7.6 Exercises

1. The standard deviation of the weights of elephants is known to be approximately 15 pounds. We wish to construct a 95% confidence interval for the mean weight of newborn elephant calves. Fifty newborn elephants are weighed. The sample mean is 244 pounds. The sample standard deviation is 11 pounds.

  1. Construct a 95% confidence interval for the population mean weight of newborn elephants.
  2. Interpret the confidence interval found in part (a).
  3. What will happen to the confidence interval obtained, if 500 newborn elephants are weighed instead of 50? Why?

2. The U.S. Census Bureau conducts a study to determine the time needed to complete the short form. The Bureau surveys 200 people. The sample mean is 8.2 minutes. There is a known standard deviation of 2.2 minutes. The population distribution is assumed to be normal.

  1. Construct a 90% confidence interval for the population mean time to complete the forms.
  2. Interpret the confidence interval found in part (a).
  3. Is it reasonable to conclude the mean time to complete the forms is 10 minutes?  Explain.
  4. If the Census wants to increase its level of confidence and keep the error bound the same by taking another survey, what changes should it make?
  5. If the Census did another survey, kept the error bound the same, and surveyed only 50 people instead of 200, what would happen to the level of confidence? Why?
  6. Suppose the Census needed to be 98% confident of the population mean length of time. Would the Census have to survey more people? Why or why not?

3. A sample of 20 heads of lettuce was selected. Assume that the population distribution of head weight is normal. The weight of each head of lettuce was then recorded. The mean weight was 2.2 pounds with a standard deviation of 0.1 pounds. The population standard deviation is known to be 0.2 pounds.

  1.  Construct a 90% confidence interval for the population mean weight of the heads of lettuce.
  2. Interpret the confidence interval found in part (a).
  3. Construct a 95% confidence interval for the population mean weight of the heads of lettuce.
  4. In complete sentences, explain why the confidence interval in part (a) is larger than in part (c).
  5. What would happen if 40 heads of lettuce were sampled instead of 20, and the error bound remained the same?
  6. What would happen if 40 heads of lettuce were sampled instead of 20, and the confidence level remained the same?

4. The mean age for all Foothill College students for a recent Fall term was 33.2. The population standard deviation has been pretty consistent at 15. Suppose that twenty-five Winter students were randomly selected. The mean age for the sample was 30.4. We are interested in the true mean age for Winter Foothill College students.

  1. Construct a 99% confidence interval for the mean age of students at Foothill College.
  2. Interpret the confidence interval found in part (a).
  3. Is it reasonable for the college to claim that the mean age of its students is 35?  Explain.
  4. Using the same mean, standard deviation, and level of confidence, suppose that [latex]n[/latex] were 69 instead of 25. Would the error bound become larger or smaller? How do you know?
  5. Using the same mean, standard deviation, and sample size, how would the error bound change if the confidence level were reduced to 90%? Why?

5. Among various ethnic groups, the standard deviation of heights is known to be approximately three inches. We wish to construct a 95% confidence interval for the mean height of male Swedes. Forty-eight male Swedes are surveyed. The sample mean is 71 inches. The sample standard deviation is 2.8 inches.

  1. Construct a 95% confidence interval for the population mean height of male Swedes.
  2. Interpret the confidence interval found in part (a).
  3. Is it reasonable to claim that the mean height of male Swedes is 75 inches?  Explain.
6. Announcements for 84 upcoming engineering conferences were randomly picked from a stack of IEEE Spectrum magazines. The mean length of the conferences was 3.94 days, with a standard deviation of 1.28 days. Assume the underlying population is normal.
  1. Construct a 97% confidence interval for the population mean length of engineering conferences.
  2. Interpret the confidence interval found in part (a).
  3. Is it reasonable to claim that the mean length of the conferences is 3 days?  Explain.

7. Suppose that an accounting firm does a study to determine the time needed to complete one person’s tax forms. It randomly surveys 100 people. The sample mean is 23.6 hours. There is a known standard deviation of 7.0 hours. The population distribution is assumed to be normal.

  1. Construct a 90% confidence interval for the population mean time to complete the tax forms.
  2. If the firm wished to increase its level of confidence and keep the error bound the same by taking another survey, what changes should it make?
  3. If the firm did another survey, kept the error bound the same, and only surveyed 49 people, what would happen to the level of confidence? Why?
  4. Suppose that the firm decided that it needed to be at least 96% confident of the population mean length of time to within one hour. How would the number of people the firm surveys change? Why?

8. A sample of 16 small bags of the same brand of candies was selected. Assume that the population distribution of bag weights is normal. The weight of each bag was then recorded. The mean weight was two ounces with a standard deviation of 0.12 ounces. The population standard deviation is known to be 0.1 ounce.

  1. Construct a 90% confidence interval for the population mean weight of the candies.
  2. Construct a 98% confidence interval for the population mean weight of the candies.
  3. In complete sentences, explain why the confidence interval in part (b) is larger than the confidence interval in part (a).
  4. In complete sentences, give an interpretation of what the interval in part (b) means.

9. What is meant by the term “90% confident” when constructing a confidence interval for a mean?

  1. If we took repeated samples, approximately 90% of the samples would produce the same confidence interval.
  2. If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the sample mean.
  3. If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the true value of the population mean.
  4. If we took repeated samples, the sample mean would equal the population mean in approximately 90% of the samples.

10. The average height of young adult males has a normal distribution with standard deviation of 2.5 inches. You want to estimate the mean height of students at your college or university to within one inch with 93% confidence. How many male students must you measure?

11.A hospital is trying to cut down on emergency room wait times. It is interested in the amount of time patients must wait before being called back to be examined. An investigation committee randomly surveyed 70 patients. The sample mean was 1.5 hours with a sample standard deviation of 0.5 hours.

  1. Construct a 99% confidence interval for the population mean time spent waiting.
  2. Interpret the confidence interval found in part (a).
  3. Is it reasonable to claim that the mean time spent waiting is 2 hours?  Explain.

12. 108 Americans were surveyed to determine the number of hours they spend watching television each month. It was revealed that they watched an average of 151 hours each month with a standard deviation of 32 hours. Assume that the underlying population distribution is normal.

  1. Construct a 99% confidence interval for the population mean hours spent watching television per month.
  2. Interpret the confidence interval found in part (a).
  3. Why would the error bound change if the confidence level were lowered to 95%?

13. In six packages of “The Flintstones® Real Fruit Snacks” there were five Bam-Bam snack pieces. The total number of snack pieces in the six bags was 68.

  1. Construct a 96% confidence interval for the proportion of Bam-Bam snack pieces per bag.
  2. Interpret the confidence interval found in part (a).

14. A random survey of enrollment at 35 community colleges across the United States yielded the following figures: 6,414; 1,550; 2,109; 9,350; 21,828; 4,300; 5,944; 5,722; 2,825; 2,044; 5,481; 5,200; 5,853; 2,750; 10,012; 6,357; 27,000; 9,414; 7,681; 3,200; 17,500; 9,200; 7,380; 18,314; 6,557; 13,713; 17,768; 7,493; 2,771; 2,861; 1,263; 7,285; 28,165; 5,080; 11,622. Assume the underlying population is normal.

  1. Construct a 95% confidence interval for the mean enrollment at community colleges in the United States.
  2. Interpret the confidence interval found in part (a).
  3. Is it reasonable to conclude that the mean enrollment at community colleges in the U.S. is 15,000?  Explain.
  4. What will happen to the error bound and confidence interval if 500 community colleges were surveyed? Why?

15. Suppose that a committee is studying whether or not there is waste of time in our judicial system. It is interested in the mean amount of time individuals waste at the courthouse waiting to be called for jury duty. The committee randomly surveyed 81 people who recently served as jurors. The sample mean wait time was eight hours with a sample standard deviation of four hours.

  1. Construct a 98% confidence interval for the population mean time wasted.
  2. Explain in a complete sentence what the confidence interval means.

16. A pharmaceutical company makes tranquilizers. It is assumed that the distribution for the length of time they last is approximately normal. Researchers in a hospital used the drug on a random sample of nine patients. The effective period of the tranquilizer for each patient (in hours) was as follows: 2.7; 2.8; 3.0; 2.3; 2.3; 2.2; 2.8; 2.1; and 2.4.

  1. Construct a 95% confidence interval for the mean length of time the tranquilizers last.
  2. What does it mean to be “95% confident” in this problem?

17. Suppose that 14 children, who were learning to ride two-wheel bikes, were surveyed to determine how long they had to use training wheels. It was revealed that they used them an average of six months with a sample standard deviation of three months. Assume that the underlying population distribution is normal.

  1. Construct a 99% confidence interval for the mean length of time children use training wheels.
  2. Interpret the confidence interval found in part (a).
  3. Why would the error bound change if the confidence level were lowered to 90%?

18. The Federal Election Commission (FEC) collects information about campaign contributions and disbursements for candidates and political committees each election cycle. A political action committee (PAC) is a committee formed to raise money for candidates and campaigns. A Leadership PAC is a PAC formed by a federal politician (senator or representative) to raise money to help other candidates’ campaigns.  The FEC has reported financial information for 556 Leadership PACs that operating during the 2011–2012 election cycle. The following table shows the total receipts during this cycle for a random selection of 20 Leadership PACs.

$46,500.00 $0 $40,966.50 $105,887.20 $5,175.00
$29,050.00 $19,500.00 $181,557.20 $31,500.00 $149,970.80
$2,555,363.20 $12,025.00 $409,000.00 $60,521.70 $18,000.00
$61,810.20 $76,530.80 $119,459.20 $0 $63,520.00
$6,500.00 $502,578.00 $705,061.10 $708,258.90 $135,810.00
$2,000.00 $2,000.00 $0 $1,287,933.80 $219,148.30
  1. Construct a 96% confidence interval for the mean amount of money raised by all Leadership PACs during the 2011–2012 election cycle.
  2. Interpret the confidence interval found in part (a).

19. Unoccupied seats on flights cause airlines to lose revenue. Suppose a large airline wants to estimate its mean number of unoccupied seats per flight over the past year. To accomplish this, the records of 225 flights are randomly selected and the number of unoccupied seats is noted for each of the sampled flights. The sample mean is 11.6 seats and the sample standard deviation is 4.1 seats.

  1. Construct a 92% confidence interval for the mean number of unoccupied seats per flight.
  2. Interpret the confidence interval found in part (a).
  3. Is it reasonable for the airlines to claim that the mean number of unoccupied seats per flight is 20?  Exlain.

20. In a recent sample of 84 used car sales costs, the sample mean was $6,425 with a standard deviation of $3,156. Assume the underlying distribution is approximately normal.

  1. Construct a 95% confidence interval for the mean cost of a used car.
  2. Explain what a “95% confidence interval” means for this study.

21. A survey of the mean number of cents off that coupons give was conducted by randomly surveying one coupon per page from the coupon sections of a recent San Jose Mercury News. The following data were collected: 20¢; 75¢; 50¢; 65¢; 30¢; 55¢; 40¢; 40¢; 30¢; 55¢; $1.50; 40¢; 65¢; 40¢. Assume the underlying distribution is approximately normal.

  1. Construct a 95% confidence interval for the mean worth of coupons.
  2. Interpret the confidence interval found in part (a).
  3. If many random samples were taken of size 14, what percent of the confidence intervals constructed should contain the population mean worth of coupons? Explain why.

    22. Marketing companies are interested in knowing the percent of women who make the majority of household purchasing decisions.

    1. When designing a study to determine this population proportion, what is the minimum number you would need to survey to be 90% confident that the population proportion is estimated to within 0.05?
    2. If it were later determined that it was important to be more than 90% confident and a new survey were commissioned, how would it affect the minimum number you need to survey? Why?

    23. Suppose the marketing company did do a survey. They randomly surveyed 200 households and found that in 120 of them, the woman made the majority of the purchasing decisions. We are interested in the population proportion of households where women make the majority of the purchasing decisions.

    1. Construct a 95% confidence interval for the proportion of households where the women make the majority of the purchasing decisions. 
    2. Interpret the confidence interval found in part (a).
    3. Is it reasonable for the marketing company to claim that women make the majority of purchasing decisions in 70% of households?  Explain.

    24. A poll of 1,200 voters asked what the most significant issue was in the upcoming election. Sixty-five percent answered the economy. We are interested in the population proportion of voters who feel the economy is the most important.

    1. Construct a 90% confidence interval for the proportion of voters who believe the economy is the most significant issue in the upcoming election.
    2. Interpret the confidence interval found in part (a).
    3. Is it reasonable to claim that 60% of voters believe the economy is the most significant issue in the upcoming election?  Explain.
    4. What would happen to the confidence interval if the level of confidence were 95%?

    25. The Ice Chalet offers dozens of different beginning ice-skating classes. All of the class names are put into a bucket. The 5 P.M., Monday night, ages 8 to 12, beginning ice-skating class was picked. In that class were 64 girls and 16 boys. Suppose that we are interested in the true proportion of girls, ages 8 to 12, in all beginning ice-skating classes at the Ice Chalet. Assume that the children in the selected class are a random sample of the population.

    1. Construct a 92% confidence interval for the proportion of girls in the ages 8 to 12 beginning ice-skating classes at the Ice Chalet.
    2. Interpret the confidence interval found in part (a).

    26. Insurance companies are interested in knowing the percent of drivers who always buckle up before riding in a car.

    1. When designing a study to determine this population proportion, what is the minimum number you would need to survey to be 95% confident that the population proportion is estimated to within 0.03?
    2. If it were later determined that it was important to be more than 95% confident and a new survey was commissioned, how would that affect the minimum number you would need to survey? Why?
    3. Suppose that the insurance companies did do a survey. They randomly surveyed 400 drivers and found that 320 claimed they always buckle up. We are interested in the population proportion of drivers who claim they always buckle up.  Construct a 95% confidence interval for the proportion who claim they always buckle up.

    27. Stanford University conducted a study of whether running is healthy for men and women over age 50. During the first eight years of the study, 1.5% of the 451 members of the 50-Plus Fitness Association died. We are interested in the proportion of people over 50 who ran and died in the same eight-year period.

    1. Construct a 97% confidence interval for the proportion of people over 50 who ran and died in the same eight–year period.
    2. Explain what a “97% confidence interval” means for this study.

    28. A telephone poll of 1,000 adult Americans was reported in an issue of Time Magazine. One of the questions asked was “What is the main problem facing the country?” Twenty percent answered “crime.” We are interested in the population proportion of adult Americans who feel that crime is the main problem.

    1. Construct a 93% confidence interval for the proportion of adult Americans who feel that crime is the main problem.
    2. Interpret the confidence interval found in part (a).
    3. Is it reasonable to claim that 30% of Americans feel crime is the main problem?  Explain.

    29. According to a Field Poll, 79% of California adults (actual results are 400 out of 506 surveyed) feel that “education and our schools” is one of the top issues facing California. We wish to construct a 90% confidence interval for the true proportion of California adults who feel that education and the schools is one of the top issues facing California.

    1. Construct a 90% confidence interval for the proportion of California adults who feel education and schools is one of the top issues facing California.
    2. Interpret the confidence interval found in part (a).
    3. Is it reasonable to claim that 90% of California adults feel education and schools is one of the top issues facing California?  Explain.

    30. Public Policy Polling recently conducted a survey asking adults across the U.S. about music preferences. When asked, 80 of the 571 participants admitted that they have illegally downloaded music.

    1. Construct a 99% confidence interval for the proportion of American adults who have illegally downloaded music.
    2. Interpret the confidence interval found in part (a).
    3. Without performing any calculations, describe how the confidence interval would change if the confidence level changed from 99% to 90%.

    31. You plan to conduct a survey on your college campus to learn about the political awareness of students. You want to estimate the true proportion of college students on your campus who voted in the 2012 presidential election with 95% confidence and a margin of error no greater than five percent. How many students must you interview?

    Attribution

    “Chapter 3 Homework” and “Chapter 8 Practice” in Introductory Statistics by OpenStax is licensed under a Creative Commons Attribution 4.0 International License.

    License

    Icon for the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License

    Introduction to Statistics Copyright © 2022 by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.