Problem
A car company says that the mean gas mileage for its luxury sedan is at least 22 miles per gallon $(\mathrm{mpg})$. You believe the claim is incorrect and find that a random sample of 6 cars has a mean gas mileage of $21 \mathrm{mpg}$ and a standard deviation of $5 \mathrm{mpg}$. At $\alpha=0.05$, test the company's claim. Assume the population is normally distributed. Click here to view the t-distribution table. Click here to view page 1 of the normal table. Click here to view page 2 of the normal table. Which sampling distribution should be used and why? A. Use a normal sampling distribution because $n>30$. C. Use a t-sampling distribution because the population is normal, and $\sigma$ is known. E. Use a normal sampling distribution because the population is normal, and $\sigma$ is known. B. Use a normal sampling distribution because the population is normal, and $\sigma$ is unknown. D. Use a t-sampling distribution because the population is normal, and $\sigma$ is unknown. F. Use a t-sampling distribution because $n<30$.
Solution
Step 1 ：The question is asking which sampling distribution should be used given the information provided.
Step 2 ：The sample size is less than 30, the population is normally distributed, and the population standard deviation is unknown.
Step 3 ：Therefore, we should use a t-sampling distribution because the population is normal, and the standard deviation is unknown.
Step 4 ：This is because the t-distribution is typically used when the sample size is small and the population standard deviation is unknown.
Step 5 ：Final Answer: \boxed{\text{D. Use a t-sampling distribution because the population is normal, and } \sigma \text{ is unknown.}}
