Statistics and Probability Answers

A large manufacturing company is in the process of training its personnel in quality-control

procedures. At present 40% of the assembly lines use control charts, 40% use inspection

techniques and the remaining assembly lines do not use any method for controlling quality.

Assume that an assembly line can only use a maximum of 1 method, if any, for quality

control.

The assembly lines that use control charts have a 1% defective rate. The assembly lines

that use inspection techniques have a 5% defective rate. The assembly lines that do not use

any quality-control techniques have a 12% defective rate. The probability that an item

produced by this company is defective is (correct to 3 decimal places):

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

f(x, y ) = k e ^{− (8x^2− 6xy − 18y^2)}

Find

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

(b) the joint moment generating function of Z = 2X − Y and W = 3X + 2Y

Let X, Y independent and identically distributed random variables from a distribution having probability density function

f(x) = 1, 0 < x < 1

Further let Z = max(X, Y). Using the distribution function technique, find the probability density function of Z. Hence find its mean and variance values.

State the null hypothesis 𝐻0 and alternative hypothesis 𝐻𝐴 for the following experiments

2. Does fresh fruit juice have effect on colds in the group who regularly drink it compared to the group who do not? 

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.