Question

The yield of a chemical process is being studied. From previous experience with this process the standard deviation of yield is known to be 2.8. The past 5 days of plant operation have resulted in the following yields (given in percentages): 91.6, 88.75, 90.8, 89.95 and 91.3.
Use alpha=.05

(a) Is there evidence that the mean yield is not 90%? What is the P-value of the test?

(b)What sample size would be required to detect a true mean yield of 85% with probability 0.95?

(c) What is the type II error probability if the true mean yield is 92%?

(d) Find a 95% two-sided confidence interval on the true mean yield.

(e) Use the confidence interval found in part (d) to test the hypothesis.

Please help with equations and table use.

Answer 1

The sample mean is

(a)

Hypotheses are:

The test statistics is

The p-value is:

p-value= 2P(z > 0.36) = 0.7188

No, p-value is greater than 0.05.

(b)

Test is two tailed so for we have

and

and

So sample size is

(c)

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

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

The z-score for and is

The z-score for and is

Type II error:

(d)

The required confidence interval is

Answer: (87.9, 93.1)

(e)

Since confidence interval contains 90 so we fail to reject the null hypothesis.