1. Let Xi, , X10 be a random sample of size 10 from a N(p, 02) population. Suppose Y = E(X; — p)2 i=1 Find the probability that the random interval Y Y (20 5' 3 25) includes the point 02. Hint: Note...

1. Let Xi, , X10 be a random sample of size 10 from a N(p, 02) population. Suppose
Y = E(X; — p)2
i=1
Find the probability that the random interval Y Y (20 5' 3 25)
includes the point 02.
Hint: Note that 1; is A (why?), and the question is equivalent to computing P(3.25

2. Let the random variable X have the pdf f (r) = e-x for 0

— xn+1 S
has a t distribution. 6. In the framework of Exercise 5 suppose it = 8. Determine k such that P(g — kS

8. Let Si and s3 denote the variances of random samples, of sizes it and m respectively, from two independent distributions which are N(µ3,02) and N(µ2,02), respectively. Note that the two populations have different means but the same variances. You may use the fact that nS? + ntS3 has the distribution x2 with (n + m — 2) df to help you to find a confidence interval for the common unknown variance a2
9. Please do Problem 5 on page 276 of the textbook 10. Please do Problem 22 on page 284 of the textbook 11. Please do Problem 30, parts (a) and (b), AND Problem 31, parts (a) and (b), on page 292 of the textbook
May 18, 2022
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