Question

The McCollough Corporation, a producer of various kinds of batteries, has been producing "D" size batteries with a life expectancy of 87 hours. Due to an improved production process, the management believes that there has been an increase in the life expectancy of their "D" size batteries. A sample of 36 batteries showed an average life of 88.5 hours. Assume from past information that it is known that the standard deviation of the population is 9 hours. Test management's belief and make a decision such that probability of making a type 1 error is no more than 0.01.

1.The null & alternative hypothesis are:

2.Significance level of the test is:

3.Explain the appropriate test statistic is Z or t. Then explain why.

4.Calculated value of the test statistic is:

5.The critical value of the test is:

6.The decision about the test is:

7.The decision using p-value criterion is:

Answer 1

Solution:

Given: An average life of "D" size batteries is 87 hours. That is hours

Claim: Due to an improved production process, the management believes that there has been an increase in the life expectancy of their "D" size batteries.

Sample size = n = 36

Sample mean = hours

Population standard deviation = hours.

probability of making a type 1 error is no more than 0.01.

Part 1) The null & alternative hypothesis are:

Vs

Part 2) Significance level of the test is

We have Probability of making a type 1 error is no more than 0.01.

Thus Significance level of the test is = 0.01

Part 3) Explain the appropriate test statistic is Z or t. Then explain why.

We use Z test statistic.

Since sample size n is large as well as population standard deviation is known.

Part 4) Calculated value of the test statistic is:

Part 5) The critical value of the test is:

We have Significance level of the test is = 0.01 and H1 is > type , so this is right tailed test.

Thus look in z table for Area = 1 - 0.01 = 0.99 and find corresponding z value.

Area 0.9901 is closest to 0.9900 , thus corresponding z value is 2.3 and 0.03

Thus Z_critical = 2.33

Part 6) The decision about the test is:

Decision Rule: reject H0 , if z test statistic value > z critical value = 2.33, otherwise we fail to reject H0.

Since z test statistic value = 1.00 < z critical value = 2.33, we fail to reject H0.

Thus there is not sufficient evidence to support the management's claim that: there has been an increase in the life expectancy of their "D" size batteries.

Part 7) The decision using p-value criterion is:

p-value = P( Z > z test statistic value)

p-value = P(Z > 1.00)

p-value = 1 - P( Z< 1.00)

Look in z table for z = 1.0 and 0.00 and find area.

Thus from z table we get:

P( Z< 1.00) = 0.841

Thus

p-value = 1 - P( Z< 1.00)

p-value = 1 - 0.8413

p-value = 0.1587

Decision rule: Reject H0 , if p-value < 0.01 level of significance, otherwise we fail to reject H0.

Since p-value = 0.1587 > 0.01 level of significance, we fail to reject H0.