Decide whether you should use the standard normal sampling...

Decide whether you should use the standard normal sampling distribution or a $t$ -sampling distribution to perform the hypothesis test. Justify your decision. Then use the distribution to test the claim. Write a short paragraph about the results of the test and what you can conclude about the claim. Gas Mileage A car company claims that the mean gas mileage for its Iuxury sedan is at least 23 miles per gallon. You believe the claim is incorrect and find that a random sample of 5 cars has a mean gas mileage of 22 miles per gallon and a standard deviation of 4 miles per gallon. At $\alpha=0.05,$ test the company's claim. Assume the population is normally distributed.

Decide whether you should use the standard normal sampling distribution or a $t$ -sampling distribution to perform the hypothesis test. Justify your decision. Then use the distribution to test the claim. Write a short paragraph about the results of the test and what you can conclude about the claim.
Gas Mileage A car company claims that the mean gas mileage for its Iuxury sedan is at least 23 miles per gallon. You believe the claim is incorrect and find that a random sample of 5 cars has a mean gas mileage of 22 miles per gallon and a standard deviation of 4 miles per gallon.
At $\alpha=0.05,$ test the company's claim. Assume the population is normally distributed.

Key Concepts

t-distribution

The t-distribution is used when estimating population parameters from a small sample size, particularly when the population standard deviation is unknown. It accounts for additional variability by having thicker tails compared to the normal distribution, which provides more accurate critical values for hypothesis tests in these scenarios.

Standard Normal Distribution

The standard normal distribution, often used when the population standard deviation is known or the sample size is large, offers a set of critical values based on the assumption of a normal shape. It is not appropriate for small sample sizes with an unknown population standard deviation because it underestimates the natural variability in the estimate.

Small Sample Size Consideration

When the sample size is small (typically n < 30), the reliability of variance estimates decreases, making the t-distribution a better choice. The t-distribution adjusts for the estimation of the sample standard deviation, leading to more appropriate confidence intervals and hypothesis tests under these conditions.

Hypothesis Testing

Hypothesis testing involves comparing a test statistic to a critical value to determine whether to reject the null hypothesis. In this context, the test is used to evaluate the company's claim about mean gas mileage by setting up null and alternative hypotheses, computing a test statistic based on the sample, and comparing it to the critical t-value at a given significance level.

Degrees of Freedom

Degrees of freedom, calculated as the sample size minus one (n - 1), are crucial in determining the shape of the t-distribution. They are used to adjust the distribution to better represent the variability inherent in small samples, affecting both the critical values and the resulting p-value in hypothesis tests.

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