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Please show me the answer of question 1 and 2

Please show me the answer of question 1 and 2

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Please show me the answer of question 1 and 2

Suppose we are interested in the mean of a normal RV X ~ N(μ, o2) with known o. The null hypothesis is Ho Ho and we test it against the alternative hypothesis : µ H₁ : µ > µ。 at some significance level a, using the sample mean à as our test statistic. (i) If the null hypothesis is true, how often would the test reject the null hypothesis? μ (ii) If the null hypothesis is not true, but instead = μ₁ for some ₁ > Mo, how often would the test reject the null hypothesis? Express your answer in terms of the standard normal distribution. It will involve the significance level a, the sample size n, the standard deviation o, and effect size p₁ - Mo. (Hint: What is the rejection threshold for this test? See also the end of lecture note on Fri. 3/3.) = (iii) Use the answer to part (ii) to show that if the alternative hypothesis is true, then the probability of rejecting the null hypothesis converges to 1 as the sample size n goes to infinity. (iv) In lecture, we saw that the p-value p(X) as a RV is uniformly distributed under the null hypothesis. Show that p(X) is not uniformly distributed under the alternative hypothesis. More precisely, show that P(p(X) ≤ α; µ = µ₁) > a for any a € (0, 1) and any µ > µ₁. Graphically, this means the CDF of p(X) when μμo always lies above the CDF of a uniform distribution. (Hint: use part (ii) and the equivalent formulation of the test using p-value as a statistic.)
Transcribed Image Text:Suppose we are interested in the mean of a normal RV X ~ N(μ, o2) with known o. The null hypothesis is Ho Ho and we test it against the alternative hypothesis : µ H₁ : µ > µ。 at some significance level a, using the sample mean à as our test statistic. (i) If the null hypothesis is true, how often would the test reject the null hypothesis? μ (ii) If the null hypothesis is not true, but instead = μ₁ for some ₁ > Mo, how often would the test reject the null hypothesis? Express your answer in terms of the standard normal distribution. It will involve the significance level a, the sample size n, the standard deviation o, and effect size p₁ - Mo. (Hint: What is the rejection threshold for this test? See also the end of lecture note on Fri. 3/3.) = (iii) Use the answer to part (ii) to show that if the alternative hypothesis is true, then the probability of rejecting the null hypothesis converges to 1 as the sample size n goes to infinity. (iv) In lecture, we saw that the p-value p(X) as a RV is uniformly distributed under the null hypothesis. Show that p(X) is not uniformly distributed under the alternative hypothesis. More precisely, show that P(p(X) ≤ α; µ = µ₁) > a for any a € (0, 1) and any µ > µ₁. Graphically, this means the CDF of p(X) when μμo always lies above the CDF of a uniform distribution. (Hint: use part (ii) and the equivalent formulation of the test using p-value as a statistic.)
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