For part (b), I don't understand why when sigma is unknown, we do not reject the H0 in part (b), but when sigma is known, we reject H0 in part (b). I understand why we use t distribution when sigma is unknown, but why this result in different conclusions about the hypothesis testing? Please explain this and draw distribution graph to illustrate when necessary. Thanks!

4. A random sample of 16 observations from the population N(u, 2) yields the sample mean x = 9.31 and the sample variance s2 = 0.375. At the 5% significance level, test the following hypotheses by obtaining critical values: (a) Ho : M = 9 vs. H1 : M > 9. (b) Ho : M = 9 vs. H1 : M 1:035:15 = 1.753, against H1 2 pt > 9. (b) t {0325} 15 = 2.131, against H1 1 ,1), 7E 9. For the given sample, t = 2.02. Hence we reject H0 against the alternative H1 : p, > 9, but we will not reject H0 against the two other alternative hypotheses. When 02 is known, we use the test statistic T = JarLO? 9)/0'. Now under H0, T N N(0, 1). With 0: = 0.05, we reject H0 if: (a) t > 21335 = 1.645, against H1 : p > 9. (b) t 740325 = 1.96, against H1 : ,u 75 9. For the given sample, t = 2.02. Hence we reject H0 against the alternative H1 : ,u > 9 and H1 : p 75 9, but we will not reject H0 against H1 : p.