The Commute Time Study
A transportation researcher records the commute times (in minutes) of 40 randomly selected employees at a large company. The sample mean is 32.5 minutes and the sample standard deviation is 8.2 minutes. A histogram of the data shows a roughly symmetric distribution with no outliers. The researcher plans to construct a 95% confidence interval for the population mean commute time. However, a colleague points out that the population standard deviation is unknown and suggests using a t-interval instead of a z-interval. The researcher also wonders if the sample size is large enough to satisfy the conditions for inference.
1. Explain why a t-interval is more appropriate than a z-interval in this situation.
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A t-interval is more appropriate because the population standard deviation is unknown and is estimated by the sample standard deviation. The t-distribution accounts for the additional uncertainty introduced by estimating the population standard deviation from the sample. Using a z-interval would underestimate the variability and produce an interval that is too narrow, leading to incorrect coverage probability.
2. Check the conditions for constructing a t-interval for the population mean. Are they satisfied? Justify your answer.
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The conditions are: (1) Random sample: The employees were randomly selected, so this condition is met. (2) Independence: The sample size (40) is less than 10% of all employees (assuming the company has more than 400 employees), so individual observations can be considered independent. (3) Normality: The sample size is 40, which is greater than 30, and the histogram is roughly symmetric with no outliers, so the sampling distribution of the sample mean is approximately normal by the Central Limit Theorem. Thus, all conditions are satisfied.