Final answer:
The correct null and alternative hypotheses to test the claim that the population mean μ is greater than or equal to 72 are H0: μ ≥ 72 and Ha: μ < 72, respectively
Step-by-step explanation:
To test the claim about the population mean μ, we need to set up the correct null hypothesis (H0) and alternative hypothesis (Ha). Based on the given claim that the population mean μ is greater than or equal to 72, the hypotheses for our t-test would be:
- H0: μ ≥ 72 (the population mean is 72 or higher)
- Ha: μ < 72 (the population mean is less than 72)
Since the claim involves a ≥ sign, and we are testing if the mean is less than this value, it becomes a one-tailed test. However, the given solution in the book incorrectly states H0: μ = 65, which is not related to the student's question. Therefore, the correct setup for our hypothesis test based on the claim and given sample statistics (xbar = 74.7, s = 3.5, n = 27) should instead be the hypotheses mentioned above, and since α = 0.05, we are working at a 5% level of significance.
With these given sample statistics, we would use a Student's t-distribution since the population standard deviation is unknown, and the sample size of 27 is not large enough to rely on the normal z-distribution. The use of a t-test is appropriate here because the population is normally distributed. If the p-value from the t-test is less than 0.05, we will reject the null hypothesis in favor of the alternative hypothesis that μ is less than 72.