2.1 Identification of the Statistical Test

The objective of this study case is to estimate the population mean when the population standard deviation is known and the sample size is sufficiently large. Under these conditions, the appropriate statistical method is the Z-Confidence Interval for the population mean.

This method is justified because the sampling distribution of the sample mean follows a normal distribution, as guaranteed by the Central Limit Theorem.

2.2 Mathematical Computation of the Confidence Intervals

Given the following parameters:

\[ \bar{x} = 12.6, \quad \sigma = 3.2, \quad n = 100 \]

The general form of the confidence interval for the population mean when the population variance is known is:

\[ \bar{x} \pm z_{\alpha/2}\left(\frac{\sigma}{\sqrt{n}}\right) \]

First, compute the standard error:

\[ \frac{\sigma}{\sqrt{n}} = \frac{3.2}{\sqrt{100}} = \frac{3.2}{10} = 0.32 \]

2.3 Comparison Visualization of the Confidence Intervals

x_bar <- 12.6

ci_plot <- data.frame(
  Confidence_Level = c("90%", "95%", "99%"),
  Mean  = rep(x_bar, 3),
  Lower = c(12.0736, 11.9728, 11.7757),
  Upper = c(13.1264, 13.2272, 13.4243)
)

colors <- c("#1E90FF", "#FF8C00", "#DC143C")
par(
  mar = c(6, 6, 5, 2),
  bg = "#F8F9FA"
)

plot(
  x = 1:3,
  y = ci_plot$Mean,
  ylim = range(ci_plot[, c("Lower", "Upper")]),
  pch = 21,
  bg = colors,
  col = "black",
  cex = 3,
  xaxt = "n",
  xlab = "Confidence Level",
  ylab = "Mean Daily Transactions",
  main = "Comparison of Confidence Intervals",
  cex.lab = 1.8,
  cex.main = 2,
  font.main = 2
)

# GRID 
grid(col = "gray85", lty = "dotted")

# ERROR BAR 
arrows(
  x0 = 1:3,
  y0 = ci_plot$Lower,
  x1 = 1:3,
  y1 = ci_plot$Upper,
  angle = 90,
  code = 3,
  length = 0.15,
  col = colors,
  lwd = 4
)

axis(
  1,
  at = 1:3,
  labels = ci_plot$Confidence_Level,
  cex.axis = 1.6,
  font = 2
)

# GARIS MEAN 
abline(
  h = x_bar,
  lty = 2,
  col = "#2E8B57",
  lwd = 3
)

2.4 Interpretation

An error bar plot was used to visualize the 90%, 95%, and 99% confidence intervals of the mean daily transactions. It clearly shows the sample mean as a central point and the margin of error as vertical lines, making it easy to compare how interval width increases with higher confidence levels. The simplicity of the plot keeps the focus on the relationship between confidence level and interval width.

The 90 percent confidence interval provides a relatively narrow range, indicating a more precise estimate of the average daily transactions. This level of confidence is suitable for short-term operational decisions, such as daily monitoring and routine resource allocation, where higher precision is prioritized and a small level of uncertainty is acceptable.

The 95 percent confidence interval offers a balanced combination of precision and reliability. This level is widely used in business analytics and provides strong confidence that the true average daily transactions fall within the estimated range. It is most appropriate for medium-term planning decisions, including performance evaluation, budgeting, and capacity planning.

The 99 percent confidence interval results in the widest range, reflecting the highest level of certainty but lower precision. This conservative estimate is best suited for long-term and high-risk strategic decisions, such as infrastructure investment and system scalability planning, where decision errors could have significant consequences.

Overall, the results indicate that higher confidence levels provide greater certainty but result in wider intervals, while lower confidence levels offer more precise estimates with higher uncertainty. Therefore, the choice of confidence level should be adjusted to the risk level and decision horizon of the business context.

3.1 Identification of the Statistical Test

Since the population standard deviation is unknown and the sample size is small (n<30), the appropriate method is a t-Confidence Interval. The sample mean follows a t-Student distribution with \(n-1 = 11\) degrees of freedom.

3.2 Mathematical Computation of the Confidence Intervals

Given data:

\[ x = [8.4, 7.9, 9.1, 8.7, 8.2, 9.0, 7.8, 8.5, 8.9, 8.1, 8.6, 8.3], \quad n = 12 \]

Compute sample mean and standard deviation:

\[ \bar{x} = 8.525, \quad s = 0.431 \]

Compute standard error:

\[ SE = \frac{s}{\sqrt{n}} = \frac{0.431}{\sqrt{12}} \approx 0.124 \]

3.3 Comparison Visualization of the Confidence Intervals

# Data
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)
x_bar <- mean(x)
s <- sd(x)
SE <- s / sqrt(n)

# Confidence levels and t-values
conf_levels <- c(0.90, 0.95, 0.99)
t_values <- qt(1 - (1 - conf_levels)/2, df = n - 1)
margins <- t_values * SE

# CI results
ci_plot <- data.frame(
  Confidence_Level = c("90%", "95%", "99%"),
  Mean = rep(x_bar, 3),
  Lower = x_bar - margins,
  Upper = x_bar + margins
)

colors <- c("#1E90FF", "#FF8C00", "#DC143C")

# Plot
par(mar=c(6,6,5,2), bg="#F8F9FA")
plot(
  1:3,
  ci_plot$Mean,
  ylim = range(ci_plot[, c("Lower", "Upper")]),
  pch=21,
  bg=colors,
  col="black",
  cex=3,
  xaxt="n",
  xlab="Confidence Level",
  ylab="Mean Task Completion Time (minutes)",
  main="Comparison of Confidence Intervals (t-distribution)",
  cex.lab=1.8,
  cex.main=2,
  font.main=2
)

grid(col="gray85", lty="dotted")

arrows(
  x0=1:3,
  y0=ci_plot$Lower,
  x1=1:3,
  y1=ci_plot$Upper,
  angle=90,
  code=3,
  length=0.15,
  col=colors,
  lwd=4
)

axis(1, at=1:3, labels=ci_plot$Confidence_Level, cex.axis=1.6, font=2)
abline(h=x_bar, lty=2, col="#2E8B57", lwd=3)

3.4 Interpretation

An error bar plot was used to visualize the 90%, 95%, and 99% confidence intervals of the mean task completion time. It displays the sample mean as the central point and the intervals as vertical lines, making it easy to compare the widths and see how higher confidence levels result in wider intervals. The plot focuses on the uncertainty around the sample mean and clearly fulfills the task objective.

The width of a confidence interval is influenced by both the sample size and the confidence level. A larger sample size reduces the standard error, resulting in a narrower interval, because the estimate of the mean becomes more precise. Conversely, a higher confidence level increases the critical t-value, which widens the interval, reflecting greater certainty that the true mean lies within the range. Therefore, narrower intervals occur with larger samples or lower confidence levels, while wider intervals result from smaller samples or higher confidence levels.

4.1 Identification of the Statistical Method

The goal is to estimate the true proportion of users who click the CTA.
Since we are working with categorical (binary) data, the appropriate method is a proportion-based confidence interval (using normal approximation because \(np\) and \(n(1-p)\) are sufficiently large).

4.2 Mathematical Computation

Compute the sample proportion:

\[ \hat{p} = \frac{x}{n} = \frac{156}{400} = 0.39 \]

Standard error:

\[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.39 \times 0.61}{400}} \approx 0.0244 \]

4.3 Visualization of the Three Intervals

# Data
p_hat <- 0.39
ci_levels <- c("90%", "95%", "99%")
z_vals <- c(1.645, 1.96, 2.576)
SE <- sqrt(p_hat * (1 - p_hat) / 400)
Margins <- z_vals * SE

ci_plot <- data.frame(
  Level = ci_levels,
  Mean = rep(p_hat, 3),
  Lower = p_hat - Margins,
  Upper = p_hat + Margins
)

colors <- c("#1E90FF", "#FF8C00", "#DC143C")

plot(1:3, ci_plot$Mean, ylim=c(min(ci_plot$Lower)-0.01, max(ci_plot$Upper)+0.01),
     xaxt="n", pch=19, col="black", cex=2,
     xlab="Confidence Level", ylab="Proportion of Users Clicking CTA",
     main="Comparison of Confidence Intervals for CTA Clicks")
axis(1, at=1:3, labels=ci_plot$Level)
for(i in 1:3){
  lines(c(i,i), c(ci_plot$Lower[i], ci_plot$Upper[i]), lwd=3, col=colors[i])
}
points(1:3, rep(p_hat,3), pch=19, col="black")

4.4 Interpretation

The widths of confidence intervals differ because they depend on the confidence level and sample size, not just the data itself. Even with similar data, a higher confidence level requires a larger critical value (z or t), which widens the interval to ensure greater certainty that the true parameter lies within it. Conversely, a lower confidence level uses a smaller critical value, producing a narrower interval. Additionally, small variations in sample statistics (like sample mean or proportion) can slightly change the interval bounds, but the main driver of width differences is the chosen confidence level.

An error bar plot was chosen to visualize the confidence intervals because it clearly shows the sample mean or proportion as a central point and the margin of error as vertical lines. This allows for an immediate comparison of interval widths across different confidence levels. The simplicity of the chart keeps the focus on how the interval changes with confidence level, which is the main objective of the analysis, without distractions from the underlying data distribution.

5.1 Identification of the Statistical Test

• Team A: Population standard deviation is known → Z-Confidence Interval.
  
• Team B: Population standard deviation unknown → t-Confidence Interval with \(df = n-1 = 35\).

5.2 Mathematical Computation

5.3 Comparison Visualization of the Confidence Intervals

Compare API latency measured by two teams. Team A uses a Z-Test (known σ), and Team B uses a t-Test (unknown σ).

# Parameters
n <- 36
x_bar <- 210
sigma <- 24   # Team A
s <- 24       # Team B
df <- n-1

# Standard Errors
SE_z <- sigma / sqrt(n)
SE_t <- s / sqrt(n)

# Sequence for plotting
x_vals <- seq(190, 230, length.out=500)

# Density curves
density_z <- dnorm(x_vals, mean=x_bar, sd=SE_z)
density_t <- dt((x_vals - x_bar)/SE_t, df=df)/SE_t  # scaled t to x-axis

# Colors
col_blue <- "#1E90FF55"   # semi-transparent
col_orange <- "#FF8C0055"
line_blue <- "#1E90FF"
line_orange <- "#FF8C00"

# Plot empty canvas
plot(x_vals, density_z, type="n",
     ylim=c(0, max(density_z, density_t)*1.2),
     xlab="API Latency (ms)", ylab="Density",
     main="Density Plot: Z-Test vs t-Test with Vertical CI Lines")

# Fill area under curves
polygon(c(x_vals, rev(x_vals)), c(density_z, rep(0, length(density_z))),
        col=col_blue, border=line_blue, lwd=2)
polygon(c(x_vals, rev(x_vals)), c(density_t, rep(0, length(density_t))),
        col=col_orange, border=line_orange, lwd=2)

# Confidence Intervals
CI_levels <- c(0.90, 0.95, 0.99)
z_values <- c(1.645, 1.96, 2.576)
t_values <- qt(1 - (1 - CI_levels)/2, df = df)

# Draw vertical CI lines for Team A (Z-Test)
for(i in 1:3){
  CI_z <- c(x_bar - z_values[i]*SE_z, x_bar + z_values[i]*SE_z)
  segments(CI_z[1], 0, CI_z[1], dnorm(CI_z[1], mean=x_bar, sd=SE_z), col=line_blue, lwd=3)
  segments(CI_z[2], 0, CI_z[2], dnorm(CI_z[2], mean=x_bar, sd=SE_z), col=line_blue, lwd=3)
  text(mean(CI_z), max(density_z)*0.95 - 0.02*i,
       paste0(CI_levels[i]*100, "%"), col=line_blue, font=2)
}

# Draw vertical CI lines for Team B (t-Test)
for(i in 1:3){
  CI_t <- c(x_bar - t_values[i]*SE_t, x_bar + t_values[i]*SE_t)
  segments(CI_t[1], 0, CI_t[1], dt((CI_t[1]-x_bar)/SE_t, df=df)/SE_t, col=line_orange, lwd=3)
  segments(CI_t[2], 0, CI_t[2], dt((CI_t[2]-x_bar)/SE_t, df=df)/SE_t, col=line_orange, lwd=3)
  text(mean(CI_t), max(density_t)*0.95 - 0.02*i,
       paste0(CI_levels[i]*100, "%"), col=line_orange, font=2)
}

# Legend
legend("topright", legend=c("Team A (Z-Test)", "Team B (t-Test)"),
       fill=c(col_blue, col_orange), border=c(line_blue, line_orange), bty="n")

5.4 Interpretation

Even with the same sample mean and size, interval widths differ because Team A (Z-Test) uses the known σ, giving a narrower interval, while Team B (t-Test) uses the sample s, producing a wider t-interval due to heavier tails. Higher confidence levels also widen intervals, so 90% is narrowest, 95% medium, and 99% widest. The Z-Test is more precise, whereas the t-Test accounts for extra uncertainty.

Density plots show the shape of the estimated statistic (API latency mean). With Team A (Z-Test) and Team B (t-Test), we can see the difference in spread and peak. Z-Test appears sharper (narrower) while t-Test is wider because the t-distribution has heavier tails, especially when the population variance is unknown.

6.1 Identification of the Statistical Method

We are estimating a population proportion (p) from categorical (binary) data.
Since \(n\hat{p}\) and \(n(1-\hat{p})\) are large enough, we can use normal approximation for one-sided confidence intervals.

6.2 Mathematical Computation

Sample proportion:

\[ \hat{p} = \frac{x}{n} = \frac{185}{250} = 0.74 \]

Standard error:

\[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.74 \times 0.26}{250}} \approx 0.0277 \]

6.3 Visualization the lower bounds for all confidence levels.

# Parameters
n <- 250
x <- 185
p_hat <- x / n
SE <- sqrt(p_hat * (1 - p_hat) / n)

# Confidence levels
CI_levels <- c(0.90, 0.95, 0.99)
z_values <- qnorm(CI_levels)  # one-sided lower

# Compute lower bounds
lower_bounds <- p_hat - z_values * SE

# Data frame for plotting
ci_plot <- data.frame(
  Level = c("90%", "95%", "99%"),
  Lower = lower_bounds
)

# Bar colors
bar_colors <- c("#1E90FF", "#FF8C00", "#DC143C")

# Plot with proper margins
par(bg="#F8F9FA", mar=c(6,6,5,4))  # tambah ruang kanan

# Create barplot
bp <- barplot(height=ci_plot$Lower,
        names.arg=ci_plot$Level,
        col=bar_colors,
        ylim=c(0,1.1),
        main="One-Sided Lower Confidence Interval for Premium Feature Usage",
        ylab="Lower Bound Proportion",
        cex.lab=1.5,
        cex.main=1.8,
        cex.names=1.4)

# Add target line
target <- 0.70
abline(h=target, col="#2E8B57", lwd=3, lty=2)

# Label for target line OUTSIDE plotting area (right side)
mtext("Target = 70%", side=4, line=2, col="#2E8B57", font=2, cex=1.4)

# Add exact lower bound values above each bar
text(x=bp, y=ci_plot$Lower + 0.02, labels=round(ci_plot$Lower,3), col="black", font=2, cex=1.2)

6.4 Interpretation

The SaaS company’s target is that at least 70% of weekly active users utilize the premium feature. Based on the one-sided lower confidence intervals:

• 90% CI lower bound = 0.705 → above 70%, target statistically satisfied.
  
• 95% CI lower bound = 0.694 → slightly below 70%, target marginally not satisfied.
  
• 99% CI lower bound = 0.677 → below 70%, target not satisfied.

Conclusion: Meeting the 70% target depends on the confidence level. Higher confidence levels provide more conservative estimates, which can fall below the target even if the sample proportion is above 70%.