Final answer:
To test the claim that the standard deviation is greater than $15, a hypothesis test is performed using a one-sample chi-square test. The null hypothesis (H0) is that the standard deviation is not greater than $15 and the alternative hypothesis (Ha) is that the standard deviation is greater than $15. The test statistic is calculated and compared to the critical value from the chi-square distribution table, and since the test statistic is less than the critical value, the null hypothesis is not rejected.
Step-by-step explanation:
To test the claim that the standard deviation is greater than $15, we need to use a hypothesis test. The null hypothesis (H0) is that the standard deviation is not greater than $15, and the alternative hypothesis (Ha) is that the standard deviation is greater than $15. We can use a one-sample chi-square test to perform this hypothesis test.
Step 1: Set up the hypotheses:
- H0: The standard deviation is not greater than $15
- Ha: The standard deviation is greater than $15
Step 2: Calculate the test statistic:
- degrees of freedom = n - 1 = 43 - 1 = 42
- test statistic = (n - 1) * sample standard deviation^2 / hypothesized standard deviation^2 = 42 * 12^2 / 15^2 = 40.32
Step 3: Determine the critical value:
We need to compare the test statistic to the critical value from the chi-square distribution table, with 42 degrees of freedom and a significance level of 5%.
- If the test statistic is greater than the critical value, we reject the null hypothesis.
- If the test statistic is less than or equal to the critical value, we fail to reject the null hypothesis.
After performing the calculations, we find that the test statistic is less than the critical value. Therefore, we fail to reject the null hypothesis. This means that there is not enough evidence to support the claim that the standard deviation is greater than $15.