Homework

8.1 A Single Population Mean Using the Normal Distribution

95.

Among various ethnic groups, the standard deviation of heights is known to be approximately three inches. We wish to construct a 95 percent 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 in.

    1. x¯x¯ =________
    2. σ =________
    3. n =________
  1. In words, define the random variables X and X¯X¯.
  2. Which distribution should you use for this problem? Explain your choice.
  3. Construct a 95 percent confidence interval for the population mean height of male Swedes.
    1. State the confidence interval.
    2. Sketch the graph.
    3. Calculate the error bound.
  4. What will happen to the level of confidence obtained if 1,000 male Swedes are surveyed instead of 48? Why?
96.

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. In words, define the random variables X and X¯X¯.
  2. Which distribution should you use for this problem? Explain your choice.
  3. Construct a 95 percent confidence interval for the population mean length of engineering conferences.
    1. State the confidence interval.
    2. Sketch the graph.
    3. Calculate the error bound.
97.

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. x¯x¯ =________
    2. σ =________
    3. n =________
  1. In words, define the random variables X and X¯X¯.
  2. Which distribution should you use for this problem? Explain your choice.
  3. Construct a 90 percent confidence interval for the population mean time to complete the tax forms.
    1. State the confidence interval.
    2. Sketch the graph.
    3. Calculate the error bound.
  4. If the firm wished to increase its level of confidence and keep the error bound the same by taking another survey, which changes should it make?
  5. 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?
  6. Suppose that the firm decided that it needed to be at least 96 percent confident of the population mean length of time to within one hour. How would the number of people the firm surveys change? Why?
98.

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. x¯x¯ =________
    2. σ =________
    3. sx =________
  1. In words, define the random variable X.
  2. In words, define the random variable X¯X¯.
  3. Which distribution should you use for this problem? Explain your choice.
  4. Construct a 90 percent confidence interval for the population mean weight of the candies.
    1. State the confidence interval.
    2. Sketch the graph.
    3. Calculate the error bound.
  5. Construct a 98 percent confidence interval for the population mean weight of the candies.
    1. State the confidence interval.
    2. Sketch the graph.
    3. Calculate the error bound.
  6. In complete sentences, explain why the confidence interval in Part f is larger than the confidence interval in Part e.
  7. In complete sentences, give an interpretation of what the interval in Part f means.
99.

A camp director is interested in the mean number of letters each child sends during his or her camp session. The population standard deviation is known to be 2.5. A survey of 20 campers is taken. The mean from the sample is 7.9, with a sample standard deviation of 2.8.

    1. x¯x¯ =________
    2. σ =________
    3. n =________
  1. Define the random variables X and X¯X¯ in words.
  2. Which distribution should you use for this problem? Explain your choice.
  3. Construct a 90 percent confidence interval for the population mean number of letters campers send home.
    1. State the confidence interval.
    2. Sketch the graph.
    3. Calculate the error bound.
  4. What will happen to the error bound and confidence interval if 500 campers are surveyed? Why?
100.

What is meant by the term 90 percent confident when constructing a confidence interval for a mean?

  1. If we took repeated samples, approximately 90 percent of the samples would produce the same confidence interval.
  2. If we took repeated samples, approximately 90 percent of the confidence intervals calculated from those samples would contain the sample mean.
  3. If we took repeated samples, approximately 90 percent 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 percent of the samples.
101.

The Federal Election Commission collects information about campaign contributions and disbursements for candidates and political committees during each election cycle. During the 2012 campaign season, there were 1,619 candidates for the House of Representatives across the United States who received contributions from individuals. Table 8.11 shows the total receipts from individuals for a random selection of 40 House candidates rounded to the nearest $100. The standard deviation for this data to the nearest hundred is σ = $909,200.

$3,600 $1,243,900 $10,900 $385,200 $581,500
$7,400 $2,900 $400 $3,714,500 $632,500
$391,000 $467,400 $56,800 $5,800 $405,200
$733,200 $8,000 $468,700 $75,200 $41,000
$13,300 $9,500 $953,800 $1,113,500 $1,109,300
$353,900 $986,100 $88,600 $378,200 $13,200
$3,800 $745,100 $5,800 $3,072,100 $1,626,700
$512,900 $2,309,200 $6,600 $202,400 $15,800
Table 8.11
  1. Find the point estimate for the population mean.
  2. Using 95 percent confidence, calculate the error bound.
  3. Create a 95 percent confidence interval for the mean total individual contributions.
  4. Interpret the confidence interval in the context of the problem.
102.

The American Community Survey (ACS), part of the U.S. Census Bureau, conducts a yearly census similar to the one taken every 10 years, but with a smaller percentage of participants. The most recent survey estimates with 90 percent confidence that the mean household income in the United States falls between $69,720 and $69,922. Find the point estimate for mean U.S. household income and the error bound for mean U.S. household income.

103.

The average height of young adult males has a normal distribution with standard deviation of 2.5 in. You want to estimate the mean height of students at your college or university to within 1 in. with 93 percent confidence. How many male students must you measure?

8.2 A Single Population Mean Using the Student's t-Distribution

104.

In six packages of multicolored fruit snacks, there were five red snack pieces. The total number of snack pieces in the six bags was 68. We wish to calculate a 96 percent confidence interval for the population proportion of red snack pieces.

  1. Define the random variables X and P′ in words.
  2. Which distribution should you use for this problem? Explain your choice
  3. Calculate p′.
  4. Construct a 96 percent confidence interval for the population proportion of red snack pieces per bag.
    1. State the confidence interval.
    2. Sketch the graph.
    3. Calculate the error bound.
  5. Do you think that six packages of fruit snacks yield enough data to give accurate results? Why or why not?
105.

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. x¯x¯ = __________
    2. sxsx = __________
    3. n = __________
    4. n – 1 = __________
  1. Define the random variables XX and X¯X¯ in words.
  2. Which distribution should you use for this problem? Explain your choice.
  3. Construct a 95 percent confidence interval for the population mean enrollment at community colleges in the United States.
    1. State the confidence interval.
    2. Sketch the graph.
    3. Calculate the error bound.
  4. What will happen to the error bound and confidence interval if 500 community colleges are surveyed? Why?
106.

Suppose that a committee is studying whether there is wasted 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 8 hr, with a sample standard deviation of 4 hr.

    1. x¯x¯ = __________
    2. sxsx = __________
    3. n = __________
    4. n – 1 = __________
  1. Define the random variables XX and X¯X¯ in words.
  2. Which distribution should you use for this problem? Explain your choice.
  3. Construct a 95 percent confidence interval for the population mean time wasted.
    1. State the confidence interval.
    2. Sketch the graph.
    3. Calculate the error bound.
  4. Explain in a complete sentence what the confidence interval means.
107.

A pharmaceutical company makes a drug used during surgery. It is assumed that the distribution for the length of time the drug lasts is approximately normal. Researchers in a hospital used the drug on a random sample of nine patients. The effective period of the antibiotic drug 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. x¯x¯ = __________
    2. sxsx = __________
    3. n = __________
    4. n – 1 = __________
  1. Define the random variable XX in words.
  2. Define the random variable X¯X¯ in words.
  3. Which distribution should you use for this problem? Explain your choice.
  4. Construct a 95 percent confidence interval for the population mean length of time.
    1. State the confidence interval.
    2. Sketch the graph.
    3. Calculate the error bound.
  5. What does it mean to be 95 percent confident in this problem?
108.

Suppose that 14 children8 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. x¯x¯ = __________
    2. sxsx = __________
    3. n = __________
    4. n – 1 = __________
  1. Define the random variable XX in words.
  2. Define the random variableX¯X¯ in words.
  3. Which distribution should you use for this problem? Explain your choice.
  4. Construct a 99 percent confidence interval for the population mean length of time using training wheels.
    1. State the confidence interval.
    2. Sketch the graph.
    3. Calculate the error bound.
  5. Why would the error bound change if the confidence level were lowered to 90 percent?
109.

The Federal Election Commission (FEC) collects information about campaign contributions and disbursements for candidates and political committees during 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 operated 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
Table 8.12

x¯=$251,854.23x¯=$251,854.23

s= $521,130.41s= $521,130.41

Use the sample data to construct a 96 percent confidence interval for the mean amount of money raised by all Leadership PACs during the 2011–2012 election cycle. Use the Student's t-distribution.

110.

A major business magazine published data on the best small firms in 2012. These were firms that have been publicly traded for at least a year, have a stock price of at least $5 per share, and have reported annual revenue between $5 million and $1 billion. Table 8.13 shows the ages of the corporate CEOs for a random sample of these firms.

48 58 51 61 56
59 74 63 53 50
59 60 60 57 46
55 63 57 47 55
57 43 61 62 49
67 67 55 55 49
Table 8.13

Use the sample data to construct a 90 percent confidence interval for the mean age of CEOs for these top small firms. Use the Student's t-distribution.

111.

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. x¯x¯ = __________
    2. sxsx = __________
    3. n = __________
    4. n – 1 = __________
  1. Define the random variables XX and X¯X¯ in words.
  2. Which distribution should you use for this problem? Explain your choice.
  3. Construct a 92 percent confidence interval for the population mean number of unoccupied seats per flight.
    1. State the confidence interval.
    2. Sketch the graph.
    3. Calculate the error bound.
112.

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. Which distribution should you use for this problem? Explain your choice.
  2. Define the random variable X¯X¯ in words.
  3. Construct a 95 percent confidence interval for the population mean cost of a used car.
    1. State the confidence interval.
    2. Sketch the graph.
    3. Calculate the error bound.
  4. Explain what a 95 percent confidence interval means for this study.
113.

Six different national brands of chocolate chip cookies were randomly selected at the supermarket. The grams of fat per serving are as follows: 8, 8, 10, 7, 9, 9. Assume the underlying distribution is approximately normal.

  1. Construct a 90 percent confidence interval for the population mean grams of fat per serving of chocolate chip cookies sold in supermarkets.
    1. State the confidence interval.
    2. Sketch the graph.
    3. Calculate the error bound.
  2. If you wanted a smaller error bound while keeping the same level of confidence, what should have been changed in the study before it was done?
  3. Go to the store and record the grams of fat per serving of six brands of chocolate chip cookies.
  4. Calculate the mean.
  5. Is the mean within the interval you calculated in Part a? Did you expect it to be? Why or why not?
114.

A survey of the mean number of cents off given by coupons was conducted by randomly surveying one coupon per page from the coupons section of a local newspaper. 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. x¯x¯ = __________
    2. sxsx = __________
    3. n = __________
    4. n – 1 = __________
  1. Define the random variables XX and X¯X¯ in words.
  2. Which distribution should you use for this problem? Explain your choice.
  3. Construct a 95 percent confidence interval for the population mean worth of coupons.
    1. State the confidence interval.
    2. Sketch the graph.
    3. Calculate the error bound.
  4. If many random samples were collected with 14 samples as the size, which percentage of the confidence intervals constructed should contain the population mean worth of coupons? Explain why.

 


 
Use the following information to answer the next two exercises: A quality control specialist for a restaurant chain takes a random sample of size 12 to check the amount of soda served in the 16-oz serving size. The sample mean is 13.30, with a sample standard deviation of 1.55. Assume the underlying population is normally distributed.
115.

Find the 95 percent confidence interval for the true population mean for the amount of soda served.

  1. (12.42, 14.18)
  2. (12.32, 14.29)
  3. (12.50, 14.10)
  4. Impossible to determine
116.

Which of the following is the error bound?

  1. 0.87
  2. 1.98
  3. 0.99
  4. 1.74

8.3 A Population Proportion

117.

Insurance companies are interested in knowing the population percentage 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 percent 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 percent confident and a new survey was commissioned, how would that affect the minimum number you would need to survey? Why?
118.

Suppose that the insurance companies did conduct 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.

    1. x = __________
    2. n = __________
    3. p′ = __________
  1. Define the random variables X and P′ in words.
  2. Which distribution should you use for this problem? Explain your choice.
  3. Construct a 95 percent confidence interval for the population proportion who claim they always buckle up.
    1. State the confidence interval.
    2. Sketch the graph.
    3. Calculate the error bound.
  4. If this survey were done by telephone, list three difficulties the companies might have in obtaining random results.
119.

According to a recent survey of 1,200 people, 61 percent believe that the president is doing an acceptable job. We are interested in the population proportion of people who believe the president is doing an acceptable job.

  1. Define the random variables X and P′ in words.
  2. Which distribution should you use for this problem? Explain your choice.
  3. Construct a 90 percent confidence interval for the population proportion of people who believe the president is doing an acceptable job.
    1. State the confidence interval.
    2. Sketch the graph.
    3. Calculate the error bound.
120.

An article regarding dating and marriage recently appeared in a major newspaper. Of the 1,709 randomly selected adults, 315 identified themselves as ethnicity A, 323 identified themselves as ethnicity B, 254 identified themselves as ethnicity C, and 779 identified themselves as ethnicity D. In this survey, 86 percent of ethnicity B said that they would welcome a person of ethnicity A into their families. Among ethnicity C, 77 percent would welcome a person of ethnicity D into their families, 71 percent would welcome a person of ethnicity A, and 66 percent would welcome a person of ethnicity B.

  1. We are interested in finding the 95 percent confidence interval for the percent of all ethnicity B adults who would welcome a person of ethnicity D into their families. Define the random variables X and P′ in words.
  2. Which distribution should you use for this problem? Explain your choice.
  3. Construct a 95 percent confidence interval.
    1. State the confidence interval.
    2. Sketch the graph.
    3. Calculate the error bound.
121.

Refer to the information in Exercise 8.120.

  1. Construct three 95 percent confidence intervals:
    1. percentage of all ethnicity C adults who would welcome a person of ethnicity D into their families
    2. percentage of all ethnicity C adults who would welcome a person of ethnicity A into their families
    3. percentage of all ethnicity C adults who would welcome a person of ethnicity B into their families
  2. Even though the three point estimates are different, do any of the confidence intervals overlap? Which?
  3. For any intervals that do overlap, in words, what does this imply about the significance of the differences in the true proportions?
  4. For any intervals that do not overlap, in words, what does this imply about the significance of the differences in the true proportions?
122.

Stanford University conducted a study of whether running is healthy for men and women over age 50. During the first 8 years of the study, 1.5 percent 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. Define the random variables X and P′ in words.
  2. Which distribution should you use for this problem? Explain your choice.
  3. Construct a 97 percent confidence interval for the population proportion of people over 50 who ran and died in the same eight-year period.
    1. State the confidence interval.
    2. Sketch the graph.
    3. Calculate the error bound.
  4. Explain what a 97 percent confidence interval means for this study.
123.

A telephone poll of 1,000 adult Americans was reported in an issue of a national magazine. One of the questions asked, “What is the main problem facing the country?” Twenty percent responded, "crime." We are interested in the population proportion of adult Americans who believe that crime is the main problem.

  1. Define the random variables X and P′ in words.
  2. Which distribution should you use for this problem? Explain your choice.
  3. Construct a 95 percent confidence interval for the population proportion of adult Americans who believe that crime is the main problem.
    1. State the confidence interval.
    2. Sketch the graph.
    3. Calculate the error bound.
  4. Suppose we want to lower the sampling error. What is one way to accomplish that?
  5. The sampling error given by the group of researchers who conducted the poll is ±3 percent. In one to three complete sentences, explain what the ±3 percent represents.
124.

Refer to Exercise 8.123. Another question in the poll asked, “[How much are] you worried about the quality of education in our schools?” Sixty-three percent responded, “A lot.” We are interested in the population proportion of adult Americans who are worried a lot about the quality of education in our schools.

  1. Define the random variables X and P′ in words.
  2. Which distribution should you use for this problem? Explain your choice.
  3. Construct a 95 percent confidence interval for the population proportion of adult Americans who are worried a lot about the quality of education in our schools.
    1. State the confidence interval.
    2. Sketch the graph.
    3. Calculate the error bound.
  4. The sampling error given by the group of researchers who conducted the poll is ±3 percent. In one to three complete sentences, explain what the ±3 percent represents.

 


 
Use the following information to answer the next three exercises: According to a Field Poll, 79 percent of California adults (actual results are 400 out of 506 surveyed) believe that education and our schools is one of the top issues facing California. We wish to construct a 90 percent confidence interval for the true proportion of California adults who believe that education and the schools is one of the top issues facing California.
125.

A point estimate for the true population proportion is _______.

  1. 0.90
  2. 1.27
  3. 0.79
  4. 400
126.

A 90 percent confidence interval for the population proportion is _______.

  1. (0.761, 0.820)
  2. (0.125, 0.188)
  3. (0.755, 0.826)
  4. (0.130, 0.183)
127.

The error bound is approximately _____.

  1. 1.581
  2. 0.791
  3. 0.059
  4. 0.030

 


 
Use the following information to answer the next two exercises: Five hundred eleven (511) homes in a certain southern California community are randomly surveyed to determine whether they meet minimal earthquake preparedness recommendations. One hundred seventy-three (173) of the homes surveyed meet the minimum recommendations for earthquake preparedness, and 338 do not.
128.

Find the confidence interval at the 90 percent confidence level for the true population proportion of southern California community homes meeting at least the minimum recommendations for earthquake preparedness.

  1. (0.2975, 0.3796)
  2. (0.6270, 0.6959)
  3. (0.3041, 0.3730)
  4. (0.6204, 0.7025)
129.

The point estimate for the population proportion of homes that do not meet the minimum recommendations for earthquake preparedness is ______.

  1. 0.6614
  2. 0.3386
  3. 173
  4. 338
130.

On May 23, 2013, a polling group reported that of the 1,005 people surveyed, 76 percent of U.S. workers believe that they will continue working past retirement age. The confidence level for this study was reported at 95 percent with a ±3 percent margin of error.

  1. Determine the estimated proportion from the sample.
  2. Determine the sample size.
  3. Identify CL and α.
  4. Calculate the error bound based on the information provided.
  5. Compare the error bound in Part d to the margin of error reported by the polling group. Explain any differences between the values.
  6. Create a confidence interval for the results of this study.
  7. A reporter is covering the release of this study for a local news station. How should she explain the confidence interval to her audience?
131.

A national survey of 1,000 adults was conducted on May 13, 2013, by a group of researchers. It concluded with 95 percent confidence that 49 percent to 55 percent of Americans believe that big-time college sports programs corrupt the process of higher education.

  1. Find the point estimate and the error bound for this confidence interval.
  2. Can we (with 95 percent confidence) conclude that more than half of all American adults believe this?
  3. Use the point estimate from Part a and n = 1,000 to calculate a 75 percent confidence interval for the proportion of American adults who believe that major college sports programs corrupt higher education.
  4. Can we (with 75 percent confidence) conclude that at least half of all American adults believe this?
132.

A polling group recently conducted a survey asking adults across the United States about music preferences. When asked, 80 of the 571 participants download music weekly.

  1. Create a 99 percent confidence interval for the true proportion of American adults who download music weekly.
  2. This survey was conducted through automated telephone interviews on May 6 and 7, 2013. The error bound of the survey compensates for sampling error, or natural variability among samples. List some factors that could affect the survey’s outcome that are not covered by the margin of error.
  3. Without performing any calculations, describe how the confidence interval would change if the confidence level decreased from 99 percent to 90 percent.
133.

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 percent confidence and a margin of error no greater than 5 percent. How many students must you interview?

134.

In a recent poll, 9 of 48 respondents rated the likelihood of a certain event occurring in their community as likely or very likely. Use the plus-four method to create a 97 percent confidence interval for the proportion of American adults who believe that the event is likely or very likely. Explain what this confidence interval means in the context of the problem. A local poll in a New England town found that nine of 48 households think winter-proofing their cars is very important. Use the plus-four method to create a 97 percent confidence interval for the proportion of town residents who think winter-proofing their cars is very important. Explain what this confidence interval means in the context of this scenario.