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 Ho be rejected or not?

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Chapter1: Starting With Matlab
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Suppose that a random (i.i.d.) sample of nine observations X1, ..., X, is taken from the
normal distribution with unknown mean µ and unknown variance o², and it is desired to
test the following hypotheses: Ho µ = 20 vs. Hị : µ + 20. Suppose also that the sample
data are such that X; = 198 and (X; - Xn)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 Ho 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 dis-
tribution of the test statistic (under Ho)? (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 Ho?
19
Transcribed Image Text:Suppose that a random (i.i.d.) sample of nine observations X1, ..., X, is taken from the normal distribution with unknown mean µ and unknown variance o², and it is desired to test the following hypotheses: Ho µ = 20 vs. Hị : µ + 20. Suppose also that the sample data are such that X; = 198 and (X; - Xn)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 Ho 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 dis- tribution of the test statistic (under Ho)? (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 Ho? 19
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