---
title: "Normal Distribution: Practical Industry Applications"
subtitle: "Bridging Statistical Theory to Managerial Decision-Making"
author: "Professor Bongo Adi"
date: today
date-format: "MMMM D, YYYY"
format:
html:
theme: cosmo
toc: true
toc-depth: 3
toc-location: left
toc-title: "📋 Table of Contents"
number-sections: true
number-depth: 3
code-fold: true
code-tools: true
code-summary: "Show Code"
highlight-style: github
fig-width: 8
fig-height: 5
fig-align: center
smooth-scroll: true
anchor-sections: true
citations-hover: true
footnotes-hover: true
html-math-method: mathjax
self-contained: true
css: styles.css
execute:
echo: true
warning: false
message: false
cache: true
#bibliography: references.bib
---
```{r}
#| label: setup
#| include: false
# Load required libraries
library(tidyverse)
library(knitr)
library(kableExtra)
library(ggplot2)
library(patchwork)
library(scales)
# Set global theme
theme_set(
theme_minimal(base_size = 13) +
theme(
plot.title = element_text(face = "bold", color = "#2C3E50", size = 15),
plot.subtitle = element_text(color = "#7F8C8D", size = 11),
axis.title = element_text(color = "#2C3E50"),
panel.grid.minor = element_blank(),
plot.background = element_rect(fill = "white", color = NA)
)
)
# Color palette
palette <- c(
primary = "#2980B9",
secondary = "#E74C3C",
accent = "#27AE60",
warning = "#F39C12",
light = "#ECF0F1"
)
```
::: {.callout-note appearance="minimal"}
> *"The goal is not to teach students to calculate probabilities — it is to teach them to make better decisions under uncertainty."*
:::
***
# Course Overview {.unnumbered}
::: {.callout-tip icon=true}
## Learning Objectives
By the end of these exercises, students will be able to:
- **Apply** normal distribution concepts to real-world business scenarios
- **Interpret** probabilities as actionable managerial risk statements
- **Recognize** that\(\sigma\) is a managerial lever, not merely a statistical parameter
- **Develop** data-driven recommendations under uncertainty
- **Evaluate** trade-offs using distributional thinking
:::
***
# Manufacturing & Quality Control {#sec-manufacturing}
::: {.callout-caution collapse="false" appearance="default"}
## 🏭 Industry Context
Manufacturing quality control relies heavily on process distribution analysis.
Understanding variation is the foundation of operational excellence.
:::
## The Defective Product Dilemma {#sec-defective}
### Scenario
> A bottling plant fills cola bottles with a **mean of 500ml** and **standard deviation of 5ml**.
> FDA regulations require bottles to contain **between 490ml and 515ml**.
### Tasks & Solutions
```{r}
#| label: fig-bottling
#| fig-cap: "Cola Bottle Fill Distribution with FDA Compliance Bounds"
# Parameters
mu_bottle <- 500
sd_bottle <- 5
lower <- 490
upper <- 515
# Probability calculation
p_compliant <- pnorm(upper, mu_bottle, sd_bottle) -
pnorm(lower, mu_bottle, sd_bottle)
daily_production <- 50000
expected_failures <- daily_production * (1 - p_compliant)
# Visualization
x <- seq(mu_bottle - 4 * sd_bottle,
mu_bottle + 4 * sd_bottle, length.out = 1000)
df_bottle <- tibble(x = x, y = dnorm(x, mu_bottle, sd_bottle))
ggplot(df_bottle, aes(x, y)) +
# Fail zones
geom_area(
data = filter(df_bottle, x < lower),
aes(x, y), fill = palette["secondary"], alpha = 0.4
) +
geom_area(
data = filter(df_bottle, x > upper),
aes(x, y), fill = palette["warning"], alpha = 0.4
) +
# Pass zone
geom_area(
data = filter(df_bottle, x >= lower & x <= upper),
aes(x, y), fill = palette["accent"], alpha = 0.4
) +
geom_line(color = palette["primary"], linewidth = 1.2) +
geom_vline(xintercept = c(lower, upper),
linetype = "dashed", color = palette["secondary"], linewidth = 0.9) +
geom_vline(xintercept = mu_bottle,
linetype = "solid", color = palette["primary"], linewidth = 1) +
annotate("text", x = 495, y = 0.06, label = "✅ FDA\nCompliant",
color = "#27AE60", fontface = "bold", size = 3.5) +
annotate("text", x = 487, y = 0.025, label = "❌ Under\nFill",
color = palette["secondary"], fontface = "bold", size = 3) +
annotate("text", x = 518, y = 0.025, label = "⚠️ Over\nFill",
color = palette["warning"], fontface = "bold", size = 3) +
labs(
title = "Bottle Fill Distribution — FDA Compliance Analysis",
subtitle = paste0("μ = ", mu_bottle, "ml | σ = ", sd_bottle,
"ml | Compliance Rate = ",
round(p_compliant * 100, 2), "%"),
x = "Fill Volume (ml)",
y = "Density"
)
```
```{r}
#| label: tbl-bottling-results
#| tbl-cap: "Bottling Plant Compliance Summary"
tibble(
Metric = c(
"Compliance Probability",
"Daily Expected Failures",
"Annual Expected Failures (250 days)"
),
Value = c(
paste0(round(p_compliant * 100, 3), "%"),
format(round(expected_failures), big.mark = ","),
format(round(expected_failures * 250), big.mark = ",")
),
`Business Impact` = c(
"FDA regulatory threshold",
"Units requiring rework or disposal",
"Annual waste exposure"
)
) |>
kbl(align = c("l", "c", "l")) |>
kable_styling(
bootstrap_options = c("striped", "hover", "condensed", "responsive"),
full_width = FALSE
) |>
column_spec(1, bold = TRUE, color = "#2C3E50") |>
column_spec(2, color = "#2980B9", bold = TRUE)
```
::: {.callout-important}
## 💬 Managerial Discussion Prompt
**Task 3:** The plant manager wants to reduce waste below **1%**. Should they:
- **(A)** Adjust the process **mean** (recalibrate machinery)?
- **(B)** Reduce process **variance** (invest in precision equipment)?
Use your calculations to justify the recommendation. What are the **cost trade-offs** of each approach?
:::
***
## Six Sigma Process Capability {#sec-sixsigma}
### Scenario
> A circuit board manufacturer measures component resistance:
> **μ = 100 ohms**, **σ = 2 ohms**. Acceptable range: **94–106 ohms**.
```{r}
#| label: fig-sixsigma
#| fig-cap: "Circuit Board Resistance Distribution and Process Capability"
mu_circuit <- 100
sd_circuit <- 2
lsl <- 94 # Lower Spec Limit
usl <- 106 # Upper Spec Limit
# Process Capability Index
cp <- (usl - lsl) / (6 * sd_circuit)
cpk <- min((usl - mu_circuit) / (3 * sd_circuit),
(mu_circuit - lsl) / (3 * sd_circuit))
p_defect <- (1 - (pnorm(usl, mu_circuit, sd_circuit) -
pnorm(lsl, mu_circuit, sd_circuit))) * 1e6
x2 <- seq(mu_circuit - 5 * sd_circuit,
mu_circuit + 5 * sd_circuit, length.out = 1000)
df_circuit <- tibble(x = x2, y = dnorm(x2, mu_circuit, sd_circuit))
ggplot(df_circuit, aes(x, y)) +
geom_area(
data = filter(df_circuit, x >= lsl & x <= usl),
aes(x, y), fill = palette["accent"], alpha = 0.35
) +
geom_area(
data = filter(df_circuit, x < lsl | x > usl),
aes(x, y), fill = palette["secondary"], alpha = 0.5
) +
geom_line(color = palette["primary"], linewidth = 1.2) +
geom_vline(xintercept = c(lsl, usl),
linetype = "dashed", color = palette["secondary"], linewidth = 0.9) +
annotate("text", x = 100, y = 0.15,
label = paste0("Cp = ", round(cp, 3),
"\nCpk = ", round(cpk, 3)),
size = 4, fontface = "bold", color = "#2C3E50") +
labs(
title = "Six Sigma Process Capability — Resistance Distribution",
subtitle = paste0("Defect Rate: ", round(p_defect, 1), " DPMO"),
x = "Resistance (Ohms)",
y = "Density"
)
```
::: {.callout-tip}
## 📐 Key Formula Reference
\(C_p = \frac{USL - LSL}{6\sigma} \qquad C_{pk} = \min\left(\frac{USL - \mu}{3\sigma},\ \frac{\mu - LSL}{3\sigma}\right)\)
A **Six Sigma process** achieves\(C_{pk} \geq 2.0\), yielding only **3.4 DPMO** (defects per million opportunities).
:::
***
# Healthcare & Operations Management {#sec-healthcare}
::: {.callout-caution collapse="false"}
## 🏥 Industry Context
In healthcare, distributional decisions have direct human consequences.
Operations managers must balance cost, staffing, and patient outcomes simultaneously.
:::
## Emergency Room Staffing Model {#sec-er}
### Scenario
> Patient wait times: **μ = 35 minutes**, **σ = 8 minutes**.
> Hospital policy: **95% of patients** seen within a target threshold.
```{r}
#| label: fig-er-staffing
#| fig-cap: "Emergency Room Wait Time Analysis — Staffing Decision Scenarios"
mu_er <- 35
sd_er <- 8
sd_new <- 5 # After hiring additional nurse
threshold_95 <- qnorm(0.95, mu_er, sd_er)
threshold_30 <- qnorm(0.30, mu_er, sd_er)
# Comparative visualization
x_er <- seq(0, 70, length.out = 1000)
df_er <- tibble(
x = rep(x_er, 2),
y = c(dnorm(x_er, mu_er, sd_er),
dnorm(x_er, mu_er, sd_new)),
Staffing = rep(c("Current (σ = 8 min)", "With New Nurse (σ = 5 min)"),
each = length(x_er))
)
p_over45_current <- 1 - pnorm(45, mu_er, sd_er)
p_over45_new <- 1 - pnorm(45, mu_er, sd_new)
ggplot(df_er, aes(x, y, color = Staffing, fill = Staffing)) +
geom_line(linewidth = 1.2) +
geom_area(alpha = 0.15) +
geom_vline(xintercept = threshold_95,
linetype = "dashed", color = "#2C3E50", linewidth = 0.8) +
geom_vline(xintercept = 45,
linetype = "dotted", color = palette["secondary"], linewidth = 0.9) +
annotate("text", x = threshold_95 + 1, y = 0.045,
label = paste0("95th pct\n", round(threshold_95, 1), " min"),
size = 3, color = "#2C3E50") +
annotate("text", x = 47, y = 0.04,
label = "45 min\nthreshold",
size = 3, color = palette["secondary"]) +
scale_color_manual(values = c(palette["primary"], palette["accent"])) +
scale_fill_manual(values = c(palette["primary"], palette["accent"])) +
labs(
title = "ER Wait Time Distributions — Staffing Scenarios",
subtitle = paste0(
"P(wait > 45 min) | Current: ",
round(p_over45_current * 100, 1), "% → With New Nurse: ",
round(p_over45_new * 100, 1), "%"
),
x = "Wait Time (minutes)",
y = "Density",
color = "Staffing Scenario",
fill = "Staffing Scenario"
) +
theme(legend.position = "bottom")
```
```{r}
#| label: tbl-er-decision
#| tbl-cap: "Hospital Staffing Investment Analysis"
tibble(
Metric = c(
"95th Percentile Threshold",
"Fast-Track Cutoff (30th pct)",
"P(Wait > 45 min) — Current",
"P(Wait > 45 min) — With Nurse",
"Improvement",
"Annual Nurse Cost"
),
Result = c(
paste0(round(threshold_95, 1), " minutes"),
paste0(round(threshold_30, 1), " minutes"),
paste0(round(p_over45_current * 100, 2), "%"),
paste0(round(p_over45_new * 100, 2), "%"),
paste0(round((p_over45_current - p_over45_new) * 100, 2),
" percentage points"),
"$80,000"
)
) |>
kbl(col.names = c("Decision Metric", "Calculated Result")) |>
kable_styling(
bootstrap_options = c("striped", "hover", "condensed"),
full_width = FALSE
) |>
column_spec(1, bold = TRUE, color = "#2C3E50") |>
column_spec(2, color = palette["primary"], bold = TRUE) |>
row_spec(6, background = "#FDEBD0", bold = TRUE)
```
***
# Supply Chain & Inventory Management {#sec-supplychain}
## The Reorder Point Problem {#sec-reorder}
### Scenario
> Daily demand: **μ = 200 units**, **σ = 25 units**.
> Lead time = **5 days** | Stockout cost = **$50/unit** | Holding cost = **$2/unit/day**
```{r}
#| label: fig-reorder
#| fig-cap: "Safety Stock vs. Service Level Trade-off Analysis"
mu_daily <- 200
sd_daily <- 25
lead_time <- 5
# Lead time demand distribution
mu_lt <- mu_daily * lead_time
sd_lt <- sd_daily * sqrt(lead_time)
service_levels <- c(0.90, 0.95, 0.99)
reorder_df <- tibble(
service_level = service_levels,
z_score = qnorm(service_levels),
reorder_point = mu_lt + qnorm(service_levels) * sd_lt,
safety_stock = qnorm(service_levels) * sd_lt,
holding_cost = qnorm(service_levels) * sd_lt * 2 * 365,
label = paste0(service_levels * 100, "%")
)
# Visualization
p1 <- ggplot(reorder_df, aes(x = label, y = safety_stock, fill = label)) +
geom_col(width = 0.5, show.legend = FALSE) +
geom_text(aes(label = round(safety_stock, 0)),
vjust = -0.5, fontface = "bold", size = 4) +
scale_fill_manual(values = c("#3498DB", "#E67E22", "#E74C3C")) +
labs(title = "Safety Stock by Service Level",
x = "Service Level", y = "Safety Stock (Units)") +
theme(plot.title = element_text(size = 12))
p2 <- ggplot(reorder_df, aes(x = label, y = holding_cost / 1000, fill = label)) +
geom_col(width = 0.5, show.legend = FALSE) +
geom_text(aes(label = paste0("$", round(holding_cost / 1000, 1), "K")),
vjust = -0.5, fontface = "bold", size = 4) +
scale_fill_manual(values = c("#3498DB", "#E67E22", "#E74C3C")) +
labs(title = "Annual Holding Cost by Service Level",
x = "Service Level", y = "Annual Holding Cost ($000s)") +
theme(plot.title = element_text(size = 12))
p1 + p2 +
plot_annotation(
title = "Inventory Decision Trade-off: Service Level vs. Cost",
subtitle = "Higher service levels protect against stockouts but increase holding costs",
theme = theme(plot.title = element_text(face = "bold", size = 14))
)
```
```{r}
#| label: tbl-inventory
#| tbl-cap: "Reorder Point Decision Matrix"
reorder_df |>
mutate(
across(c(reorder_point, safety_stock), ~ round(., 0)),
holding_cost = dollar(round(holding_cost, 0)),
z_score = round(z_score, 3),
service_level = paste0(service_level * 100, "%")
) |>
select(
`Service Level` = service_level,
`Z-Score` = z_score,
`Reorder Point (units)` = reorder_point,
`Safety Stock (units)` = safety_stock,
`Annual Holding Cost` = holding_cost
) |>
kbl(align = c("c", "c", "c", "c", "c")) |>
kable_styling(
bootstrap_options = c("striped", "hover", "condensed", "responsive")
) |>
column_spec(1, bold = TRUE) |>
row_spec(2, background = "#D5F5E3") |>
add_header_above(c(
" " = 2,
"Inventory Metrics" = 2,
"Cost Impact" = 1
))
```
::: {.callout-important}
## 💬 Managerial Discussion Prompt
Which service level would you choose? **Does your answer change** if this analysis is for:
- 🛒 Retail consumer goods?
- 💊 Hospital medications?
- ✈️ Aerospace components?
:::
***
# Finance & Risk Management {#sec-finance}
## Portfolio Risk Assessment {#sec-portfolio}
### Scenario
> Annual stock returns: **μ = 12%**, **σ = 8%**
```{r}
#| label: fig-portfolio
#| fig-cap: "Stock Return Distribution — Risk and Value at Risk Analysis"
mu_stock <- 0.12
sd_stock <- 0.08
var_5 <- qnorm(0.05, mu_stock, sd_stock)
p_loss <- pnorm(0, mu_stock, sd_stock)
p_loss10 <- pnorm(-0.10, mu_stock, sd_stock)
x_stock <- seq(mu_stock - 4 * sd_stock,
mu_stock + 4 * sd_stock, length.out = 1000)
df_stock <- tibble(x = x_stock, y = dnorm(x_stock, mu_stock, sd_stock))
ggplot(df_stock, aes(x, y)) +
# Catastrophic loss region
geom_area(
data = filter(df_stock, x < -0.10),
aes(x, y), fill = "#922B21", alpha = 0.7
) +
# Loss region
geom_area(
data = filter(df_stock, x >= -0.10 & x < 0),
aes(x, y), fill = palette["secondary"], alpha = 0.4
) +
# VaR region
geom_area(
data = filter(df_stock, x >= 0 & x < -var_5),
aes(x, y), fill = palette["warning"], alpha = 0.3
) +
# Gain region
geom_area(
data = filter(df_stock, x >= -var_5),
aes(x, y), fill = palette["accent"], alpha = 0.25
) +
geom_line(color = palette["primary"], linewidth = 1.2) +
geom_vline(xintercept = var_5,
linetype = "dashed", color = palette["warning"], linewidth = 0.9) +
geom_vline(xintercept = 0,
linetype = "dashed", color = palette["secondary"], linewidth = 0.9) +
annotate("text", x = var_5 - 0.003, y = 3.5,
label = paste0("VaR (5%)\n", round(var_5 * 100, 1), "%"),
size = 3.2, hjust = 1, color = "#E67E22", fontface = "bold") +
scale_x_continuous(labels = percent_format()) +
labs(
title = "Portfolio Return Distribution — Risk Analysis",
subtitle = paste0(
"P(Loss) = ", round(p_loss * 100, 1),
"% | P(Loss > 10%) = ", round(p_loss10 * 100, 1), "%",
" | VaR(5%) = ", round(var_5 * 100, 1), "%"
),
x = "Annual Return",
y = "Density"
)
```
***
# Marketing & Consumer Behavior {#sec-marketing}
## Pricing Strategy Optimization {#sec-pricing}
### Scenario
> Customer willingness-to-pay: **μ = $85/month**, **σ = $18**
```{r}
#| label: fig-pricing
#| fig-cap: "Revenue Optimization Across Price Points"
mu_wtp <- 85
sd_wtp <- 18
prices <- seq(50, 130, by = 5)
pricing_df <- tibble(
Price = prices,
pct_buy = 1 - pnorm(prices, mu_wtp, sd_wtp),
revenue_index = prices * (1 - pnorm(prices, mu_wtp, sd_wtp))
)
optimal_price <- pricing_df$Price[which.max(pricing_df$revenue_index)]
p1 <- ggplot(pricing_df, aes(Price, pct_buy)) +
geom_line(color = palette["primary"], linewidth = 1.2) +
geom_point(color = palette["primary"], size = 2) +
scale_y_continuous(labels = percent_format()) +
scale_x_continuous(labels = dollar_format()) +
labs(title = "Purchase Probability by Price",
x = "Price ($/month)", y = "% Customers Who Buy")
p2 <- ggplot(pricing_df, aes(Price, revenue_index)) +
geom_line(color = palette["accent"], linewidth = 1.2) +
geom_point(color = palette["accent"], size = 2) +
geom_vline(xintercept = optimal_price,
linetype = "dashed", color = palette["secondary"], linewidth = 0.9) +
annotate("text", x = optimal_price + 2, y = max(pricing_df$revenue_index) * 0.9,
label = paste0("Optimal\n$", optimal_price),
color = palette["secondary"], fontface = "bold", size = 3.5) +
scale_x_continuous(labels = dollar_format()) +
labs(title = "Revenue Index by Price Point",
x = "Price ($/month)", y = "Revenue Index")
p1 + p2 +
plot_annotation(
title = "Willingness-to-Pay Analysis — Pricing Strategy",
theme = theme(plot.title = element_text(face = "bold", size = 14))
)
```
```{r}
#| label: tbl-pricing
#| tbl-cap: "Price Point Analysis Table"
library(dplyr)
library(kableExtra)
library(scales)
# 1. Prepare and identify the optimal row index first
# Using near() prevents "it didn't work" due to floating point precision issues
pricing_summary <- pricing_df |>
filter(Price %in% c(60, 75, 85, 99, 110, optimal_price)) |>
distinct() |>
arrange(Price)
opt_row_idx <- which(near(pricing_summary$Price, optimal_price))
# 2. Build the table
pricing_summary |>
mutate(
pct_buy = percent(pct_buy, accuracy = 0.1),
revenue_index = round(revenue_index, 2),
Optimal = if_else(near(Price, optimal_price), "⭐ Optimal", "")
) |>
kbl(
col.names = c("Price", "% Who Buy", "Revenue Index", ""),
align = "cccc",
caption = "Price Point Analysis"
) |>
kable_styling(
bootstrap_options = c("striped", "hover", "condensed"),
full_width = FALSE
) |>
row_spec(
opt_row_idx,
background = "#D5F5E3",
bold = TRUE
)
```
***
# Human Resources Management {#sec-hr}
## Employee Performance & Compensation Tiers {#sec-performance}
### Scenario
> Performance scores: **μ = 72**, **σ = 12** | Company size: **500 employees**
```{r}
#| label: fig-hr
#| fig-cap: "Employee Performance Distribution — Before and After Training Intervention"
mu_hr <- 72
sd_hr <- 12
mu_hr_new <- 78
n_emp <- 500
# Thresholds
top10 <- qnorm(0.90, mu_hr, sd_hr)
top25 <- qnorm(0.75, mu_hr, sd_hr)
bot15 <- qnorm(0.15, mu_hr, sd_hr)
x_hr <- seq(30, 115, length.out = 1000)
df_hr <- tibble(
x = rep(x_hr, 2),
y = c(dnorm(x_hr, mu_hr, sd_hr),
dnorm(x_hr, mu_hr_new, sd_hr)),
Program = rep(c("Before Training (μ=72)",
"After Training (μ=78)"), each = length(x_hr))
)
ggplot(df_hr, aes(x, y, color = Program, fill = Program)) +
geom_line(linewidth = 1.2) +
geom_area(alpha = 0.12) +
geom_vline(xintercept = c(bot15, top25, top10),
linetype = "dashed", color = "#7F8C8D", linewidth = 0.7) +
annotate("text", x = bot15, y = 0.034,
label = paste0("PIP\n<", round(bot15, 1)),
size = 2.8, color = "#E74C3C", fontface = "bold") +
annotate("text", x = top25, y = 0.034,
label = paste0("Bonus\n>", round(top25, 1)),
size = 2.8, color = "#E67E22", fontface = "bold") +
annotate("text", x = top10, y = 0.034,
label = paste0("Executive\n>", round(top10, 1)),
size = 2.8, color = "#27AE60", fontface = "bold") +
scale_color_manual(values = c(palette["primary"], palette["accent"])) +
scale_fill_manual(values = c(palette["primary"], palette["accent"])) +
labs(
title = "Employee Performance Distribution",
subtitle = "Impact of Training Intervention on Tier Classification",
x = "Performance Score",
y = "Density",
color = NULL, fill = NULL
) +
theme(legend.position = "bottom")
```
```{r}
#| label: tbl-hr-tiers
#| tbl-cap: "Performance Tier Analysis — Before vs. After Training"
hr_tiers <- function(mu, sd, n) {
tibble(
Tier = c("🏆 Executive Bonus (Top 10%)",
"✅ Standard Bonus (Top 25%)",
"📋 Performance Improvement (Bottom 15%)"),
Threshold = c(paste0("> ", round(qnorm(0.90, mu, sd), 1)),
paste0("> ", round(qnorm(0.75, mu, sd), 1)),
paste0("< ", round(qnorm(0.15, mu, sd), 1))),
Employees = c(
round(n * 0.10),
round(n * 0.15),
round(n * 0.15)
)
)
}
before_tiers <- hr_tiers(mu_hr, sd_hr, n_emp) |> rename_with(~ paste0(., " (Before)"), -Tier)
after_tiers <- hr_tiers(mu_hr_new, sd_hr, n_emp) |> rename_with(~ paste0(., " (After)"), -Tier)
left_join(before_tiers, after_tiers, by = "Tier") |>
kbl(align = c("l", "c", "c", "c", "c")) |>
kable_styling(
bootstrap_options = c("striped", "hover", "condensed", "responsive")
) |>
column_spec(1, bold = TRUE) |>
add_header_above(c(" " = 1, "Before Training" = 2, "After Training" = 2))
```
::: {.callout-warning}
## ⚖️ Ethical Discussion Prompt
Is applying a **normal distribution assumption** to evaluate human performance inherently problematic?
Consider: forced ranking systems, self-fulfilling prophecies, and diversity implications.
:::
***
# 🔧 Integrated Capstone: The Bakery Challenge {#sec-capstone}
### Scenario
> You manage a **50-store bakery chain**. Each store:
>
> - Production: **μ = 150 loaves/day**, **σ = 20 loaves**
> - Selling price: **$8/loaf** | Production cost: **$3/loaf**
> - Markdown (unsold): **$1/loaf** | Lost sale opportunity: **$4/loaf**
```{r}
#| label: fig-bakery
#| fig-cap: "Expected Daily Profit Optimization — Bakery Production Decision"
mu_bak <- 150
sd_bak <- 20
price <- 8
cost <- 3
salvage <- 1
lost <- 4
expected_profit <- function(q, mu, sd) {
# Expected sales = E[min(D, Q)]
exp_sales <- mu * pnorm(q, mu, sd) -
sd * dnorm(q, mu, sd) * sd +
q * (1 - pnorm(q, mu, sd))
# More precise: E[min(D,Q)]
exp_min_dq <- q - (q - mu) * pnorm(q, mu, sd) -
sd * dnorm(q, mu, sd)
exp_leftover <- pmax(q - exp_min_dq, 0)
exp_shortage <- pmax(mu - exp_min_dq, 0)
profit <- (price - cost) * exp_min_dq -
(cost - salvage) * exp_leftover -
lost * exp_shortage
profit
}
quantities <- seq(80, 220, by = 1)
profits <- sapply(quantities, expected_profit, mu = mu_bak, sd = sd_bak)
opt_q <- quantities[which.max(profits)]
opt_profit <- max(profits)
df_bakery <- tibble(q = quantities, profit = profits)
ggplot(df_bakery, aes(q, profit)) +
geom_line(color = palette["primary"], linewidth = 1.3) +
geom_vline(xintercept = opt_q,
linetype = "dashed", color = palette["secondary"], linewidth = 0.9) +
geom_point(data = tibble(q = opt_q, profit = opt_profit),
aes(q, profit), color = palette["secondary"], size = 4) +
annotate("text", x = opt_q + 5, y = opt_profit * 0.98,
label = paste0("Optimal: ", opt_q, " loaves\n$",
round(opt_profit, 2), "/store/day"),
color = palette["secondary"], fontface = "bold", size = 3.5) +
scale_y_continuous(labels = dollar_format()) +
labs(
title = "Expected Daily Profit by Production Quantity",
subtitle = paste0(
"50-Store Chain | Daily Chain Profit at Optimum: $",
format(round(opt_profit * 50, 0), big.mark = ",")
),
x = "Loaves Produced per Store",
y = "Expected Daily Profit per Store"
)
```
::: {.callout-tip}
## 🎯 Strategic Questions for Discussion
1. Should each store manager **decide independently** or should production be **centralized**?
2. How does your answer change if stores are in **different neighborhoods** (different demand profiles)?
3. What data would you need to **refine this model**?
:::
***
# Assessment Rubric {#sec-rubric .unnumbered}
```{r}
#| label: tbl-rubric
#| tbl-cap: "Grading Rubric — Industry Application Exercises"
tibble(
Criteria = c(
"Calculation Accuracy",
"Business Interpretation",
"Critical Thinking",
"Communication"
),
`Excellent (A)` = c(
"All z-scores & probabilities correct",
"Nuanced managerial recommendations with trade-offs",
"Challenges assumptions, explores alternatives",
"Clear, professional, publication-ready"
),
`Proficient (B)` = c(
"Minor computational errors",
"Basic interpretation present",
"Addresses main decision variables",
"Adequate clarity and structure"
),
`Developing (C)` = c(
"Conceptual misunderstandings present",
"Results not connected to decisions",
"Surface-level analysis only",
"Difficult to follow reasoning"
)
) |>
kbl(align = c("l", "l", "l", "l")) |>
kable_styling(
bootstrap_options = c("striped", "hover", "condensed", "responsive"),
full_width = TRUE
) |>
column_spec(1, bold = TRUE, color = "#2C3E50", width = "15%") |>
column_spec(2, background = "#D5F5E3", width = "28%") |>
column_spec(3, background = "#D6EAF8", width = "28%") |>
column_spec(4, background = "#FDEBD0", width = "28%")
```
***
# Teaching Notes {#sec-teaching .unnumbered}
::: {.callout-note appearance="minimal"}
## 📅 Recommended Sequencing
| Week | Exercises | Focus |
| --- | --- | --- |
| **Week 1** | Manufacturing (§1) | Concrete, visual, z-score fluency |
| **Week 2** | Healthcare + Supply Chain (§2–3) | Decision trade-offs emerge |
| **Week 3** | Finance + Marketing (§4–5) | Abstract probabilistic thinking |
| **Week 4** | HR + Capstone (§6–7) | Full synthesis and ethics |
:::
::: {.callout-tip appearance="default"}
## 🎓 Core Pedagogical Goals
- Students should recognize **σ as a managerial lever**, not just a statistical parameter
- Every probability is a **business risk statement**
- Z-scores connect directly to **actionable threshold decisions**
- Always close with: *"What would YOU decide, and what risk are you willing to accept?"*
:::
***
