9.6 Exercises

1. The known standard deviation in salary for all mid-level professionals in the financial industry is $11,000. Company A and Company B are in the financial industry. Suppose samples are taken of mid-level professionals from Company A and from Company B. The sample mean salary for mid-level professionals in Company A is $80,000. The sample mean salary for mid-level professionals in Company B is $96,000.  
  1. Construct a 99% confidence interval for the difference in the mean salary for mid-level professionals at the two companies.
  2. Interpret the confidence interval in part (a).
  3. Is it reasonable to claim that mean salary for mid-level professionals the same at the two companies?  Explain.

 

2. It is believed that the average grade on an English essay in a particular school system for females is higher than for males. A random sample of 31 females had a mean score of 82 with a standard deviation of three, and a random sample of 25 males had a mean score of 76 with a standard deviation of four.  At the 5% significance level test if the average grade on an English essay is higher for females than males.

 

3. In a random sample of 100 forests in the United States, 56 were coniferous or contained conifers. In a random sample of 80 forests in Mexico, 40 were coniferous or contained conifers. At the 5% significance level, is the proportion of conifers in the United States greater than the proportion of conifers in Mexico?

 

4. A study is done to determine which of two soft drinks has more sugar. There are 13 cans of Beverage A in a sample and six cans of Beverage B. The mean amount of sugar in Beverage A is 36 grams with a standard deviation of 0.6 grams. The mean amount of sugar in Beverage B is 38 grams with a standard deviation of 0.8 grams. Both populations have normal distributions.  At the 5% significance level, determine if the average amount of sugar in Beverage B is greater than Beverage A.

 

5. The mean number of English courses taken in a two–year time period by male and female college students is believed to be about the same. An experiment is conducted and data are collected from 29 males and 16 females. The males took an average of three English courses with a standard deviation of 0.8. The females took an average of four English courses with a standard deviation of 1.0. Are the means statistically the same?  Use a 5% significance level.

 

6. A student at a four-year college claims that mean enrollment at four–year colleges is higher than at two–year colleges in the United States. Two surveys are conducted. Of the 35 two–year colleges surveyed, the mean enrollment was 5,068 with a standard deviation of 4,777. Of the 35 four-year colleges surveyed, the mean enrollment was 5,466 with a standard deviation of 8,191.  At the 5% significance level, is the mean enrollment at four-year colleges higher than at two-year colleges?

 

7. Mean entry-level salaries for college graduates with mechanical engineering degrees and electrical engineering degrees are believed to be approximately the same. A recruiting office thinks that the mean mechanical engineering salary is actually lower than the mean electrical engineering salary. The recruiting office randomly surveys 50 entry level mechanical engineers and 60 entry level electrical engineers. Their mean salaries were $46,100 and $46,700, respectively. Their standard deviations were $3,450 and $4,210, respectively. Conduct a hypothesis test to determine if you agree that the mean entry-level mechanical engineering salary is lower than the mean entry-level electrical engineering salary.  Use a 5% significance level.

 

8. Marketing companies have collected data implying that teenage girls use more ring tones on their cellular phones than teenage boys do. In one particular study of 40 randomly chosen teenage girls and boys (20 of each) with cellular phones, the mean number of ring tones for the girls was 3.2 with a standard deviation of 1.5. The mean for the boys was 1.7 with a standard deviation of 0.8. Conduct a hypothesis test to determine if the means are approximately the same or if the girls’ mean is higher than the boys’ mean.  Use a 5% significance level.

 

 

9. Researchers interviewed street prostitutes in Canada and the United States. The mean age of the 100 Canadian prostitutes upon entering prostitution was 18 with a standard deviation of six. The mean age of the 130 United States prostitutes upon entering prostitution was 20 with a standard deviation of eight. Is the mean age of entering prostitution in Canada lower than the mean age in the United States? Test at a 1% significance level.

 

10. A powder diet is tested on 49 people, and a liquid diet is tested on 36 different people. Of interest is whether the liquid diet yields a higher mean weight loss than the powder diet. The powder diet group had a mean weight loss of 42 pounds with a standard deviation of 12 pounds. The liquid diet group had a mean weight loss of 45 pounds with a standard deviation of 14 pounds.  Test at a 5% significance level.

  1. Construct a 94% confidence interval for the difference in the mean weight loss for the powder and liquid diets.
  2. Interpret the confidence interval in part (a).
  3. Is it reasonable to claim that the mean weight loss with the powder diet is less than the liquid diet?  Explain.

 

11.  The mean speeds of fastball pitches from two different baseball pitchers are to be compared. A sample of 14 fastball pitches is measured from each pitcher. The populations have normal distributions. The table shows the result. Scouters believe that Rodriguez pitches a speedier fastball.  At the 1% significance level, what is your conclusion?

Pitcher Sample Mean Speed of Pitches (mph) Population Standard Deviation
Wesley 86 3
Rodriguez 91 7

12. A researcher is testing the effects of plant food on plant growth. Nine plants have been given the plant food. Another nine plants have not been given the plant food. The heights of the plants are recorded after eight weeks. The populations have normal distributions. The following table is the result. The researcher thinks the food makes the plants grow taller.  At the 1% significance level, what is your conclusion?
Plant Group Sample Mean Height of Plants (inches) Population Standard Deviation
Food 16 2.5
No food 14 1.5

13. Two metal alloys are being considered as material for ball bearings. The mean melting point of the two alloys is to be compared. 15 pieces of each metal are being tested. Both populations have normal distributions. The following table is the result. It is believed that Alloy Zeta has a different melting point.  At the 1% significance level, what is your conclusion?

Sample Mean Melting Temperatures (°F) Population Standard Deviation
Alloy Gamma 800 95
Alloy Zeta 900 105

14. A study is done to determine if students in the California state university system take longer to graduate, on average, than students enrolled in private universities. One hundred students from both the California state university system and private universities are surveyed. The California state university system students took on average 4.5 years with a standard deviation of 0.8. The private university students took on average 4.1 years with a standard deviation of 0.3.  Suppose that from years of research, it is known that the population standard deviations are 1.5811 years and 1 year, respectively. At the 5% significance level, what is your conclusion?

 

15. Parents of teenage boys often complain that auto insurance costs more, on average, for teenage boys than for teenage girls. A group of concerned parents examines a random sample of insurance bills. The mean annual cost for 36 teenage boys was $679. For 23 teenage girls, it was $559. From past years, it is known that the population standard deviation for each group is $180. Determine whether or not you believe that the mean cost for auto insurance for teenage boys is greater than that for teenage girls.  Use a 5% significance level.

 

16. A group of transfer bound students wondered if they will spend the same mean amount on texts and supplies each year at their four-year university as they have at their community college. They conducted a random survey of 54 students at their community college and 66 students at their local four-year university. The sample means were $947 and $1,011, respectively. The population standard deviations are known to be $254 and $87, respectively.
  1. Construct a 96% confidence interval for the difference in the mean amount students spend on texts at university and community college.
  2. Interpret the confidence interval in part (a).
  3. Is it reasonable to claim that the mean amount students spend on texts is the same at university and community college?  Explain.

17. Some manufacturers claim that non-hybrid sedan cars have a lower mean miles-per-gallon (mpg) than hybrid ones. Suppose that consumers test 21 hybrid sedans and get a mean of 31 mpg with a standard deviation of seven mpg. Thirty-one non-hybrid sedans get a mean of 22 mpg with a standard deviation of four mpg.

  1. Construct a 95% confidence interval for the difference in the average miles-per-gallon in hybrid and non-hybrid cars.
  2. Interpret the confidence interval in part (a).
  3. Is it reasonable to claim that the average mpg for hybrid cars is higher than non-hybrid cars?  Explain.

18. One of the questions in a study of marital satisfaction of dual-career couples was to rate the statement “I’m pleased with the way we divide the responsibilities for childcare.” The ratings went from one (strongly agree) to five (strongly disagree). The table below contains ten of the paired responses for husbands and wives. Conduct a hypothesis test to see if the mean difference in the husband’s versus the wife’s satisfaction level is negative (meaning that, within the partnership, the husband is happier than the wife).  Use a 5% significance level.

Wife’s Score 2 2 3 3 4 2 1 1 2 4
Husband’s Score 2 2 1 3 2 1 1 1 2 4

19. Two types of phone operating system are being tested to determine if there is a difference in the proportions of system failures (crashes). Fifteen out of a random sample of 150 phones with OS1 had system failures within the first eight hours of operation. Nine out of another random sample of 150 phones with OS2 had system failures within the first eight hours of operation. OS2 is believed to be more stable (have fewer crashes) than OS1.  At the 5% significance level, is there a difference in the proportions of system failures?

 

20. A recent drug survey showed an increase in the use of drugs and alcohol among local high school seniors as compared to the national percent. Suppose that a survey of 100 local seniors and 100 national seniors is conducted to see if the proportion of drug and alcohol use is higher locally than nationally. Locally, 65 seniors reported using drugs or alcohol within the past month, while 60 national seniors reported using them.  At the 5% significance level, is the proportion of drug and alcohol abuse higher locally than nationally?

 

 

21. Neuroinvasive West Nile virus is a severe disease that affects a person’s nervous system . It is spread by the Culex species of mosquito. In the United States in 2010 there were 629 reported cases of neuroinvasive West Nile virus out of a total of 1,021 reported cases and there were 486 neuroinvasive reported cases out of a total of 712 cases reported in 2011. Is the 2011 proportion of neuroinvasive West Nile virus cases more than the 2010 proportion of neuroinvasive West Nile virus cases? Using a 1% level of significance, conduct an appropriate hypothesis test.

 

22. Adults aged 18 years old and older were randomly selected for a survey on obesity. Adults are considered obese if their body mass index (BMI) is at least 30. The researchers wanted to determine if the proportion of women who are obese in the south is less than the proportion of southern men who are obese. The results are shown in table. Test at the 1% level of significance.

Number who are obese Sample size
Men 42,769 155,525
Women 67,169 248,775

23. Two computer users were discussing tablet computers. A higher proportion of people ages 16 to 29 use tablets than the proportion of people age 30 and older. The table details the number of tablet owners for each age group. Test at the 1% level of significance.

16–29 year olds 30 years old and older
Own a Tablet 69 231
Sample Size 628 2,309

24. A group of friends debated whether more men use smartphones than women. They consulted a research study of smartphone use among adults. The results of the survey indicate that of the 973 men randomly sampled, 379 use smartphones. For women, 404 of the 1,304 who were randomly sampled use smartphones. Test at the 5% level of significance.

  1. Construct a 93% confidence interval for the difference in the proportion of men and women who use smartphones.
  2. Interpret the confidence interval in part (a).
  3. Is it reasonable to claim that the proportion of men who use smartphones is higher than the proportion of women?  Explain.

25. We are interested in whether children’s educational computer software costs less, on average, than children’s entertainment software. Thirty-six educational software titles were randomly picked from a catalog. The mean cost was $31.14 with a standard deviation of $4.69. Thirty-five entertainment software titles were randomly picked from the same catalog. The mean cost was $33.86 with a standard deviation of $10.87.  At the 5% significance level, determine if children’s educational software costs less, on average, than children’s entertainment software.

 

26. Joan Nguyen recently claimed that the proportion of college-age males with at least one pierced ear is as high as the proportion of college-age females. She conducted a survey in her classes. Out of 107 males, 20 had at least one pierced ear. Out of 92 females, 47 had at least one pierced ear.

  1. Construct a 98% confidence interval for the difference in the proportion of college-age males and females with at least one pierced ear.
  2. Interpret the confidence interval in part (a).
  3. Is it reasonable to claim that the proportion of college-age males with at least one pierced ear equals the proportion of college-age females?  Explain.

27. A study was conducted to test the effectiveness of a software patch in reducing system failures over a six-month period. Results for randomly selected installations are shown in the table below. The “before” value is matched to an “after” value, and the differences are calculated. The differences have a normal distribution.

Installation A B C D E F G H
Before 3 6 4 2 5 8 2 6
After 1 5 2 0 1 0 2 2
  1. Construct a 97% confidence interval for the mean difference in the number of failures before and after the software patch was installed.
  2. Interpret the confidence interval in part (a).
  3. Is it reasonable to claim that the average number of failures did not change after the software patch was installed?  Explain.

28. A study was conducted to test the effectiveness of a juggling class. Before the class started, six subjects juggled as many balls as they could at once. After the class, the same six subjects juggled as many balls as they could. The differences in the number of balls are calculated. The differences have a normal distribution.
Subject A B C D E F
Before 3 4 3 2 4 5
After 4 5 6 4 5 7
  1. Construct a 99% confidence interval for the mean difference in the number of balls a subject can juggle after the class.
  2. Interpret the confidence interval in part (a).
  3. Is it reasonable to claim that the average number of balls a subject can juggle higher after the class?  Explain.

29. A doctor wants to know if a blood pressure medication is effective. Six subjects have their blood pressures recorded. After twelve weeks on the medication, the same six subjects have their blood pressure recorded again. For this test, only systolic pressure is of concern. At the 1% significance level, did the medication, on average, lower the patients blood pressure?

Patient A B C D E F
Before 161 162 165 162 166 171
After 158 159 166 160 167 169

 

30. Ten individuals went on a low–fat diet for 12 weeks to lower their cholesterol. The data are recorded in the table below. Do you think that their cholesterol levels were significantly lowered?  Use a 5% significance level.

Starting cholesterol level Ending cholesterol level
140 140
220 230
110 120
240 220
200 190
180 150
190 200
360 300
280 300
260 240

 

 

31. A local cancer support group believes that the estimate for new female breast cancer cases in the south is higher in 2013 than in 2012. The group compared the estimates of new female breast cancer cases by southern state in 2012 and in 2013. The results are in the table.  At the 5% significance level, determine if the average number of breast cancer cases is higher in 2013 than in 2012.

Southern States 2012 2013
Alabama 3,450 3,720
Arkansas 2,150 2,280
Florida 15,540 15,710
Georgia 6,970 7,310
Kentucky 3,160 3,300
Louisiana 3,320 3,630
Mississippi 1,990 2,080
North Carolina 7,090 7,430
Oklahoma 2,630 2,690
South Carolina 3,570 3,580
Tennessee 4,680 5,070
Texas 15,050 14,980
Virginia 6,190 6,280

 

32. A traveler wanted to know if the prices of hotels are different in the ten cities that he visits the most often. The list of the cities with the corresponding hotel prices for his two favorite hotel chains is in the table. Test at the 1% level of significance.

Cities Hyatt Regency prices in dollars Hilton prices in dollars
Atlanta 107 169
Boston 358 289
Chicago 209 299
Dallas 209 198
Denver 167 169
Indianapolis 179 214
Los Angeles 179 169
New York City 625 459
Philadelphia 179 159
Washington, DC 245 239

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