Introduction to Probability · Problem 9.1.14
Central Limit Theorem — Grinstead & Snell
Mathematics — Probability · central-limit-theorem-bernoulli-trials
Problem
A restaurant feeds 400 customers per day. On the average 20 percent of the customers order apple pie. (a) Give a range (called a 95 percent confidence interval) for the number of pieces of apple pie ordered on a given day such that you can be 95 percent sure that the actual number will fall in this range. (b) How many customers must the restaurant have, on the average, to be at least 95 percent sure that the number of customers ordering pie on that day falls in the 19 to 21 percent range?Intuitive explanation
This problem hinges on the normal approximation to the binomial distribution and understanding how confidence intervals scale with sample size. When you have a large number of independent events (customers ordering pie), the count follows approximately a normal distribution centered at the expected value with spread determined by the standard deviation . A 95% confidence interval extends roughly 2 standard deviations on each side of the mean—the key insight is recognizing that to tighten your interval (part b), you need more data, which shrinks the standard deviation and thus the margin of error.
Intuitive explanation
This problem hinges on the normal approximation to the binomial distribution and understanding how confidence intervals scale with sample size. When you have a large number of independent events (customers ordering pie), the count follows approximately a normal distribution centered at the expected value with spread determined by the standard deviation . A 95% confidence interval extends roughly 2 standard deviations on each side of the mean—the key insight is recognizing that to tighten your interval (part b), you need more data, which shrinks the standard deviation and thus the margin of error.