PLEASE ANSWER PART 7 Exercise 2.1 Parts 1-5 appeared on Quiz #1, 2010-11, Term 2

Question

PLEASE ANSWER PART 7

Exercise 2.1 Parts 1-5 appeared on Quiz #1, 2010-11, Term 2 Suppose a random sample of size n-2 is drawn from a N (, 2) distribution to estimate when 2 is unknown There is a big impact on the 95% confidence interval for from using the t distribution instead of the standard normal. > qnorm (0.975) [1] 1.959964 > qt (0.975, df1) [1] 12.7062 Thus, a confidence interval for based on the t distribution will be much, much wider. Why? And why does the t distribution have 1 df here (and not 2 from n = 2)? The exercise sheds some light on these questions Let Y, and ½ be independent random variables sampled from a N (, 2) distribution. For such a sample of size n = 2 it is easily shown that the sample variance. S2 simplifies to You may use this result without proof. 1. Let V = (Yi-Y2)/V2. Show the following properties of V. Carefully state any result you are using (no proof of the result required) and how it is applied here (a) E(V)=0. (b) Var(V)=g2. (c) V has a normal distribution. 2. Explain why the distribution of V/o is standard normal. 3. Hence argue that when n = 2, the distribution of S2/2 is . (A result on the connection between N (0,1) and without proof random variables may be stated and used

Explanation / Answer

Answer:

Part 7

R code

l<-qchisq(.025, df=1)

l

u<-qchisq(.975, df=1)

u

R output:

l<-qchisq(.025, df=1)

> l

[1] 0.0009820691

> u<-qchisq(.975, df=1)

> u

[1] 5.023886

Quantile l=0.0009820691

Quantile u=5.023886

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