Question

A mixture of pulverized fuel ash and Portland cement used for grouting should have a compressive strength of more than 1300 KN/m². The mixture will not be used unless experimental evidence indicates conclusively that the strength specification has been met, that is, the true mean compressive strength is more than 1300. Suppose compressive strength for specimens of this mixture is normally distributed with standard deviation of 60. A sample of 10 randomly chosen specimens has a sample mean compressive strength of 1331.26.

a) What are the appropriate null and alternative hypotheses?

b) Carry out the test from part a) at 5% level of significance stating clearly the conclusion in the context of the question.

c) What is the probability of making a Type I error in part b) and describe it in the context of the question?

d) Compute an appropriate one-sided 95% confidence bound for the true mean compressive strength and explain why you chose this bound in the context of the question.

THE ANSWERS FOR THIS PROBLEM ARE AS FOLLOWS:

a) Null: ?=1300, Alternative: ?>1300?

b) z=1.6475, z0.05 = 1.645, z>z0.05 at 5% and mean>1300, so we reject the null.

c) ? = 0.05, Reject null when true. Conclude ?>1300 when it is not.

d) lower: 1300.04 and 1300 is below. ?>1300.

Please explain the steps for how to solve this problem. Thank you!

Answer 1

a) Null and alternative hypothesis:

b) Test statistic:

Critical value:

At = 0.05, right tailed critical value, = 1.645

As z = 1.6475 > 1.645, we reject the null hypothesis.

c) Type I error is: rejecting null hypothesis when it is true.

Concluding that the compressive strength is more than 1300 when in fact it is 1300 or less.

Probability of making a type I error , = 0.05

d) We will compute lower one-sided interval.

lower one sided 95% confidence interval:

As lower confidence interval is more than 1300. we reject the null hypothesis.