Statistics and Probability Answers

Present complete solutions for each item. Illustrate the normal curve if possible.

1. A population consists of six values (6, 9, 12, 15, 18, and 21).

a. Select a random sample of size 3. Explain the random sampling that you used.

b. How many possible samples can be drawn?

c. List all possible samples and compute the mean of each sample.

d. Construct a frequency distribution of the sample means obtained in step 2 including

x̅; f; P(x̅); ̅x ⋅ P(x̅); ̅x²⋅ P(x̅); ΣP(x̅); Σx̅⋅ P(x̅) and Σx̅²⋅ P(x̅).

e. Determine the mean, variance and standard deviation of the sample mean.

A researcher is testing the hypothesis that all teenagers spend an average of 8 hours on their computers during the weekends. He knows that the standard deviation is 0.3 hour. He selects a sample of 144 teenagers and decides to reject the null hypothesis when the sample mean is 8.5 hours or less.

A. What us the probability of that the researcher commits a type I error?

B. If the true population mean is 7 hours, what is the probability that the he commits a type II error?

C. Determine the power of test.

Let X1, X2 have joint probability density function

f(x1, x2) = {1/8e^−(8x1+x2) , x1,x2>0

0, elsewhere

Find the probability density function of Y =1/2 (X1 + X2)

Let X, Y have joint probability density function

f(x, y) = {1/8e^−(8x+y). , x,y>0

0, elsewhere

Find the probability density function of Z=1/2 (X + Y)

Suppose that X and Y have bivariate normal distribution with the probability density function

f(x, y) = k exp ^− (8X^2 − 6XY − 18Y^2 )

Find

(a) Pr(X + Y > 1/2)

(b) the joint moment generating function of Z1 = 2X − XY and Z2 = 3X + 2Y

Suppose that X and Y have bivariate normal distribution with the probability density function

f(x, y) = k exp ^− (8X^2 − 6XY − 18Y^2 )

Find

(a) Pr(X + Y > 1/2)

(b) the joint moment generating function of Z1 = 2X − XY and Z2 = 3X + 2Y