Statistics 1

In-Class Exercise 10

Yu-You Liou

Shih Chien University

2026-09-16

Question 1

The Central Limit Theorem

A population has \mu=50 and \sigma=12. A sample of n=36 is drawn.

1. State the mean and the standard error of \bar{X}.
2. Find P(\bar{X}>53) with pnorm().
3. Explain in one sentence why P(\bar{X}>53) is much smaller than P(X>53) for a single observation.

Question 2

Seeing the Central Limit Theorem

The exponential distribution is strongly skewed to the right.

set.seed(2026)
means <- replicate(1000, mean(rexp(30, rate = 0.5)))

1. Draw a histogram of means with ggplot() and geom_histogram().
2. Report mean(means) and sd(means).
3. The population mean is 1/0.5=2 and the population standard deviation is also 2. Compare sd(means) with \sigma/\sqrt{n} and comment in one sentence.

Question 3

Normal Approximation to the Binomial

In a city, 30% of households own an electric vehicle. A sample of 200 households is taken.

1. Verify that np\ge5 and nq\ge5.
2. Find P(X\ge70) using the normal approximation with the continuity correction 69.5.
3. Compute the exact binomial probability with sum(dbinom(70:200, 200, 0.30)) and compare the two answers.