Final answer:
To test the car company's claim about gas mileage, since the sample size is small and the population standard deviation is unknown, we use the t-distribution for hypothesis testing. The null hypothesis states that the mean gas mileage is at least 20 mpg, while the alternative hypothesis is that it's less than 20 mpg.
Step-by-step explanation:
Hypothesis Testing and Sampling Distribution
To address the student's question, we must conduct a hypothesis test to evaluate the car company's claim that their luxury sedan has a mean gas mileage of at least 20 miles per gallon (mpg). Given a sample mean of 17 mpg, a sample standard deviation of 5 mpg, and a sample size of 6, we first need to determine which sampling distribution to use.
Because the sample size is less than 30 and the population standard deviation is unknown, we must use the t-distribution rather than the z-distribution. The reason is that the t-distribution better accommodates smaller sample sizes and more accurately reflects the uncertainty in the standard deviation estimate.
Next, we'll conduct the hypothesis test. Our null hypothesis (H0) is that the mean mileage is at least 20 mpg, while the alternative hypothesis (Ha) is that the mean mileage is less than 20 mpg. Using the t-distribution, we'll calculate the t-score, compare it to the critical t-value at our significance level of α = 0.05, and then make a decision regarding the null hypothesis. If the t-score is less than the critical value, we reject the null hypothesis, indicating that there is sufficient evidence to support the claim that the mean gas mileage for luxury sedans is less than 20 mpg.