Significance Tests and Confidence Intervals quiz

The power of a significance test to reject the null hypothesis when a particular value of the alternative is true is 0.8. What is the probability that this test results in a Type II error?

0.20

A student claims that statistics students at her school spend, on average, an hour doing statistics homework each night. In an attempt to substantiate this claim, she selects a random sample of 6 of the 62 students who are currently taking statistics and asks them how much time they spend completing statistics homework each night. She would like to know if there is convincing statistical evidence that the true mean amount of time that statistics students spend doing statistics homework each night is less than 1 hour. The student plans to test the hypotheses, = 1 versus < 1, where μ = the true mean amount of time that statistics students spend doing statistics homework each night. Assume all conditions have been met. The student discovers that the power of this test to reject the null hypothesis when μ = 0.75 is not very high. Which value of the alternative hypothesis would yield the greatest power of this test?

0.25

An engineer would like to design a parking garage in the most cost-effective manner. The garage must be able to fit pickup trucks, which have an average height of 76.4 inches. To double-check this figure, the engineer employs a statistician. The statistician selects a random sample of 100 trucks, which will be used to determine if these data provide convincing evidence that the true mean height of all trucks is greater than 76.4 inches. The statistician plans to test the hypotheses, = 76.4 versus > 76.4, where μ = the true mean height of all trucks using α = 0.05. The statistician would like to increase the power of this test to reject the null hypothesis when μ = 77 inches. Which sample size would increase the power of this test?

110

A study was conducted to determine the true mean hourly wage of all working high school students. A 95% confidence interval for the true mean hourly wage of high school students is ($6.50, $13.00). Based upon this interval, what conclusion should be made about the hypotheses: = 15 versus where = the true mean hourly income of all working high school students at α = 0.05?

Reject H0. Since 15 falls outside the 95% confidence interval, there is convincing evidence that the true mean hourly income of all working high school students is different than $15.

A social scientist collects information about study time for a random sample of 40 students with the intention of testing the hypotheses = 2 hours per night versus 2 hours per night where = the true mean number of hours of study time per night for students. Rather than test these hypotheses, she computes the 90% confidence interval, (1.5, 1.8). Based upon the confidence interval, what conclusion can be made using = 0.10?

She should reject the null hypothesis. Since 2 falls outside of the 90% confidence interval, there is convincing evidence that the true mean number of hours of study time per night for students differs from 2 hours per night.

A study was conducted to determine the true mean weight of all packages shipped by a company. A 90% confidence interval for the true mean weight is 4.9 pounds to 14.8 pounds. Based upon this interval, what decision should be made about the hypotheses: =5 versus 5 where μ = the true mean weight of all packages at α = 0.10?

The null hypothesis should not be rejected, because 5 falls inside the 90% confidence interval.

An engineer would like to design a parking garage in the most cost-effective manner. The garage must be able to fit pickup trucks, which have an average height of 76.4 inches. To double-check this figure, the engineer employs a statistician. The statistician selects a random sample of 50 trucks, which will be used to determine if there is convincing evidence that the true mean height of all trucks is greater than 76.4 inches. The statistician plans to test the hypotheses, = 76.4 versus > 76.4, where μ = the true mean height of all trucks. The power of this test to reject the null hypothesis when μ = 77 inches is 0.97. What is the interpretation of 0.97?

The probability of rejecting H0: = 76.4 when = 77 inches is 0.97.

A social scientist collects information about study time for a random sample of 40 students with the intention of testing the hypotheses = 2 hours per night versus 2 hours per night where μ = the true mean number of hours of study time per night for students. The power of this test to reject the null hypothesis when μ = 2.25 is 0.35. What is the correct interpretation of the value 0.35?

The probability that this test will reject that the true mean number of hours of study time per night for students is 2 when the true mean is really 2.25 is 0.35.

A headache medication is supposed to contain 500 mg of active ingredient. A researcher would like to test the hypotheses =500 versus 500 where μ = the true amount of active ingredient for all pills. A 99% confidence interval based on a random sample of 60 pills is (505, 511). Based on the interval, what decision should be made at α = 0.01?

The researcher should reject the null hypothesis because 500 falls outside the 99% confidence interval.

A school newspaper article claims that the mean weight of student backpacks is 20 pounds. A student of this school would like to test the hypotheses = 20 versus where = the true mean weight of all backpacks. A 95% confidence interval based upon a random sample of 50 students is (18.5, 19.75). Using the interval, can the researcher reject the null hypothesis?

Yes, the null hypothesis can be rejected at the significance level = 0.05 because 20 is not contained in the 95% confidence interval.

A certain size of tires is supposed to be inflated to 35 psi. The owner of a vehicle repair shop would like to test the hypotheses =35 versus 35 where μ = the true mean tire pressure for all customers with this tire size. A 90% confidence interval based upon a random sample of 40 customers is (32.1, 34.3). Using the interval, can the owner reject the null hypothesis?

Yes, the null hypothesis can be rejected at the significance level α = 0.10 because 35 is not contained in the 90% confidence interval.

An engineer would like to design a parking garage in the most cost-effective manner. The garage must be able to fit pickup trucks, which have an average height of 76.4 inches. To double-check this figure, the engineer employs a statistician. The statistician selects a random sample of 100 trucks, which will be used to determine if these data provide convincing evidence that the true mean height of all trucks is greater than 76.4 inches. The statistician plans to test the hypotheses, = 76.4 versus > 76.4, where μ = the true mean height of all trucks using α = 0.05. The statistician would like to increase the power of this test to reject the null hypothesis when μ = 77 inches. Which combination of sample size and significance level would yield the greatest power of this test?

D

The owner of a fitness watch would like to determine if the mean number of steps he takes per day differs from the recommended 10,000 steps per day, using α = 0.05. He selects a random sample of 50 days with the intention of testing the hypotheses = 10,000 steps versus steps where μ = the true mean number of steps taken per day. Rather than test these hypotheses, he computes a 95% confidence interval for the true mean number of steps he takes per day. The 95% confidence interval is (8,250, 10,700). Based on the confidence interval, what conclusion can be made?

Fail to reject H0. Since 10,000 does not fall outside the 95% confidence interval, there is not convincing evidence that the true mean number of steps he takes per day differs from 10,000 steps.