Suppose X has a normal distribution with mean 0 and variance 1, Y

has a chi-square distribution with n degrees of freedom, W has a chi-square

distribution with p degrees of freedom, and W, X, and Y are independent.

What is the sampling distribution of the statistic V = *

X

W+Y

p+n

?

14. A random sample X1, X2, ..., Xn of size n is selected from a normal

population with mean and standard deviation 1. Later an additional independent observation Xn+1 is obtained from the same population. What

is the distribution of the statistic (Xn+1 ? )2 + n

i=1(Xi ? X)2, where X

denote the sample mean

Suppose you take 50 measurements on the speed of cars on Interstate 5, and

that these measurements follow roughly a Normal distribution. Do you expect

the standard deviation of these 50 measurements to be about 1 mph, 5 mph, 10

mph, or 20 mph? Explain.

28. Suppose that combined verbal and math SAT scores follow a normal

distribution with mean 896 and standard deviation 174. Suppose further that

Peter finds out that he scored in the top 3% of SAT scores. Determine how high

Peter's score must have been.

29. Heights of adult women in the United States are normally distributed with a

population mean of ?= 63.5 inches and a population standard deviation of ? =

2.5. A medical re- searcher is planning to select a large random sample of adult

women to participate in a future study. What is the standard value, or z-value,

for an adult woman who has a height of 68.5 inches?

30. An automobile manufacturer introduces a new model that averages 27 miles

per gallon in the city. A person who plans to purchase one of these new cars

wrote the manufacturer for the details of the tests, and found out that the

standard deviation is 3 miles per gallon. Assume that in-city mileage is

approximately normally distributed.

a. What is the probability that the person will purchase a car that averages less

than 20 miles per gallon for in-city driving?

b. What is the probability that the person will purchase a car that averages

between 25 and 29 miles per gallon for in-city driving

3, For two random variables X and Y, the joint probability density function is given by fxx(x, y) = Ce * Zu(x)u(y) Find: a. C b. Fxx (x, y), the joint cumulative distribution function. c. Fx(x) and Fy (y), marginal distribution functions d. Are X and Y independent?Exercise 6.42. Let (X, Y) be a uniform random point on the rectangle D = [0,2] x [0,3] = {(x, y) : 0 0, y>0. (a) Find the joint probability density function f(x, y). (b) Are the random variables X and Y independent? (c) Find Fx (4)