hw4_sol

Stats 120B Homework 4: Solution 1. Assume X 1 , . . . , X n form a random sample from N ( µ, σ 2 ) with both µ and σ 2 unknown. (a) What is the sampling distribution of ¯ X ? State the result. You don’t have to prove it. ¯ X ∼ N ( µ, σ 2 /n ) (b) State a result about the distribution of an expression involving the sample variance S 2 = ∑ n i =1 ( X i − ¯ X ) 2 / ( n − 1). You don’ have to prove it. ( n − 1) S 2 σ 2 ∼ χ 2 ( n − 1) Note that any other results that involves S 2 is also fine. (c) Is T a statistic? Is T a pivotal quantity? Explain. Under H 0 , T is a statistic because there is no parmater in it ( µ 0 is a fixed constant). T is a pivotal quantity since its distribution does not have any parameters. (d) Assume ( − 1 , 2) is a 90% confidence interval for µ , find a 90% confidence in- terval for 1 /µ . P ( − 1 < µ < 2) = . 90 ⇒ P ( µ − 1 > 1 / 2or µ − 1 < − 1) = . 90 So CI is ( −∞ , − 1) ∪ (1 / 2 , ∞ ). Read the following R code and output, answer questions (e)–(j). > 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 (e) Find the value of n , ¯ X , S 2 and µ 0 . 1

n = 100 , ¯ X = 1 . 49 , S 2 = 15 . 95 , µ 0 = 0 (f) State the alternative hypothesis. H a : µ ̸ = 0 (g) State your conclusion of hypothesis test at the significance level of α = . 05. p-value is 0 . 0003167, so we reject the null hypothesis at significance level of α = . 05. There is enough evidence to reject µ = 0. (h) State your conclusion of hypothesis test at the significance level of α = . 0001. p-value is above α now. So we don’t have enough evidence against the null hypothesis. (i) What are the assumptions we make on the data? (1) normality; (2) independence; (3) identically distributed. (j) Here is the QQ plot of the data. Which assumption is checked here? Does the data satisfy that assumption? -3 -2 -1 0 1 2 3 -0.2 0.0 0.2 0.4 0.6 Theoretical Quantiles Sample Quantiles QQ plot is used to check normality assumption. And it appears that normality is not satisfied for the data. (The answer here is subjective, you can say that the normality assumption looks reasonable if you think the curve looks close to the straight line. ) 2