9. Hypothesis Testing for One Sample

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.

[DATA] Family Size A random sample of 60 married couples who have been married 7 years was asked the number of children they have. The results of the survey are as follows:

Note: x̄ = 2.27, s = 1.22.

a. What is the shape of the distribution of the sample mean? Why?

To test H0: μ = 100 versus H1: μ ≠ 100, a simple random sample of size n = 23 is obtained from a population that is known to be normally distributed.

d. Will the researcher reject the null hypothesis? Why?

To test H0: μ = 100 versus H1: μ ≠ 100, a simple random sample of size n = 23 is obtained from a population that is known to be normally distributed.

e. Construct a 99% confidence interval to test the hypothesis.

SAT Verbal ScoresDo students who learned English and another language simultaneously score worse on the SAT Critical Reading exam than the general population of test takers? The mean score among all test takers on the SAT Critical Reading exam is 501. A random sample of 100 test takers who learned English and another language simultaneously had a mean SAT Critical Reading score of 485 with a standard deviation of 116. Do these results suggest that students who learn English as well as another language simultaneously score worse on the SAT Critical Reading exam?

a. State the appropriate null and alternative hypotheses.

SAT Verbal ScoresDo students who learned English and another language simultaneously score worse on the SAT Critical Reading exam than the general population of test takers? The mean score among all test takers on the SAT Critical Reading exam is 501. A random sample of 100 test takers who learned English and another language simultaneously had a mean SAT Critical Reading score of 485 with a standard deviation of 116. Do these results suggest that students who learn English as well as another language simultaneously score worse on the SAT Critical Reading exam?

b. Verify that the requirements to perform the test using the t-distribution are satisfied.

SAT Verbal ScoresDo students who learned English and another language simultaneously score worse on the SAT Critical Reading exam than the general population of test takers? The mean score among all test takers on the SAT Critical Reading exam is 501. A random sample of 100 test takers who learned English and another language simultaneously had a mean SAT Critical Reading score of 485 with a standard deviation of 116. Do these results suggest that students who learn English as well as another language simultaneously score worse on the SAT Critical Reading exam?

c. Use the classical or P-value approach at the α = 0.1 level of significance to test the hypotheses in part (a).

SAT Verbal ScoresDo students who learned English and another language simultaneously score worse on the SAT Critical Reading exam than the general population of test takers? The mean score among all test takers on the SAT Critical Reading exam is 501. A random sample of 100 test takers who learned English and another language simultaneously had a mean SAT Critical Reading score of 485 with a standard deviation of 116. Do these results suggest that students who learn English as well as another language simultaneously score worse on the SAT Critical Reading exam?

d. Write a conclusion based on your results to part (c).

"[DATA] Starbucks StockThe volume of a stock is the number of shares traded for a given day. In 2011, Starbucks stock had a mean daily volume of 7.52 million shares according to Yahoo!Finance. A random sample of 40 trading days in 2018 was obtained and the volume of shares traded on those days was recorded. Go to www.pearsonhighered.com/sullivanstats to obtain the data file 10_3_27 using the file format of your choice for the version of the text you are using.

d. Does the evidence suggest that the volume of Starbucks stock has changed since 2011? Use an α = 0.05 level of significance."