"Burger King’s Drive-Thru Suppose that cars arrive at Burger King’s drive-thru at the rate of 20 cars every hour between 12:00 noon and 1:00 P.M. A random sample of 40 one-hour time periods between 12:00 noon and 1:00 P.M. is selected and has 22.1 as the mean number of cars arriving.
a. Why is the sampling distribution of x_bar approximately normal?"

Verified step by step guidance

1. Identify the type of distribution for the number of cars arriving per hour. Since cars arrive randomly and independently over time, this can be modeled as a Poisson process, where the number of arrivals in a fixed interval follows a Poisson distribution.
2. Recall that the sampling distribution of the sample mean \( \overline{x} \) is the distribution of the means of many samples drawn from the population.
3. Apply the Central Limit Theorem (CLT), which states that for a sufficiently large sample size, the sampling distribution of the sample mean \( \overline{x} \) will be approximately normal regardless of the shape of the population distribution.
4. Note that the sample size here is 40, which is generally considered large enough for the CLT to hold, making the sampling distribution of \( \overline{x} \) approximately normal.
5. Therefore, the reason the sampling distribution of \( \overline{x} \) is approximately normal is because the sample size is large, allowing the Central Limit Theorem to apply even though the original distribution (Poisson) is not normal.

Key Concepts

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean describes the distribution of means calculated from many random samples of the same size drawn from a population. It shows how the sample mean varies from sample to sample and is fundamental for making inferences about the population mean.

Central Limit Theorem (CLT)

The Central Limit Theorem states that, regardless of the population's distribution, the sampling distribution of the sample mean approaches a normal distribution as the sample size becomes large (usually n ≥ 30). This explains why the sample mean's distribution is approximately normal in this problem.

Sample Size and Its Effect on Normality

A sufficiently large sample size (like 40 in this case) ensures the sampling distribution of the mean is approximately normal, even if the original data are not normal. Larger samples reduce variability and make the normal approximation more accurate.