Please show me the answer of question 3 and 4 Exercise 6Suppose we are interested in the mean µ of a normal RV X ~ N(μ, o2) with knowno. The null hypothesis is Ho μ = μo and we test it against the alternative hypothesisH₁ μμo at some significance level a, using the sample mean as our test statistic.:µ(i) If the null hypothesis is true, how often would the test reject the null hypothesis?(ii) If the null hypothesis is not true, but instead μ = μ₁ for some μ₁ > μo, how often wouldμι μιthe test reject the null hypothesis? Express your answer in terms of the standardnormal distribution. It will involve the significance level a, the sample size n, thestandard deviation o, and effect size ₁ o. (Hint: What is the rejection thresholdfor 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 theprobability of rejecting the null hypothesis converges to 1 as the sample size n goes toinfinity.(iv) In lecture, we saw that the p-value p(X) as a RV is uniformly distributed under thenull hypothesis. Show that p(X) is not uniformly distributed under the alternativehypothesis. More precisely, show thatP(p(X) ≤ α; μ = μ₁) > αfor any a € (0, 1) and any μ> ₁. Graphically, this means the CDF of p(X) whenµ > µo always lies above the CDF of a uniform distribution. (Hint: use part (ii) andthe equivalent formulation of the test using p-value as a statistic.)

Given information: Here X is the random variable, X~Nμ,σ2 with unknown σ.

Suppose we are interested in the mean of a normal RV X ~ N(μ, o2) with known o. The null hypothesis is Ho μ = μo and we test it against the alternative hypothesis H₁ : µ > µo at some significance level a, using the sample mean à as our test statistic.

Suppose we are interested in the mean of a normal RV X ~ N(μ, o2) with known o. The null hypothesis is Ho μ = μo and we test it against the alternative hypothesis H₁ : µ > µo at some significance level a, using the sample mean à as our test statistic.

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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Question

Please show me the answer of question 3 and 4

Exercise 6
Suppose we are interested in the mean µ of a normal RV X ~ N(μ, o2) with known
o. The null hypothesis is Ho μ = μo and we test it against the alternative hypothesis
H₁ μμo at some significance level a, using the sample mean as our test statistic.
:
µ
(i) If the null hypothesis is true, how often would the test reject the null hypothesis?
(ii) If the null hypothesis is not true, but instead μ = μ₁ for some μ₁ > μo, how often would
μι μι
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 ₁ o. (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) > α
for any a € (0, 1) and any μ> ₁. Graphically, this means the CDF of p(X) when
µ > µo 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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