A normally distributed random variable X is assumed to have a mean of 25. The standard deviation of this random variable is 2.25. A random sample of five observations are obtained and the sample average. Required:
a. Test the hypothesis that u (the mean of X) equals 25 (the alternative hypothesis is that the mean does not equal 25). Use α = 0.05.
b. What is the p-value for the test statistic computed in part a?
c. Assume the true mean of X is 23 (with σ = 2.25) and five new observations of X will be obtained. If a test that μ = 25 will be conducted with only the five new observations, what is the probability of Type II error (B)?
d. The sample standard deviation obtained from yet another five observations of X obtained at a different time (with sample average = 21.9) is 5. Because of this value the assumption that the standard deviation of X is 2.25 is no longer made. Using only these most recent five observations conduct a test of the hypothesis that u (the mean of X) equals 25. Use α = 0.05.