Assume X1, . . . , Xn form a random sample from N(µ, 2 ) with both µ and 2 unkno

Question

Assume X1, . . . , Xn form a random sample from N(µ, 2 ) with both µ and 2 unknown. (a) What is the sampling distribution of X¯? State the result. You don’t have to prove it. (b) State a result about the distribution of an expression involving the sample variance S 2 = n i=1(Xi X¯) 2/(n 1). You don’ have to prove it. (c) Consider a null hypothesis H0 : µ = µ0, prove that under H0, T = n(X¯ µ0) S tn1. (d) Is T a statistic? Is T a pivotal quantity? Explain. 2 (e) Assume (1, 2) is a 90% confidence interval for µ, find a 90% confidence interval for 1/µ. Read the following R code and output, answer questions (f )–(h). > n = length(x) > n [1] 100 > mean(x) [1] 1.49067 > var(x) [1] 15.95129 > stat = (mean(x) - 0)/(sqrt(var(x)/(n))) > stat [1] 3.732360 > pt(stat,n-1,lower=F) *2 [1] 0.0003167425 (f) Find the value of n,X¯, S 2 and µ0. (g) State the alternative hypothesis. (h) State your conclusion of hypothesis test at the significance level of = .05. (i) State your conclusion of hypothesis test at the significance level of = .0001. (j) What are the assumptions we make on the data? (k) Here is the QQ plot of the data. Which assumption is checked here? Does the data satisfy that assumption?

Explanation / Answer

a. The Central Limit Theorem states that if a large enough sample is taken (typically n30) then the sampling distribution of x¯ is approximately a normal distribution with a mean of and a standard deviation of / n.

QQ plot of the data missing ??

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