Question

It is known that the thread life of a certain type of tire has a normal distribution with standard deviation of 1500.

a) A sample of 16 tires is found to have an average thread life of 30960. Does this provide sufficient evidence at 1% level of significance to conclude that the true average thread life of this type of tires is more than 30000? Explain by carrying out an appropriate hypothesis test stating clearly the hypotheses.

b) What is the probability of making a Type II error in the hypothesis test in part "a" if the true average thread life is in fact 31000?

c) If a 1% level of significance is used to carry out the test in part "a" and it is also required that ?(30500) = 0.05, what sample size is necessary?

THE ANSWERS ARE AS FOLLOWS

a) ?=30000, ?>30000, z=2.56, z0.01=2.327, reject.

b) 0.3671

c) 142

Please explain the process to solve this problem. Thank you!

Answer 1

a)

Hypotheses are:

Given information:

Since population standard deviation is known so single sample z test will be used.

The test statistics is

The critical value using excel function "=NORMSINV(0.01)" is

Since z > 2.327 , so we reject H0

b)

The critical value of sample mean for which we will reject the null hypothesis is

The z-score for and is

The probability of making a Type II error in the hypothesis test in part "a" if the true average thread life is in fact 31000 is

c)

Test is right tailed so for we have

and

and

So sample size is