main .section.level2 > *{
  position: relative;
  z-index: 1;
}

/* layer blur warna di belakang konten */
#main .section.level2::before{
  content:"";
  position:absolute;
  inset: -70px;                  /* biar blur ga kepotong */
  z-index: 0;
  background:
    radial-gradient(circle at 12% 18%, var(--caseA) 0%, rgba(255,255,255,0) 62%),
    radial-gradient(circle at 88% 20%, var(--caseB) 0%, rgba(255,255,255,0) 62%),
    radial-gradient(circle at 70% 92%, var(--caseC) 0%, rgba(255,255,255,0) 66%);
  filter: blur(46px) saturate(140%);
  opacity: 0.95;
  transform: scale(1.15);
}

/* ===== WARNA PER CASE (pakai palet kamu) ===== */
#main .section.level2:has(> #case1){ --caseA: rgba(248,249,186,0.95); --caseB: rgba(143,186,177,0.55); --caseC: rgba(255,221,226,0.55); border-left-color: var(--snow-fur) !important; }
#main .section.level2:has(> #case2){ --caseA: rgba(255,221,226,0.95); --caseB: rgba(226,229,233,0.65); --caseC: rgba(248,249,186,0.55); border-left-color: var(--ear-pink) !important; }
#main .section.level2:has(> #case3){ --caseA: rgba(226,229,233,0.85); --caseB: rgba(143,186,177,0.55); --caseC: rgba(255,221,226,0.55); border-left-color: var(--label-gray) !important; }
#main .section.level2:has(> #case4){ --caseA: rgba(143,186,177,0.75); --caseB: rgba(248,249,186,0.55); --caseC: rgba(226,229,233,0.55); border-left-color: var(--btn-explore) !important; }
#main .section.level2:has(> #case5){ --caseA: rgba(214,169,170,0.75); --caseB: rgba(255,221,226,0.65); --caseC: rgba(226,229,233,0.55); border-left-color: var(--btn-relax) !important; }


</style>
```{=html}
<div id="sidebar-toggle">
  <span></span>
</div>
<div id="mode-toggle" title="Toggle light / dark">
  <span class="sun" aria-hidden="true"></span>
  <span class="moon" aria-hidden="true"></span>
</div>

<div class="logo-wrapper">
  <div class="logo-frame">
    <div class="logo-inner">
      <img class="profile-photo no-lightbox" src="fotoku.jpg" alt="Nurul Iffah">
    </div>
  </div>
  <div class="logo-caption">NURUL IFFAH</div>
</div>

<div class="info-badges">
  <div class="badge-card badge-primary">
    <span class="badge-label">Program Studi</span>
    <span class="badge-value">Sains Data</span>
  </div>
  <div class="badge-card badge-secondary">
    <span class="badge-label">Universitas</span>
    <span class="badge-value">INSTITUT TEKNOLOGI SAINS BANDUNG</span>
  </div>
  <div class="badge-card badge-tertiary">
    <span class="badge-label">Dosen Pengampu</span>
    <span class="badge-value">Bakti Siregar, M Sc., CSD</span>
  </div>
</div>

<div class="hero-title">STUDY CASES</div>
<div class="hero-subtitle">Confidence Interval - Week 13</div>

<script>
document.addEventListener("DOMContentLoaded", function() {
  var body = document.body;

  /* ===== SIDEBAR: default CLOSED ===== */
  var savedSidebar = null;
  try {
    savedSidebar = localStorage.getItem("cute-theme-sidebar"); // "open" / "closed"
  } catch (e) {}

  // default ketutup, kecuali user pernah nyimpen "open"
  if (savedSidebar === "open") {
    body.classList.remove("sidebar-collapsed");
  } else {
    body.classList.add("sidebar-collapsed");
  }

  var menuBtn = document.getElementById("sidebar-toggle");
  if (menuBtn) {
    menuBtn.addEventListener("click", function() {
      body.classList.toggle("sidebar-collapsed");
      try {
        localStorage.setItem(
          "cute-theme-sidebar",
          body.classList.contains("sidebar-collapsed") ? "closed" : "open"
        );
      } catch (e) {}
    });
  }

  /* ===== DARK MODE: keep punyamu ===== */
  var modeBtn = document.getElementById("mode-toggle");
  if (modeBtn) {
    var savedMode = null;
    try {
      savedMode = localStorage.getItem("cute-theme-mode");
    } catch (e) {}

    if (savedMode === "dark") body.classList.add("dark-mode");

    modeBtn.addEventListener("click", function() {
      body.classList.toggle("dark-mode");
      try {
        localStorage.setItem(
          "cute-theme-mode",
          body.classList.contains("dark-mode") ? "dark" : "light"
        );
      } catch (e) {}
    });
  }
});
</script>
<style>
/* ====== A) BACKGROUND FULL (mirip gambar Snowball) ====== */
:root{
  --bg1:#f9f9f9;   /* atas kiri */
  --bg2:#f7f1f0;   /* transisi */
  --bg3:#e4c9c8;   /* pink lembut */
  --bg4:#cbaeae;   /* bawah kanan */
}

html, body{
  min-height:100%;
  margin:0;
  padding:0;
}

body{
  background: radial-gradient(circle at 15% 0%,
    var(--bg1) 0%,
    var(--bg2) 45%,
    var(--bg3) 78%,
    var(--bg4) 100%
  ) fixed !important;
}

/* wrapper readthedown sering “nutup” background */
#page, #content,
.container-fluid.main-container,
body > .container-fluid,
.page-inner, .row-fluid,
.tocify, .toc-content{
  background: transparent !important;
}

/* ====== B) WARNA CARD TIAP CASE (Case 1 beda sendiri, dst) ======
   CATATAN:
   - Default id dari "## Case Study 1" biasanya: #case-study-1
   - Kalau kamu pakai custom: "## Case Study 1 {#case1}" maka id-nya #case1
   Jadi aku tembak dua-duanya biar pasti kena.
*/
div.section.level2{
  border-radius:22px !important;
  padding:1.6rem 1.6rem !important;
}

/* Case 1 */
#case1.section.level2, #case-study-1.section.level2{
  background: linear-gradient(135deg, rgba(255,251,212,.92), rgba(255,255,255,.97)) !important;
  border: 1px solid rgba(255, 214, 102, .35) !important;
}
/* Case 2 */
#case2.section.level2, #case-study-2.section.level2{
  background: linear-gradient(135deg, rgba(255,221,226,.88), rgba(255,255,255,.97)) !important;
  border: 1px solid rgba(255, 180, 196, .45) !important;
}
/* Case 3 */
#case3.section.level2, #case-study-3.section.level2{
  background: linear-gradient(135deg, rgba(233,225,255,.88), rgba(255,255,255,.97)) !important;
  border: 1px solid rgba(190, 170, 255, .45) !important;
}
/* Case 4 */
#case4.section.level2, #case-study-4.section.level2{
  background: linear-gradient(135deg, rgba(208,244,234,.88), rgba(255,255,255,.97)) !important;
  border: 1px solid rgba(140, 218, 198, .45) !important;
}
/* Case 5 */
#case5.section.level2, #case-study-5.section.level2{
  background: linear-gradient(135deg, rgba(212,235,255,.88), rgba(255,255,255,.97)) !important;
  border: 1px solid rgba(155, 195, 235, .45) !important;
}
</style>

```

---

## Case Study 1

**Confidence Interval for Mean, $\sigma$ Known:** An **e-commerce platform** wants to estimate the **average number of daily transactions per user** after launching a new feature. Based on large-scale historical data, the **population standard deviation** is known.

$$
\begin{eqnarray*}
\sigma &=& 3.2 \quad \text{(population standard deviation)} \\
n &=& 100 \quad \text{(sample size)} \\
\bar{x} &=& 12.6 \quad \text{(sample mean)}
\end{eqnarray*}
$$


### **Identify the appropriate statistical test and justify your choice.**

Karena tujuan analisisnya adalah **mengestimasi mean populasi \(\mu\) dan nilai standar deviasi populasi \(\sigma\)** diketahui (\(\sigma = 3.2\)), maka metode yang sesuai adalah **confidence interval berbasis Z untuk mean:**

$$
\bar{x} \pm z_{\alpha/2} \left(\frac{\sigma}{\sqrt{n}}\right)
$$

---

### **Compute the Confidence Intervals for:**
   - $90\%$
   - $95\%$
   - $99\%$

**Data:**

\[
\sigma = 3.2, \quad n = 100, \quad \bar{x} = 12.6
\]

- **Hitung Standard Error (SE):**

\[
SE = \frac{\sigma}{\sqrt{n}}
    = \frac{3.2}{\sqrt{100}}
    = \frac{3.2}{10}
    = 0.32
\]

- **Rumus CI (karena \(\sigma\) known → Z):**

\[
CI = \bar{x} \pm z_{\alpha/2} \cdot SE
\]

---

**1. Confidence Interval 90%**

- **Nilai kritis:**
\[
z_{0.95} = 1.645
\]

- **Margin of Error:**
\[
ME = 1.645 \times 0.32 = 0.526
\]

- **CI 90\%:**
\[
12.6 \pm 0.526 \;\Rightarrow\; (12.074,\; 13.126)
\]

---

**2. Confidence Interval 95%**

- **Nilai kritis:**
\[
z_{0.975} = 1.960
\]

- **Margin of Error:**
\[
ME = 1.960 \times 0.32 = 0.627
\]

- **CI 95\%:**
\[
12.6 \pm 0.627 \;\Rightarrow\; (11.973,\; 13.227)
\]

---

**3. Confidence Interval 99%**

- **Nilai kritis:**
\[
z_{0.995} = 2.576
\]

- **Margin of Error:**
\[
ME = 2.576 \times 0.32 \approx 0.824
\]

- **CI 99\%:**
\[
12.6 \pm 0.824 \;\Rightarrow\; (11.776,\; 13.424)
\]

---

### **comparison visualization of the three confidence intervals.**

```{r cache=TRUE, echo=FALSE, message=FALSE, warning=FALSE, fig.width=20, fig.height=10, fig.align="center"}
library(ggplot2)
library(grid)

# ====== INPUT ======
sigma <- 3.2
n     <- 100
xbar  <- 12.6
se    <- sigma / sqrt(n)

z <- c(`90%` = qnorm(0.95),
       `95%` = qnorm(0.975),
       `99%` = qnorm(0.995))

ci <- data.frame(
  level = names(z),
  z     = unname(z),
  ME    = unname(z) * se
)
ci$lower <- xbar - ci$ME
ci$upper <- xbar + ci$ME

# ====== Kurva Normal untuk xbar ======
x <- seq(xbar - 4*se, xbar + 4*se, length.out = 2500)
df <- data.frame(x = x, y = dnorm(x, mean = xbar, sd = se))
maxy <- max(df$y)

l90 <- ci$lower[ci$level=="90%"]; u90 <- ci$upper[ci$level=="90%"]
df90 <- subset(df, x >= l90 & x <= u90)

# ====== PALETTE SNOWBALL (sesuai foto) ======
butter <- "#F8F9BA"
blush  <- "#FFDDE2"
mist   <- "#E2E5E9"

ink    <- "#374151"   # teks/garis lembut (bukan hitam pekat)
teal   <- "#8CC1B6"   # mint-teal soft
teal_d <- "#6EAFA1"   # mint-teal lebih tegas
pink_d <- "#D7B6B5"   # dusty blush (buat garis)
grey_d <- "#9CA3AF"   # abu lembut (buat 99%)

# Warna garis CI (dibikin pastel tapi tetap kebaca)
cols <- c(`90%` = teal_d,
          `95%` = pink_d,
          `99%` = grey_d)

# Mean line
mean_col <- teal

# ====== Label kotak kanan ======
fmt3 <- function(v) sprintf("%.3f", v)
lab_x <- 13.78
lab <- data.frame(
  level = c("99%","95%","90%"),
  x = lab_x,
  y = c(0.86, 0.72, 0.58) * maxy,
  seg_col = c(grey_d, pink_d, teal_d)
)

get_ci_text <- function(L){
  lo <- fmt3(ci$lower[ci$level==L])
  up <- fmt3(ci$upper[ci$level==L])
  paste0("CL ", L, " = (", lo, ", ", up, ")")
}
lab$text <- c(get_ci_text("99%"), get_ci_text("95%"), get_ci_text("90%"))

lab$x0 <- xbar + 0.18
lab$y0 <- lab$y

# ====== Plot ======
ggplot(df, aes(x, y)) +
  # arsiran area (vibe butter seperti foto)
  geom_area(data = df90, fill = butter, alpha = 0.55) +
  geom_line(color = ink, linewidth = 1.25) +

  # mean line + titik
  geom_vline(xintercept = xbar, color = mean_col, linewidth = 1.4) +
  geom_point(aes(xbar, 0), color = mean_col, size = 4) +

  # garis putus-putus batas CI
  geom_vline(xintercept = c(ci$lower[ci$level=="90%"], ci$upper[ci$level=="90%"]),
             linetype="dashed", color=cols["90%"], linewidth=1) +
  geom_vline(xintercept = c(ci$lower[ci$level=="95%"], ci$upper[ci$level=="95%"]),
             linetype="dashed", color=cols["95%"], linewidth=1) +
  geom_vline(xintercept = c(ci$lower[ci$level=="99%"], ci$upper[ci$level=="99%"]),
             linetype="dashed", color=cols["99%"], linewidth=1) +

  # bracket bawah
  geom_segment(aes(x=ci$lower[ci$level=="99%"], xend=ci$upper[ci$level=="99%"],
                   y=0.16*maxy, yend=0.16*maxy),
               color=cols["99%"], linewidth=4, lineend="round") +
  geom_segment(aes(x=ci$lower[ci$level=="95%"], xend=ci$upper[ci$level=="95%"],
                   y=0.11*maxy, yend=0.11*maxy),
               color=cols["95%"], linewidth=4, lineend="round") +
  geom_segment(aes(x=ci$lower[ci$level=="90%"], xend=ci$upper[ci$level=="90%"],
                   y=0.06*maxy, yend=0.06*maxy),
               color=cols["90%"], linewidth=4, lineend="round") +

  # tulisan μ dan persen di dalam kurva
  annotate("text", x=xbar, y=1.05*maxy, label=paste0("μ = ", xbar),
           color=mean_col, fontface="bold", size=5) +
  annotate("text", x=xbar, y=0.86*maxy, label="99%", color=cols["99%"], fontface="bold", size=5) +
  annotate("text", x=xbar, y=0.72*maxy, label="95%", color=cols["95%"], fontface="bold", size=5) +
  annotate("text", x=xbar, y=0.58*maxy, label="90%", color=cols["90%"], fontface="bold", size=5) +

  # garis penghubung ke kotak kanan
  geom_segment(data = lab,
               aes(x = x0, y = y0, xend = x - 0.18, yend = y),
               linewidth = 4, lineend="round", color = lab$seg_col, inherit.aes = FALSE) +

  # kotak kanan (bukan hitam lagi, jadi mist + teks ink)
  geom_label(data = lab,
             aes(x = x, y = y, label = text),
             inherit.aes = FALSE,
             fill = mist, color = ink,
             fontface = "bold", size = 4,
             label.size = 0,
             label.r = unit(0.35, "lines"),
             label.padding = unit(0.35, "lines")) +

  labs(
    title = "Confidence Intervals dengan Kurva Normal",
    subtitle = sprintf("σ = %.1f, n = %d, x̄ = %.1f, SE = %.2f", sigma, n, xbar, se),
    x = "Nilai", y = "Density",
    caption = "Semakin besar tingkat kepercayaan (99%), interval makin lebar."
  ) +

  coord_cartesian(xlim = c(11.25, 14.00), ylim = c(0, 1.28*maxy)) +
  scale_x_continuous(breaks = seq(11.25, 14.00, by = 0.25)) +
  scale_y_continuous(breaks = c(0, 0.5, 1.0), expand = c(0, 0)) +

  theme_minimal(base_size = 12) +
  theme(
    plot.title = element_text(face="bold", size=18, hjust=0.5, color = ink),
    plot.subtitle = element_text(hjust=0.5, color = ink),
    plot.caption = element_text(size=10, color = "#6B7280"),
    axis.title = element_text(color = ink),
    axis.text  = element_text(color = "#6B7280"),
    panel.grid.minor = element_blank(),
    panel.grid.major = element_line(color = "#E5E7EB"),
    plot.background  = element_rect(fill = "transparent", color = NA),
    panel.background = element_rect(fill = "transparent", color = NA)
  )
```

---

### **Interpret the results in a business analytics context.**

Visualisasi ini menunjukkan **perkiraan rata-rata transaksi harian per user** berada di sekitar \(\bar{x} = 12.6\) (garis vertikal/titik tengah).lalu tiga confidence interval (garis horizontal) memberi rentang nilai mean populasi yang masuk akal. yaitu:
- **CL 90\% =** (12.074, 13.126) **paling sempit.**
- **CL 95\% =** (11.973, 13.227) **lebih lebar.** 
- **CL 99\% =** (11.776, 13.424) **paling lebar.**

**Intinya**, semakin **tinggi tingkat kepercayaan**, **rentang makin melebar** karena kita ingin "lebih yakin" **mean sebenarnya ada di dalam rentang itu.**

**jadi,** toleransi **ketidakpastian diperbesar.** akibatnya **90\% paling presisi tapi kurang konservatif**, sedangkan **99\% paling aman/konservatif tapi paling tidak presisi.**

---

## Case Study 2 

**Confidence Interval for Mean, $\sigma$ Unknown:** A **UX Research team** analyzes **task completion time (in minutes)** for a new mobile application. The data are collected from **12 users**:

$$
8.4,\; 7.9,\; 9.1,\; 8.7,\; 8.2,\; 9.0,\;
7.8,\; 8.5,\; 8.9,\; 8.1,\; 8.6,\; 8.3
$$

### **Identify the appropriate statistical test**
Metode yang tepat adalah **confidence interval berbasis distribusi t (one-sample t-interval) untuk mean**, karena yang ingin **diestimasi adalah rata-rata populasi** waktu penyelesaian tugas, sementara **standar deviasi populasi (σ) tidak diketahui** sehingga harus **diganti dengan standar deviasi sampel (s)**, dan **ukuran sampel juga kecil (n = 12)** sehingga pendekatan yang paling sesuai menggunakan **t-Student** dengan derajat bebas df = 11.

### **Compute the Confidence Intervals for:**
   - $90\%$
   - $95\%$
   - $99\%$
   
**Data:**

\[
n = 12,\quad df = n - 1 = 11
\]

\[
x = (8.4,\; 7.9,\; 9.1,\; 8.7,\; 8.2,\; 9.0,\; 7.8,\; 8.5,\; 8.9,\; 8.1,\; 8.6,\; 8.3)
\]

- **Hitung rata-rata sampel (\(\bar{x}\)):**

\[
\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} = 8.4583
\]

- **Hitung standar deviasi sampel (\(s\)):**

\[
s = 0.4209
\]

- **Hitung Standard Error (SE):**

\[
SE = \frac{s}{\sqrt{n}}
     = \frac{0.4209}{\sqrt{12}}
     = 0.1215
\]

- **Rumus CI (karena \(\sigma\) unknown \(\rightarrow\) pakai distribusi t):**

\[
CI = \bar{x} \pm t_{\alpha/2,\,df} \cdot SE
\]

---

**1. Confidence Interval 90%**

- **Nilai kritis:**
\[
t_{0.95,\,11} = 1.796
\]

- **Margin of Error:**
\[
ME = 1.796 \times 0.1215 = 0.2182
\]

- **CI 90\%:**
\[
8.4583 \pm 0.2182 \;\Rightarrow\; (8.2401,\; 8.6766)
\]

---

**2. Confidence Interval 95%**

- **Nilai kritis:**
\[
t_{0.975,\,11} = 2.201
\]

- **Margin of Error:**
\[
ME = 2.201 \times 0.1215 = 0.2675
\]

- **CI 95\%:**
\[
8.4583 \pm 0.2675 \;\Rightarrow\; (8.1909,\; 8.7258)
\]

---

**3.Confidence Interval 99%**

- **Nilai kritis:**
\[
t_{0.995,\,11} = 3.106
\]

- **Margin of Error:**
\[
ME = 3.106 \times 0.1215 = 0.3774
\]

- **CI 99\%:**
\[
8.4583 \pm 0.3774 \;\Rightarrow\; (8.0809,\; 8.8357)
\]

---

### **Visualize the three intervals on a single plot.**
```{r cache=TRUE, echo=FALSE, message=FALSE, warning=FALSE, fig.width=11.5, fig.height=5.8, fig.align="center"}
library(ggplot2)

# Data (Case Study 2)
x <- c(8.4, 7.9, 9.1, 8.7, 8.2, 9.0, 7.8, 8.5, 8.9, 8.1, 8.6, 8.3)

n  <- length(x)
df_t <- n - 1
xbar <- mean(x)
s    <- sd(x)
se   <- s/sqrt(n)

# t critical values
t90 <- qt(0.95,  df_t)
t95 <- qt(0.975, df_t)
t99 <- qt(0.995, df_t)

# CI bounds (t-interval)
L90 <- xbar - t90*se; U90 <- xbar + t90*se
L95 <- xbar - t95*se; U95 <- xbar + t95*se
L99 <- xbar - t99*se; U99 <- xbar + t99*se

# ---- Kurva t dalam skala "x" ----
x_min <- xbar - 6*se
x_max <- xbar + 6*se
gridx <- seq(x_min, x_max, length.out = 4000)

curve <- data.frame(
  x = gridx,
  y = dt((gridx - xbar)/se, df = df_t) / se
)
ymax <- max(curve$y)

# =========================
# PALETTE (cocok CSS Snowball)
# =========================
butter <- "#F8F9BA"
blush  <- "#FFDDE2"
mist   <- "#E2E5E9"

ink    <- "#374151"
muted  <- "#6B7280"

teal   <- "#8CC1B6"
teal_d <- "#6EAFA1"
pink_d <- "#D7B6B5"
grey_d <- "#9CA3AF"

# Warna garis CI
c99_col  <- grey_d
c95_col  <- pink_d
c90_col  <- teal_d
mean_col <- teal

# Shading nested (lembut, sesuai palette)
fill99 <- mist
fill95 <- blush
fill90 <- butter

# =========================================================
# LEGEND KANAN (WAJIB ADA 3 BARIS) — aman dari kepotong
# =========================================================
leg <- data.frame(
  level = c("99%","95%","90%"),
  y     = ymax * c(0.78, 0.68, 0.58),
  L     = c(L99, L95, L90),
  U     = c(U99, U95, U90),
  col   = c(c99_col, c95_col, c90_col)
)

leg$x0   <- x_max + 0.10*se
leg$x1   <- x_max + 1.20*se
leg$xlab <- x_max + 1.28*se

ggplot(curve, aes(x, y)) +
  # Shading nested
  geom_area(data = subset(curve, x >= L99 & x <= U99),
            fill = fill99, alpha = 0.55) +
  geom_area(data = subset(curve, x >= L95 & x <= U95),
            fill = fill95, alpha = 0.55) +
  geom_area(data = subset(curve, x >= L90 & x <= U90),
            fill = fill90, alpha = 0.65) +

  # Outline kurva
  geom_line(linewidth = 1.1, color = ink) +

  # Garis batas CI (putus-putus)
  geom_vline(xintercept = c(L99, U99), linetype = "dashed",
             linewidth = 1.0, color = c99_col, alpha = 0.9) +
  geom_vline(xintercept = c(L95, U95), linetype = "dashed",
             linewidth = 1.0, color = c95_col, alpha = 0.9) +
  geom_vline(xintercept = c(L90, U90), linetype = "dashed",
             linewidth = 1.0, color = c90_col, alpha = 0.95) +

  # Garis mean + titik bawah
  geom_vline(xintercept = xbar, linewidth = 1.6, color = mean_col) +
  geom_point(aes(x = xbar, y = 0), size = 3.2, color = mean_col) +

  # Bar interval di bawah (stacked)
  geom_segment(aes(x = L99, xend = U99, y = ymax*0.03, yend = ymax*0.03),
               linewidth = 4, color = c99_col, lineend = "round") +
  geom_segment(aes(x = L95, xend = U95, y = ymax*0.06, yend = ymax*0.06),
               linewidth = 4, color = c95_col, lineend = "round") +
  geom_segment(aes(x = L90, xend = U90, y = ymax*0.09, yend = ymax*0.09),
               linewidth = 4, color = c90_col, lineend = "round") +

  # Label tengah
  annotate("text", x = xbar, y = ymax*0.98,
           label = paste0("\u03BC = ", round(xbar, 2)),
           color = mean_col, fontface = "bold", size = 5) +
  annotate("text", x = xbar, y = ymax*0.78, label = "99%",
           color = c99_col, fontface = "bold", alpha = 0.9, size = 6) +
  annotate("text", x = xbar, y = ymax*0.62, label = "95%",
           color = c95_col, fontface = "bold", alpha = 0.9, size = 6) +
  annotate("text", x = xbar, y = ymax*0.46, label = "90%",
           color = c90_col, fontface = "bold", alpha = 0.95, size = 6) +

  # ======= LEGEND KANAN (3 baris) =======
  geom_segment(data = leg,
               aes(x = x0, xend = x1, y = y, yend = y),
               inherit.aes = FALSE,
               linewidth = 4, color = leg$col, lineend = "round") +
  geom_label(data = leg,
             aes(x = xlab, y = y,
                 label = sprintf("CL %s = (%.3f, %.3f)", level, L, U)),
             inherit.aes = FALSE,
             hjust = 0,
             label.size = 0,
             fill = mist, color = ink, fontface = "bold") +

  labs(
    title = "Confidence Intervals dengan Kurva t",
    subtitle = paste0("n = ", n,
                      ", x\u0305 = ", round(xbar, 2),
                      ", s = ", round(s, 2),
                      ", SE = ", round(se, 2),
                      ", df = ", df_t),
    x = "Nilai", y = "Density"
  ) +

  coord_cartesian(xlim = c(x_min, x_max + 3.6*se), clip = "off") +

  theme_minimal(base_size = 13) +
  theme(
    plot.title = element_text(face = "bold", size = 20, hjust = 0.5, color = ink),
    plot.subtitle = element_text(size = 12, hjust = 0.5, color = muted),
    axis.title = element_text(face = "bold", color = ink),
    axis.text  = element_text(color = muted),
    panel.grid.major = element_line(color = "#E5E7EB"),
    panel.grid.minor = element_blank(),

    # biar nyatu sama CSS (nggak ada kotak putih)
    plot.background  = element_rect(fill = "transparent", color = NA),
    panel.background = element_rect(fill = "transparent", color = NA),

    # ruang kanan buat label (penting)
    plot.margin = margin(10, 200, 10, 10)
  )
```

---

### **Explain sample size and confidence level influence the interval width.**

Lebar confidence interval untuk mean (\(\sigma\) unknown) ditentukan oleh:

$$
CI = \bar{x} \pm t_{\alpha/2,\,df}\left(\frac{s}{\sqrt{n}}\right)
$$

Jadi “lebar” CI bergantung pada dua hal utama:

**(1) Sample size**

- Kalau **\(n\) makin besar**, maka \(\dfrac{s}{\sqrt{n}}\) **(standard error) makin kecil.** \(\rightarrow\) interval makin sempit (lebih presisi).

- Kalau **\(n\) kecil** (kayak di kasus ini \(n = 12\)), **standard error masih relatif besar** \(\rightarrow\) interval lebih lebar.

**(2) Confidence level (90% vs 95% vs 99%)**

- **Semakin tinggi confidence level**, nilai kritis \(t_{\alpha/2,\,df}\) makin besar \(\rightarrow\) interval makin lebar.
  
- **Makanya urutannya selalu:**
\[
\text{CI 90\% (paling sempit)} 
< \text{ CI 95\% } 
< \text{ CI 99\% (paling lebar)}
\]

**Intinya:** \(n\) besar bikin CI menyempit, sedangkan confidence level tinggi bikin CI melebar (lebih aman tapi kurang presisi).

---

## Case Study 3 

**Confidence Interval for a Proportion, A/B Testing:** A data science team runs an **A/B test** on a new *Call-To-Action (CTA)* button design. The experiment yields:

$$
\begin{eqnarray*}
n &=& 400 \quad \text{(total users)} \\
x &=& 156 \quad \text{(users who clicked the CTA)}
\end{eqnarray*}
$$


### **Compute the sample proportion $\hat{p}$.**

Diketahui:

\[
n = 400 \quad \text{(total users)}
\]
\[
x = 156 \quad \text{(users yang klik CTA)}
\]

Proporsi sampel:

\[
\hat{p} = \frac{x}{n} = \frac{156}{400} = 0.39
\]

Jadi sample proportion / CTR sampel = 0.39 (atau 39\%).

---

2. Compute Confidence Intervals for the proportion at:
   - $90\%$
   - $95\%$
   - $99\%$
   
**Data:**

\[
n = 400,\quad x = 156,\quad \hat{p} = \frac{156}{400} = 0.39
\]

- **Hitung Standard Error (SE) untuk proporsi:**

\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
    = \sqrt{\frac{0.39(1 - 0.39)}{400}}
    = \sqrt{\frac{0.39 \cdot 0.61}{400}}
    = \sqrt{\frac{0.2379}{400}}
    = \sqrt{0.00059475}
    \approx 0.0244
\]

- **Rumus CI proporsi (Z):**

\[
CI = \hat{p} \pm z_{\alpha/2} \cdot SE
\]

---

**1. Confidence Interval 90%**

- **Nilai kritis:**
\[
z_{0.95} = 1.645
\]

- **Margin of Error:**
\[
ME = 1.645 \times 0.0244 = 0.0401
\]

- **CI 90\%:**
\[
0.39 \pm 0.0401 \;\Rightarrow\; (0.3499,\; 0.4301)
\]

---

**2. Confidence Interval 95%**

- **Nilai kritis:**
\[
z_{0.975} = 1.960
\]

- **Margin of Error:**
\[
ME = 1.960 \times 0.0244 = 0.0478
\]

- **CI 95\%:**
\[
0.39 \pm 0.0478 \;\Rightarrow\; (0.3422,\; 0.4378)
\]

---

**3. Confidence Interval 99%**

- **Nilai kritis:**
\[
z_{0.995} = 2.576
\]

- **Margin of Error:**
\[
ME = 2.576 \times 0.0244 = 0.0628
\]

- **CI 99\%:**
\[
0.39 \pm 0.0628 \;\Rightarrow\; (0.3272,\; 0.4528)
\]

---

### **Visualize and compare the three intervals.**

```{r cache=TRUE, echo=FALSE, message=FALSE, warning=FALSE, fig.width=11.5, fig.height=5.8, fig.align="center"}
library(ggplot2)

# Input (Case Study 3)
n <- 400
x <- 156
phat <- x/n
se <- sqrt(phat*(1-phat)/n)

# Z critical values
z90 <- qnorm(0.95)
z95 <- qnorm(0.975)
z99 <- qnorm(0.995)

# CI bounds
L90 <- phat - z90*se; U90 <- phat + z90*se
L95 <- phat - z95*se; U95 <- phat + z95*se
L99 <- phat - z99*se; U99 <- phat + z99*se

# Kurva normal utk p-hat (approx)
x_min <- phat - 6*se
x_max <- phat + 6*se
gridx <- seq(x_min, x_max, length.out = 4000)

df <- data.frame(
  x = gridx,
  y = dnorm(gridx, mean = phat, sd = se)
)
ymax <- max(df$y)

# =========================
# PALETTE (cocok CSS Snowball)
# =========================
butter <- "#F8F9BA"
blush  <- "#FFDDE2"
mist   <- "#E2E5E9"

ink    <- "#374151"
muted  <- "#6B7280"

teal   <- "#8CC1B6"
teal_d <- "#6EAFA1"
pink_d <- "#D7B6B5"
grey_d <- "#9CA3AF"

# Warna CI
c99_col  <- grey_d
c95_col  <- pink_d
c90_col  <- teal_d
mean_col <- teal

# Shading nested
fill99 <- mist
fill95 <- blush
fill90 <- butter

# =========================================================
# LEGEND KANAN (WAJIB ADA 3 BARIS) — aman dari kepotong
# =========================================================
leg <- data.frame(
  level = c("99%","95%","90%"),
  y     = ymax * c(0.78, 0.68, 0.58),
  L     = c(L99, L95, L90),
  U     = c(U99, U95, U90),
  col   = c(c99_col, c95_col, c90_col)
)

leg$x0   <- x_max + 0.10*se
leg$x1   <- x_max + 1.20*se
leg$xlab <- x_max + 1.28*se

ggplot(df, aes(x, y)) +
  # Shading nested
  geom_area(data = subset(df, x >= L99 & x <= U99),
            fill = fill99, alpha = 0.55) +
  geom_area(data = subset(df, x >= L95 & x <= U95),
            fill = fill95, alpha = 0.55) +
  geom_area(data = subset(df, x >= L90 & x <= U90),
            fill = fill90, alpha = 0.65) +

  # Outline
  geom_line(linewidth = 1.1, color = ink) +

  # Batas CI (dashed)
  geom_vline(xintercept = c(L99, U99), linetype = "dashed",
             linewidth = 1.0, color = c99_col, alpha = 0.9) +
  geom_vline(xintercept = c(L95, U95), linetype = "dashed",
             linewidth = 1.0, color = c95_col, alpha = 0.9) +
  geom_vline(xintercept = c(L90, U90), linetype = "dashed",
             linewidth = 1.0, color = c90_col, alpha = 0.95) +

  # Mean line + point
  geom_vline(xintercept = phat, linewidth = 1.6, color = mean_col) +
  geom_point(aes(x = phat, y = 0), size = 3.2, color = mean_col) +

  # Bar interval di bawah
  geom_segment(aes(x = L99, xend = U99, y = ymax*0.03, yend = ymax*0.03),
               linewidth = 4, color = c99_col, lineend = "round") +
  geom_segment(aes(x = L95, xend = U95, y = ymax*0.06, yend = ymax*0.06),
               linewidth = 4, color = c95_col, lineend = "round") +
  geom_segment(aes(x = L90, xend = U90, y = ymax*0.09, yend = ymax*0.09),
               linewidth = 4, color = c90_col, lineend = "round") +

  # Label tengah
  annotate("text", x = phat, y = ymax*0.98,
           label = paste0("p̂ = ", round(phat, 2)),
           color = mean_col, fontface = "bold", size = 5) +
  annotate("text", x = phat, y = ymax*0.78, label = "99%",
           color = c99_col, fontface = "bold", alpha = 0.9, size = 6) +
  annotate("text", x = phat, y = ymax*0.62, label = "95%",
           color = c95_col, fontface = "bold", alpha = 0.9, size = 6) +
  annotate("text", x = phat, y = ymax*0.46, label = "90%",
           color = c90_col, fontface = "bold", alpha = 0.95, size = 6) +

  # ======= LEGEND KANAN (3 baris) =======
  geom_segment(data = leg,
               aes(x = x0, xend = x1, y = y, yend = y),
               inherit.aes = FALSE,
               linewidth = 4, color = leg$col, lineend = "round") +
  geom_label(data = leg,
             aes(x = xlab, y = y,
                 label = sprintf("CL %s = (%.4f, %.4f)", level, L, U)),
             inherit.aes = FALSE,
             hjust = 0,
             label.size = 0,
             fill = mist, color = ink, fontface = "bold") +

  labs(
    title = "Confidence Intervals untuk Proporsi (Normal Approx.)",
    subtitle = paste0("n = ", n, ", x = ", x, ", p̂ = ", round(phat, 2),
                      ", SE = ", round(se, 4)),
    x = "Proporsi (CTR)", y = "Density"
  ) +

  coord_cartesian(xlim = c(x_min, x_max + 3.6*se), clip = "off") +

  theme_minimal(base_size = 13) +
  theme(
    plot.title = element_text(face = "bold", size = 20, hjust = 0.5, color = ink),
    plot.subtitle = element_text(size = 12, hjust = 0.5, color = muted),
    axis.title = element_text(face = "bold", color = ink),
    axis.text  = element_text(color = muted),
    panel.grid.major = element_line(color = "#E5E7EB"),
    panel.grid.minor = element_blank(),

    # biar nyatu sama CSS
    plot.background  = element_rect(fill = "transparent", color = NA),
    panel.background = element_rect(fill = "transparent", color = NA),

    # ruang kanan buat legend
    plot.margin = margin(10, 200, 10, 10)
  )
```

---

### **Explain confidence level affects decision-making in product experiments.**

Dalam eksperimen produk (misalnya A/B testing CTA), **confidence level menentukan seberapa “ketat” kita saat menyimpulkan CTR sebenarnya**. Dari visualisasi, terlihat bahwa **CI 90% paling sempit**, **CI 95% lebih lebar**, dan **CI 99% paling lebar.** artinya **semakin tinggi confidence level, semakin lebar rentang CTR yang dianggap masuk akal**, karena kita ingin **keyakinan yang lebih tinggi.**

**Rekomendasi untuk pengambilan keputusan:**

- **Confidence level tinggi (99%)** → interval lebih lebar → keputusan lebih konservatif/aman (risiko salah klaim lebih kecil), tetapi bisa lebih sering “belum cukup bukti” **untuk mengambil keputusan cepat.**

- **Confidence level lebih rendah (90%)** → interval lebih sempit → keputusan lebih cepat dan responsif, tetapi **risikonya lebih besar dibanding 95%/99%.**

**Rekomendasi:**

- **Untuk keputusan high-risk (rollout besar, dampak revenue besar, biaya tinggi)** → lebih cocok pakai 95%–99%.

- **Untuk keputusan low-risk (iterasi UI kecil, eksplorasi awal)** → 90%–95% biasanya sudah cukup.

---

## Case Study 4 

**Precision Comparison (Z-Test vs t-Test):** Two data teams measure **API latency (in milliseconds)** under different conditions.

\begin{eqnarray*}
\text{Team A:} \\
n &=& 36 \quad \text{(sample size)} \\
\bar{x} &=& 210 \quad \text{(sample mean)} \\
\sigma &=& 24 \quad \text{(known population standard deviation)} \\[6pt]

\text{Team B:} \\
n &=& 36 \quad \text{(sample size)} \\
\bar{x} &=& 210 \quad \text{(sample mean)} \\
s &=& 24 \quad \text{(sample standard deviation)}
\end{eqnarray*}


### **Identify the statistical test used by each team.**

**Team A** menggunakan **Z confidence interval** untuk mean karena **standar deviasi populasi diketahui (\(\sigma = 24\))**.

Sedangkan **Team B** menggunakan **t confidence interval** untuk mean karena **standar deviasi populasi tidak diketahui** dan digantikan oleh standar deviasi sampel (\(s = 24\)) dengan derajat bebas
\(df = n - 1 = 35\).

### **Compute Confidence Intervals for 90%, 95%, and 99%.**

**Data**

- **Team A:** \(n = 36,\ \bar{x} = 210,\ \sigma = 24\) (\(\sigma\) diketahui)

- **Team B:** \(n = 36,\ \bar{x} = 210,\ s = 24\) (\(\sigma\) tidak diketahui)

**Standard error (keduanya sama):**

\[
SE = \frac{24}{\sqrt{36}} = \frac{24}{6} = 4
\]

---

**Team A (Z-interval, \(\sigma\) known)**

\[
CI = \bar{x} \pm z_{\alpha/2} \cdot SE
\]

- **90\%:**
\[
210 \pm 1.645(4)
\;\Rightarrow\;
(203.421,\; 216.579)
\]

- **95\%:**
\[
210 \pm 1.960(4)
\;\Rightarrow\;
(202.160,\; 217.840)
\]

- **99\%:**
\[
210 \pm 2.576(4)
\;\Rightarrow\;
(199.696,\; 220.304)
\]

---

**Team B (t-interval, \(\sigma\) unknown, \(df = 35\))**

\[
CI = \bar{x} \pm t_{\alpha/2,35} \cdot SE
\]

- **90\%:**
\[
210 \pm 1.690(4)
\;\Rightarrow\;
(203.242,\; 216.758)
\]

- **95\%:**
\[
210 \pm 2.030(4)
\;\Rightarrow\;
(201.880,\; 218.120)
\]

- **99\%:**
\[
210 \pm 2.724(4)
\;\Rightarrow\;
(199.105,\; 220.895)
\]

---

### **Visualization comparing all intervals.**
```{r cache=TRUE, echo=FALSE, message=FALSE, warning=FALSE, fig.width=12, fig.height=6, fig.align="center"}
library(ggplot2)

# ==== Input Case Study 4 ====
xbar  <- 210
n     <- 36
sigma <- 24
SE    <- sigma/sqrt(n)   # = 4
df_t  <- n - 1           # = 35

# Critical values
z90 <- qnorm(0.95);   z95 <- qnorm(0.975); z99 <- qnorm(0.995)
t90 <- qt(0.95, df_t); t95 <- qt(0.975, df_t); t99 <- qt(0.995, df_t)

# Team A (Z) bounds
A90L <- xbar - z90*SE; A90U <- xbar + z90*SE
A95L <- xbar - z95*SE; A95U <- xbar + z95*SE
A99L <- xbar - z99*SE; A99U <- xbar + z99*SE

# Team B (t) bounds
B90L <- xbar - t90*SE; B90U <- xbar + t90*SE
B95L <- xbar - t95*SE; B95U <- xbar + t95*SE
B99L <- xbar - t99*SE; B99U <- xbar + t99*SE

# ==== Curve ====
x_min <- xbar - 6*SE
x_max <- xbar + 6*SE
x <- seq(x_min, x_max, length.out = 900)
df <- data.frame(x = x, y = dnorm(x, mean = xbar, sd = SE))
ymax <- max(df$y)

# =========================================================
# SNOWBALL PALETTE — SEMUA WARNA UTAMA BEDA
# =========================================================
mist   <- "#E2E5E9"  # card/soft
blush2 <- "#FFE4E8"  # blush lembut (beda dari #FFDDE2)
butter2<- "#F8F4B3"  # butter lembut (beda dari #F8F9BA)

ink    <- "#374151"  # outline/density
mean_c <- "#8CC1B6"  # mean (mint/teal)

# Shading (Team B intervals) — 3 beda
shade99 <- mist
shade95 <- blush2
shade90 <- butter2

# Team A (Z) — 3 beda (teal family)
A99_c <- "#5F9F96"
A95_c <- "#93B2AC"
A90_c <- "#6EAFA1"

# Team B (t) — 3 beda (pink/grey family)
B99_c <- "#9CA3AF"
B95_c <- "#D7B6B5"
B90_c <- "#F1C6CE"

# ==== ruang kanan buat legend teks ====
x_right <- x_max + 6*SE

# data legend kanan (biar rapi + konsisten)
legA <- data.frame(
  lab = c("CL 99%","CL 95%","CL 90%"),
  L   = c(A99L, A95L, A90L),
  U   = c(A99U, A95U, A90U),
  col = c(A99_c, A95_c, A90_c),
  y   = ymax * c(0.80, 0.72, 0.64)
)
legB <- data.frame(
  lab = c("CL 99%","CL 95%","CL 90%"),
  L   = c(B99L, B95L, B90L),
  U   = c(B99U, B95U, B90U),
  col = c(B99_c, B95_c, B90_c),
  y   = ymax * c(0.42, 0.34, 0.26)
)

# posisi garis & kotak legend kanan
legA$x0 <- x_right; legA$x1 <- x_right + 2*SE; legA$xlab <- x_right + 2.2*SE
legB$x0 <- x_right; legB$x1 <- x_right + 2*SE; legB$xlab <- x_right + 2.2*SE

ggplot(df, aes(x, y)) +

  # Shading nested (pakai interval Team B (t))
  geom_area(data = subset(df, x >= B99L & x <= B99U), fill = shade99, alpha = 0.60) +
  geom_area(data = subset(df, x >= B95L & x <= B95U), fill = shade95, alpha = 0.60) +
  geom_area(data = subset(df, x >= B90L & x <= B90U), fill = shade90, alpha = 0.70) +

  # Outline curve
  geom_line(linewidth = 1.1, color = ink) +

  # Mean
  geom_vline(xintercept = xbar, linewidth = 1.6, color = mean_c) +
  geom_point(aes(x = xbar, y = 0), size = 3.2, color = mean_c) +

  # Team A dashed (Z) — 3 beda
  geom_vline(xintercept = c(A99L, A99U), linetype = "dashed", linewidth = 1.0, color = A99_c, alpha = 0.85) +
  geom_vline(xintercept = c(A95L, A95U), linetype = "dashed", linewidth = 1.0, color = A95_c, alpha = 0.85) +
  geom_vline(xintercept = c(A90L, A90U), linetype = "dashed", linewidth = 1.0, color = A90_c, alpha = 0.90) +

  # Team B dashed (t) — 3 beda
  geom_vline(xintercept = c(B99L, B99U), linetype = "dashed", linewidth = 1.0, color = B99_c, alpha = 0.85) +
  geom_vline(xintercept = c(B95L, B95U), linetype = "dashed", linewidth = 1.0, color = B95_c, alpha = 0.85) +
  geom_vline(xintercept = c(B90L, B90U), linetype = "dashed", linewidth = 1.0, color = B90_c, alpha = 0.95) +

  # Bars bawah — Team B (lebih bawah)
  geom_segment(aes(x = B99L, xend = B99U, y = ymax*0.03, yend = ymax*0.03), linewidth = 4, color = B99_c, lineend = "round") +
  geom_segment(aes(x = B95L, xend = B95U, y = ymax*0.06, yend = ymax*0.06), linewidth = 4, color = B95_c, lineend = "round") +
  geom_segment(aes(x = B90L, xend = B90U, y = ymax*0.09, yend = ymax*0.09), linewidth = 4, color = B90_c, lineend = "round") +

  # Bars bawah — Team A (lebih atas)
  geom_segment(aes(x = A99L, xend = A99U, y = ymax*0.12, yend = ymax*0.12), linewidth = 4, color = A99_c, lineend = "round") +
  geom_segment(aes(x = A95L, xend = A95U, y = ymax*0.15, yend = ymax*0.15), linewidth = 4, color = A95_c, lineend = "round") +
  geom_segment(aes(x = A90L, xend = A90U, y = ymax*0.18, yend = ymax*0.18), linewidth = 4, color = A90_c, lineend = "round") +

  # Center labels (pakai warna beda tiap level biar konsisten)
  annotate("text", x = xbar, y = ymax*0.98, label = paste0("\u03BC = ", xbar),
           color = mean_c, fontface = "bold", size = 5) +
  annotate("text", x = xbar, y = ymax*0.78, label = "99%", color = A99_c, fontface = "bold", alpha = 0.95, size = 6) +
  annotate("text", x = xbar, y = ymax*0.62, label = "95%", color = A95_c, fontface = "bold", alpha = 0.95, size = 6) +
  annotate("text", x = xbar, y = ymax*0.46, label = "90%", color = A90_c, fontface = "bold", alpha = 0.95, size = 6) +

  # ==== Right-side legend labels ====
  annotate("text", x = x_right, y = ymax*0.88, label = "Team A (Z, σ known)", hjust = 0, fontface = "bold", color = ink) +
  geom_segment(data = legA, aes(x = x0, xend = x1, y = y, yend = y),
               inherit.aes = FALSE, linewidth = 4, color = legA$col, lineend = "round") +
  geom_label(data = legA,
             aes(x = xlab, y = y,
                 label = sprintf("%s = (%.3f, %.3f)", lab, L, U)),
             inherit.aes = FALSE,
             hjust = 0, label.size = 0,
             fill = mist, color = ink, fontface = "bold") +

  annotate("text", x = x_right, y = ymax*0.50, label = "Team B (t, σ unknown)", hjust = 0, fontface = "bold", color = ink) +
  geom_segment(data = legB, aes(x = x0, xend = x1, y = y, yend = y),
               inherit.aes = FALSE, linewidth = 4, color = legB$col, lineend = "round") +
  geom_label(data = legB,
             aes(x = xlab, y = y,
                 label = sprintf("%s = (%.3f, %.3f)", lab, L, U)),
             inherit.aes = FALSE,
             hjust = 0, label.size = 0,
             fill = mist, color = ink, fontface = "bold") +

  labs(
    title = "Visualization Comparing All Confidence Intervals (Team A vs Team B)",
    subtitle = "n = 36, x̄ = 210, SE = 4 | Team A uses Z (σ known), Team B uses t (df = 35)",
    x = "Mean (score)", y = "Density"
  ) +

  scale_x_continuous(
    breaks = seq(x_min, x_max, by = 5),
    limits = c(x_min, x_right + 10*SE)
  ) +
  coord_cartesian(clip = "off") +

  theme_minimal(base_size = 13) +
  theme(
    plot.title = element_text(face = "bold", size = 18, hjust = 0.5, color = ink),
    plot.subtitle = element_text(size = 11, hjust = 0.5, color = "#6B7280"),
    axis.title = element_text(face = "bold", color = ink),
    axis.text  = element_text(color = "#6B7280"),
    panel.grid.major = element_line(color = "#E5E7EB"),
    panel.grid.minor = element_blank(),

    # nyatu sama CSS
    plot.background  = element_rect(fill = "transparent", color = NA),
    panel.background = element_rect(fill = "transparent", color = NA),

    # ruang kanan biar legend tidak kepotong
    plot.margin = margin(10, 240, 10, 10)
  )
```

---

### **Explain why the interval widths differ, even with similar data.**

Meskipun Team A dan Team B punya ringkasan data yang **terlihat sama** 
(\(n = 36\), \(\bar{x} = 210\), dan ukuran sebaran 24), **lebar intervalnya tetap berbeda** karena sumber ketidakpastiannya beda. **Team A memakai Z-interval karena \(\sigma\) populasi benar-benar diketahui,** sehingga ketidakpastian hanya berasal dari proses sampling. 

Sementara itu **Team B memakai t-interval karena \(\sigma\) tidak diketahui** dan harus diperkirakan dari sampel (pakai \(s\)), sehingga ada ketidakpastian tambahan. 

**Akibatnya**, nilai kritis distribusi t untuk level kepercayaan yang sama **sedikit lebih besar daripada nilai z**, sehingga **margin of error menjadi lebih besar dan interval Team B jadi lebih lebar.** 

**Jadi**, perbedaan lebar interval terutama karena t-interval lebih konservatif saat \(\sigma\) tidak diketahui.

## Case Study 5

**One-Sided Confidence Interval:** A **Software as a Service (SaaS)** company wants to ensure that **at least 70% of weekly active users** utilize a premium feature.

From the experiment:

$$
\begin{eqnarray*}
n &=& 250 \quad \text{(total users)} \\
x &=& 185 \quad \text{(active premium users)}
\end{eqnarray*}
$$

Management is only interested in the **lower bound** of the estimate.


### Identify the type of Confidence Interval and the appropriate test.

Case Study 5 menggunakan **confidence interval satu sisi (one-sided)** dengan fokus pada batas bawah itu sebagai proporsi pengguna aktif yang memakai fitur premium. 

Metode yang tepat adalah \textit{**one-proportion Z confidence interval (lower bound)**}, **karena parameter yang diestimasi adalah proporsi \(p\) dan ukuran sampel cukup besar** (\(n = 250\)) sehingga pendekatan normal dapat digunakan. 

Karena **perusahaan ingin memastikan “minimal 70\%”**, analisis ini juga sejalan dengan pengujian satu arah untuk target, yaitu
\[
H_0 : p \le 0.70
\quad \text{versus} \quad
H_1 : p > 0.70.
\]

### **Compute the one-sided lower Confidence Interval**
   - $90\%$
   - $95\%$
   - $99\%$
   
**Data:**

\[
n = 250,\quad x = 185,\quad \hat{p} = \frac{x}{n} = \frac{185}{250} = 0.74
\]

- **Hitung Standard Error (SE) proporsi:**

\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
    = \sqrt{\frac{0.74(1 - 0.74)}{250}}
    = \sqrt{\frac{0.74 \cdot 0.26}{250}}
    = \sqrt{0.0007696}
    \approx 0.02774
\]

- **Rumus one-sided lower CI (Z):**

Pada confidence interval dua sisi, kita biasanya menghitung dua batas:

\[
CI = \hat{p} \pm z_{\alpha/2} \cdot SE
\]

**Namun** pada kasus ini manajemen hanya tertarik pada batas bawah (“minimal proporsinya berapa”). 

Karena itu digunakan \textit{**one-sided lower confidence interval**}, sehingga kita **hanya menghitung batas bawah** yang dinotasikan dengan \(L\), di mana \(L\) adalah singkatan dari \textit{lower bound} **(batas bawah).**

Untuk confidence level \(C\), batas bawah one-sided ditulis:

\[
L = \hat{p} - z_C \cdot SE
\]

dan interval one-sided dapat dituliskan sebagai:

\[
(L,\, 1)
\]

karena nilai proporsi maksimum adalah 1.

---

**1. One-sided lower CI 90%**

- **Nilai kritis:**
\[
z_{0.90} = 1.282
\]

- **Margin of Error:**
\[
ME = 1.282 \times 0.02774 = 0.03555
\]

- **Lower bound 90\%:**
\[
L = 0.74 - 0.03555 = 0.70445
\]

Jadi one-sided lower CI 90\%:
\[
(0.70445,\; 1)
\]

---

**2. One-sided lower CI 95%**

- **Nilai kritis:**
\[
z_{0.95} = 1.645
\]

- **Margin of Error:**
\[
ME = 1.645 \times 0.02774 = 0.04563
\]

- **Lower bound 95\%:**
\[
L = 0.74 - 0.04563 = 0.69437
\]

Jadi one-sided lower CI 95\%:
\[
(0.69437,\; 1)
\]

---

### **Visualize the lower bounds for all confidence levels.**
```{r echo=FALSE, message=FALSE, warning=FALSE, fig.width=12, fig.height=5.2, fig.align="center"}
library(ggplot2)

# =======================
# Palet warna kamu
# =======================
butter <- "#F8F9BA"
blush  <- "#FFDDE2"
mist   <- "#E2E5E9"

ink    <- "#374151"
muted  <- "#6B7280"

teal   <- "#8CC1B6"
teal_d <- "#6EAFA1"
pink_d <- "#D7B6B5"
grey_d <- "#9CA3AF"

# =======================
# Data Case Study 5
# =======================
n <- 250
x <- 185
phat <- x/n
SE <- sqrt(phat*(1 - phat)/n)

z90 <- qnorm(0.90); z95 <- qnorm(0.95); z99 <- qnorm(0.99)
L90 <- phat - z90*SE
L95 <- phat - z95*SE
L99 <- phat - z99*SE

target <- 0.70

# =======================
# Density (Normal approx)
# =======================
xmin <- max(0, phat - 4.5*SE)
xmax <- min(1, phat + 4.5*SE)

dens_df <- data.frame(p = seq(xmin, xmax, length.out = 2000))
dens_df$y <- dnorm(dens_df$p, mean = phat, sd = SE)
maxd <- max(dens_df$y)

# =======================
# Shading: setengah kiri + 3 blok
# =======================
shade_df <- subset(dens_df, p <= phat)
shade_df$blok <- NA
shade_df$blok[shade_df$p >= L99 & shade_df$p <  L95] <- "99%"
shade_df$blok[shade_df$p >= L95 & shade_df$p <  L90] <- "95%"
shade_df$blok[shade_df$p >= L90 & shade_df$p <= phat] <- "90%"
shade_df <- subset(shade_df, !is.na(blok))
shade_df$blok <- factor(shade_df$blok, levels = c("99%","95%","90%"))

# Titik tengah buat taro teks di tiap blok
x99 <- (L99 + L95)/2
x95 <- (L95 + L90)/2
x90 <- (L90 + phat)/2

blok_label <- data.frame(
  blok = factor(c("99%","95%","90%"), levels = c("99%","95%","90%")),
  x = c(x99, x95, x90),
  y = maxd * 0.20,
  lab = c("CI 99%", "CI 95%", "CI 90%")
)


# Garis batas (beda warna + label)
lines_df <- data.frame(
  CL = factor(c("99%","95%","90%"), levels = c("99%","95%","90%")),
  L  = c(L99, L95, L90),
  y  = maxd * 1.06,
  lab = c(sprintf("L99 = %.4f", L99),
          sprintf("L95 = %.4f", L95),
          sprintf("L90 = %.4f", L90))
)

# =======================
# Warna: sesuai palet kamu
# =======================
fill_cols <- c(
  "99%" = blush,
  "95%" = butter,
  "90%" = teal
)

line_cols <- c(
  "99%" = pink_d,
  "95%" = teal_d,
  "90%" = grey_d
)

ggplot(dens_df, aes(p, y)) +
  # blok warna (setengah kiri)
  geom_area(
    data = shade_df,
    aes(x = p, y = y, fill = blok, group = blok),
    alpha = 0.62, inherit.aes = FALSE
  ) +

  # density curve
  geom_line(color = ink, size = 1.05) +

  # garis p-hat & target
  geom_vline(xintercept = phat, linetype = "dashed", size = 1, color = ink) +
  geom_vline(xintercept = target, linetype = "dotted", size = 1, color = muted) +

  # garis batas L99/L95/L90 (beda warna)
  geom_vline(
    data = lines_df,
    aes(xintercept = L, color = CL),
    linetype = "solid", size = 1.15
  ) +

  # label di atas garis (L99/L95/L90)
  geom_label(
    data = lines_df,
    aes(x = L, y = y, label = paste0(CL, "\n", lab), color = CL),
    fill = "#E2E5E9CC",  # mist transparan
    label.size = 0,
    size = 3.2, fontface = "bold",
    vjust = 0, label.r = grid::unit(0.15, "lines")
  ) +

  # teks keterangan di DALAM tiap blok
  geom_text(
    data = blok_label,
    aes(x = x, y = y, label = lab),
    size = 3.4, fontface = "bold",
    lineheight = 0.95, color = ink
  ) +

  # label p-hat dan target
  annotate(
    "text", x = phat, y = maxd*1.20,
    label = sprintf("p\u0302 = %.2f", phat),
    fontface = "bold", color = ink
  ) +
  annotate(
    "text", x = target, y = maxd*0.92,
    label = "Target 0.70",
    fontface = "bold", color = muted
  ) +

  scale_fill_manual(values = fill_cols) +
  scale_color_manual(values = line_cols) +

  labs(
    title = "Visualize & Compare One-Sided (Lower) Confidence Intervals",
    subtitle = "Blok warna = area lower-side sampai p̂, garis berwarna = lower bound tiap confidence level",
    x = "Proporsi Sampel",
    y = "Density"
  ) +

  coord_cartesian(xlim = c(max(0.62, xmin), min(0.85, xmax))) +
  theme_minimal(base_size = 12) +
  theme(
    plot.title = element_text(face = "bold", size = 15, color = ink),
    plot.subtitle = element_text(color = muted),
    axis.title = element_text(color = ink),
    axis.text  = element_text(color = muted),
    panel.grid.minor = element_blank(),
    panel.grid.major = element_line(color = "#E5E7EB"),
    legend.position = "none",
    # biar nyatu sama CSS background kamu
    panel.background = element_rect(fill = "transparent", color = NA),
    plot.background  = element_rect(fill = "transparent", color = NA)
  )
```

---

### Determine whether the **70% target** is statistically satisfied.

Untuk menentukan **apakah target 70\% “terpenuhi secara statistik”**, kita bandingkan batas bawah (lower bound) one-sided CI dengan 0.70. **Jika 
\(L \ge 0.70\)**, maka kita bisa yakin (sesuai tingkat confidence) bahwa 
proporsi sebenarnya setidaknya 70\%.

**Dari hasil yang sudah dihitung:**

- **Lower bound 90\%:**
\[
L_{90} = 0.70445
\]
lebih besar dari 0.70 \(\Rightarrow\) target \textbf{terpenuhi} pada 90\% confidence.

- **Lower bound 95\%:**
\[
L_{95} = 0.69437
\]
lebih kecil dari 0.70 \(\Rightarrow\) target \textbf{belum terpenuhi} pada 95\% confidence.

- **Lower bound 99\%:**
\[
L_{99} = 0.67546
\]
lebih kecil dari 0.70 \(\Rightarrow\) target \textbf{belum terpenuhi} pada 99\% confidence.

**Kesimpulannya,** target 70\% hanya dapat dinyatakan terpenuhi secara statistik 
pada tingkat kepercayaan 90\%, tetapi tidak terpenuhi jika perusahaan 
mensyaratkan keyakinan yang lebih ketat seperti 95\% atau 99\%.


