Suppose that a random (i.i.d.) sample of nine observations X1,,X9 is taken from the normal
distribution with unknown mean and unknown variance 2, and it is desired to test the following
hypotheses: H0:=20 vs. H1:=20. Suppose also that the sample data are such that i=19Xi=198 and
i=19(XiXn)2=72. Carry out a t-test at the level of significance (i.e., type I error rate) 0.1 such that
each tail of the rejection region has probability 0.05 . Should the hypothesis H0 be rejected or not?
(Maybe) useful facts: The 0.95 -quantile of t(8) distribution is 1.860. The value of CDF of t(8)
distribution at 2 is 0.960 . The 0.95 -quantile of t(9) distribution is 1.833 . The value of CDF of t(9)
distribution at 2 is 0.962 . Instruction: Do not simply provide a Yes/No answer Please justify your
derivation. Include in your answer the following: (1) What is the test statistic? (2) What is the
distribution of the test statistic (under H0 )? (3) What is the value of the test statistic in this
particular data set? (4) What is the rejection region with the desired type I error rate (or p-value)?
(5) What comparison leads to your decision of rejecting/not rejection H0 ?.
Suppose that a random iid sample of nine observations X.pdf
1. Suppose that a random (i.i.d.) sample of nine observations X1,,X9 is taken from the normal
distribution with unknown mean and unknown variance 2, and it is desired to test the following
hypotheses: H0:=20 vs. H1:=20. Suppose also that the sample data are such that i=19Xi=198 and
i=19(XiXn)2=72. Carry out a t-test at the level of significance (i.e., type I error rate) 0.1 such that
each tail of the rejection region has probability 0.05 . Should the hypothesis H0 be rejected or not?
(Maybe) useful facts: The 0.95 -quantile of t(8) distribution is 1.860. The value of CDF of t(8)
distribution at 2 is 0.960 . The 0.95 -quantile of t(9) distribution is 1.833 . The value of CDF of t(9)
distribution at 2 is 0.962 . Instruction: Do not simply provide a Yes/No answer Please justify your
derivation. Include in your answer the following: (1) What is the test statistic? (2) What is the
distribution of the test statistic (under H0 )? (3) What is the value of the test statistic in this
particular data set? (4) What is the rejection region with the desired type I error rate (or p-value)?
(5) What comparison leads to your decision of rejecting/not rejection H0 ?
