"Time Spent in the Drive-Thru The quality-control manager of a Long John Silver’s restaurant wants to analyze the length of time that a car spends at the drive-thru window waiting for an order. It is determined that the mean time spent at the window is 59.3 seconds with a standard deviation of 13.1 seconds. The distribution of time at the window is skewed right (data based on information provided by Danica Williams, student at Joliet Junior College).
a. To obtain probabilities regarding a sample mean using the normal model, what size sample is required?"

Verified step by step guidance

1. Understand that to use the normal model (Central Limit Theorem) for the sampling distribution of the sample mean, the sample size must be large enough to ensure the distribution of the sample mean is approximately normal, especially since the original data is skewed right.
2. Recall the Central Limit Theorem states that the sampling distribution of the sample mean will be approximately normal if the sample size is sufficiently large, often n ≥ 30 is used as a rule of thumb for skewed distributions.
3. Identify the given information: population mean \( \mu = 59.3 \) seconds, population standard deviation \( \sigma = 13.1 \) seconds, and the distribution is skewed right.
4. Since the original distribution is skewed, determine the minimum sample size \( n \) needed so that the sampling distribution of the sample mean is approximately normal. This usually involves choosing \( n \geq 30 \) to satisfy the normality condition for the sample mean.
5. Conclude that the required sample size is the smallest \( n \) such that the Central Limit Theorem applies, typically \( n \geq 30 \), to use the normal model for probabilities regarding the sample mean.

Key Concepts

Central Limit Theorem (CLT)

The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution shape. This allows us to use normal probability models for sample means when the sample size is sufficiently large, even if the original data is skewed.

Sample Size Determination

Sample size determination involves finding the minimum number of observations needed so that the sampling distribution of the sample mean can be approximated by a normal distribution. For skewed populations, a larger sample size is required to ensure the Central Limit Theorem applies and probabilities can be accurately calculated.

Skewed Distribution

A skewed distribution is one where data is not symmetrically distributed; in this case, it is skewed right, meaning there is a longer tail on the right side. Skewness affects how quickly the sampling distribution of the mean becomes normal, often requiring larger samples to apply normal-based inference methods.