## Homework 5 Due 6-18 by 10:00 pm EST

Instructions: Solve the problems in the spaces provided and save as a single PDF. Then upload the PDF to Canvas Assignments by the due date. The recommended procedure is to download and print the homework. Fill in your solutions. Then scan the document and upload to Canvas Assignments. If this is not feasible, you may solve the problems on your paper, scan your solutions, then upload to Canvas. Neatness and presentation are important.

Late homework not accepted. Show all work. Total points: 30

### 1) Let X1. . . , Xn be iid N(µ, 16) where σ2= 16 is known. We wish to test the simple hypotheses  美国统计代写

H0 : µ = 2

H1 : µ = 4

Find the sample size n for which the most powerful test will have α = β = 0.01.

Round n to the next largest integer.

Here α is the size of the test which is P(Type I Error) = P(Reject H0|H0 True)and β is P(Type II Error) = P(Fail to reject H0|H0 False).

Hint: What is the distribution of when H0 is true? When H1 is true?

What is β? Drawing a picture may help. (4 points)

### 2) Consider the simple linear regression model

yi = α + β xi + i , i N(0, σ2 ) independent

i = 1, . . . , n

Let  and be the sample means of the xi and yi , respectively. In addition, let  and be the MLEs of α and β, respectively. Let ˆyi = + xi be the fifitted values, and let ei= yi yˆi be the residuals.

a) Find Cov(, ) . (3 points)

b) Show that  = 0. (3 points)

c) Show that if  = 0, then Cov(, ) = 0. (2 points)

### 3) Consider again the simple linear model  美国统计代写

i = 1, . . . , n

a) Using the notation in our lecture notes, the sample correlation coeffiffifficient is defifined as

Derive an expression for r in terms of βˆ, sxx, and syy. Do r and βˆ have the same sign?(3 points)

b) In class, we saw that syy = SSR + SSE. That is,

Derive an expression for

in terms of βˆ, sxx, and syy and show that it equals r 2 that you derived in 3a). Hint: Start

with the expression for the fifitted values on page 4 of Lecture 15. (4 points)

c) Write the F-statistic that tests H0 : β = 0 as a function of R2. What happens as R20?What happens as R2 1? (3 points)

### 4) For a 2 × 2 table with data and cell probabilities  美国统计代写

n11, n12, n21, n22 Multinomial(n; π11, π12, π21, π22)

we found that the likelihood ratio statistic for testing independence is

a) Find (3 points)

b) Expand

in a second order Taylor series about

Then use the approximation in

to derive the Pearson chi-squared statistic X2 (see Lecture 17). (5 points)Optional problems to try for practice (do not turn in)

Page 363: 12, 13

Page 595: 15