STAT177 Homework sheet #5 Solutions

1) Ho: p ³ 1.2%

Ha: p < 1.2%

2) Here are the output results:

Z test for one proportion

Ho: population proportion is not less than 0.012

Ha: population proportion is less than 0.012

Reject Ho if test statistic < -1.645

Test statistic = -0.676

P-value = 0.25

Do not reject Ho

Conclude population proportion is not less than 0.012

95% confidence interval: 0.001 to 0.017

Since the p-value is greater than the level of significance, we do not reject the null hypothesis. We conclude the quality control measures were not effective.

3) Since we did not reject the null hypothesis, we could have committed a Type II error.

4) The p-value would still be greater than the level of significance. The answer is no.

5) Ho: p £ 1/800

Ha: p > 1/800

6) We have np = 2538(1/800) = 2538/800 = 3.1725. Since this is less than the expected number of successes of 5, the appropriate test is the exact binomial test.

7) Here is the output:

Exact binomial test for one proportion

Ho: population proportion is not greater than 0.00125

Ha: population proportion is greater than 0.00125

Reject Ho if p-value < 0.05

P-value = 0.016

Reject Ho

Conclude population proportion is greater than 0.00125

Since the p-value is less than the level of significance, we reject the null hypothesis. We conclude the marketing method was successful.

8) We can split straws here but the p-value is pretty close to the threshold value of 1% under the general rule of thumb. This provides good support for Ha allowing us to reject Ho.

9) Ho: p = 78%

Ha: p ¹ 78%

10) Since the expected number of both failures and successes is greater than 5, the Z test is the appropriate test.

11) Here is the output:

Z test for one proportion

Ho: population proportion equals 0.78

Ha: population proportion does not equal 0.78

Reject Ho if test statistic > 1.96 or < -1.96

Test statistic = 0.834

P-value = 0.404

Do not reject Ho

Conclude population proportion equals 0.78

95% confidence interval: 0.721 to 0.946

Since the p-value is greater 10% under the general rule of thumb, we do not reject the null hypothesis. We conclude there is no significant change in the percentage of students achieving an acceptable grade on provincial exams.

12) From the output, we see the maximum percentage of 94.6%

13) Ho: s ³ 1

Ha: s < 1

14) Here is the output:

Chi-square test for one standard deviation

Ho: population standard deviation is not less than 1

Ha: population standard deviation is less than 1

Reject Ho if test statistic < 6.571

Test statistic = 6.813

P-value = 0.058

Do not reject Ho

Conclude population standard deviation is not less than 1

95% confidence interval: 0.511 to 1.1

Since the p-value is greater than the level of significance, we do not reject the null hypothesis and conclude the standard deviation is not less than 1 per hour.

15) Since we did not reject the null hypothesis, the opposite conclusion would be reached if we rejected the null hypothesis. This is achieved if the p-value is less than the level of significance. Since the p-value is 5.8%, we would reject the null hypothesis for levels of significance between 5.8% and 10%.