9. Hypothesis Testing for One Sample

Hypothesis Test with Known σ

a. How do the results from Example 1 in this section change if σ is known to be 1.99240984 g? Does the knowledge of σ have much of an effect on the results of this hypothesis test?

Randomization: Testing a Claim About a Mean

In Exercises 9–12, use the randomization procedure for the indicated exercise.

Section 8-3, Exercise 21 “Lead in Medicine”

Randomization: Testing a Claim About a Mean

In Exercises 9–12, use the randomization procedure for the indicated exercise.

Section 8-3, Exercise 23 “Cell Phone Radiation”

Hypothesis Test for Lightning Deaths Refer to the sample data given in Cumulative Review Exercise 1 and consider those data to be a random sample of annual lightning deaths from recent years. Use those data with a 0.01 significance level to test the claim that the mean number of annual lightning deaths is less than the mean of 72.6 deaths from the 1980s. If the mean is now lower than in the past, identify one of the several factors that could explain the decline.

In Exercises 5–16, use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal.

Do Men Talk Less than Women? Listed below are word counts of males and females in couple relationships (from Data Set 14 “Word Counts” in Appendix B).

a. Use a 0.05 significance level to test the claim that men talk less than women.

Kiosks Yolanda opened a new fast food restaurant. From her first customer, Yolanda kept track of the time a patron needed to wait from the time placing the order to the time the customer received his/her order. Because she was unhappy with the wait time, she invested in Kiosks to take orders with the goal of decreasing wait times. In a random sample of 20 customers, it was found the wait time was 52.3 seconds.

e. Obtain 2000 simple random samples of size n=20 from the 10_3A_4 population data. Compute the mean wait time for each sample. That is, build a null model. What does each mean represent? Use the 2000 simple random samples to obtain a P-value for the hypothesis and judge whether the evidence suggests wait times have decreased. Provide an interpretation of the P-value and state a conclusion.

A simple random sample of size n = 15 is drawn from a population that is normally distributed. The sample variance is found to be 23.8. Determine whether the population variance is less than 25 at the α = 0.01 level of significance.

Reading Rates The reading speed of second-grade students is approximately normal, with a mean of 90 words per minute (wpm) and a standard deviation of 10 wpm.

e. A teacher instituted a new reading program at school. After 10 weeks in the program, it was found that the mean reading speed of a random sample of 20 second-grade students was 92.8 wpm. What might you conclude based on this result?

Effects of Alcohol on the BrainIn a study published in the American Journal of Psychiatry (157:737–744, May 2000), researchers wanted to measure the effect of alcohol on the hippocampal region, the portion of the brain responsible for long-term memory storage, in adolescents. The researchers randomly selected 12 adolescents with alcohol use disorders to determine whether the hippocampal volumes in the alcoholic adolescents were less than the normal volume of 9.02 cubic centimeters (cm³). An analysis of the sample data revealed that the hippocampal volume is approximately normal with x̄ = 8.10 cm³ and s = 0.7 cm³. Conduct the appropriate test at the α = 0.01 level of significance.

To test H0: mu = 100 versus Ha: mu > 100, a simple random sample of size n = 35 is obtained from an unknown distribution. The sample mean is 104.3 and the sample standard deviation is 12.4.

b. Use the classical or p-value approach to decide whether to reject the statement in the null hypothesis at the alpha = 0.05 level of significance.

The trade magazine QSR routinely examines fast-food drive-thru service times. Their recent research indicates that the mean time a car spends in a McDonald’s drive-thru is 167.1 seconds. A McDonald's manager in Salt Lake City feels that she has instituted a drive-thru policy that results in lower drive-thru service times. A random sample of 70 cars results in a mean service time of 163.9 seconds, with a standard deviation of 15.3 seconds. Determine whether the policy is effective in reducing drive-thru service times.

c. Conduct the appropriate test to determine if the policy is effective.

TVaholicsAccording to the American Time Use Survey, the typical American spends 154.8 minutes (2.58 hours) per day watching television. A survey of 50 Internet users results in a mean time watching television per day of 128.7 minutes, with a standard deviation of 46.5 minutes. Conduct the appropriate test to determine if Internet users spend less time watching television at the α = 0.05 level of significance.

Source: Norman H. Nie and D. Sunshine Hillygus. “Where Does Internet Time Come From? A Reconnaissance.” IT & Society, 1(2).

A simple random sample of size n = 19 is drawn from a population that is normally distributed. The sample mean is found to be 0.8, and the sample standard deviation is found to be 0.4. Test whether the population mean is less than 1.0 at the α = 0.01 level of significance.

Effects of Plastic ResinPara-nonylphenol is found in polyvinyl chloride (PVC) used in the food processing and packaging industries. Researchers wanted to determine the effect this substance had on the organ weight of first-generation mice when both parents were exposed to 50 micrograms per liter (μg/L) of para-nonylphenol in drinking water for 4 weeks. After 4 weeks, the mice were bred. After 100 days, the offspring of the exposed parents were sacrificed and the kidney weights were determined. The mean kidney weight of the 12 offspring was found to be 396.9 milligrams (mg), with a standard deviation of 45.4 mg. Is there significant evidence to conclude that the kidney weight of the offspring whose parents were exposed to 50 μg/L of para-nonylphenol in drinking water for 4 weeks is greater than 355.7 mg, the mean weight of kidneys in normal 100-day-old mice at the α = 0.05 level of significance?

Source: Vendula Kyselova et al., “Effects of p-nonylphenol and resveratrol on body and organ weight and in vitro fertility of outbred CD-1 mice,” Reproductive Biology and Endocrinology, 2003.

Credit ScoresA Fair Isaac Corporation (FICO) score is used by credit agencies (such as mortgage companies and banks) to assess the creditworthiness of individuals. Values range from 300 to 850, with a FICO score over 700 considered to be a quality credit risk. According to Fair Isaac Corporation, the mean FICO score is 703.5. A credit analyst wondered whether high-income individuals (incomes in excess of \$100,000 per year) had higher credit scores. He obtained a random sample of 40 high-income individuals and found the sample mean credit score to be 714.2 with a standard deviation of 83.2. Conduct the appropriate test to determine if high-income individuals have higher FICO scores at the α = 0.05 level of significance.