# 8.7: Chapter Homework

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## 8.2 A Confidence Interval for a Population Standard Deviation Unknown, Small Sample Case

102.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. We wish to calculate a 96% confidence interval for the population proportion of Bam-Bam snack pieces.

- 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 30 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 \(\PageIndex{3}\) \(s=\$ 521,130.41\)

Use this sample data to construct a 95% 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.

108.*Forbes*magazine published data on the best small firms in 2012. These were firms that had 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. The Table \(\PageIndex{4}\) 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

**8.4**Use this sample data to construct a 90% confidence interval for the mean age of CEO’s for these top small firms. Use the Student's t-distribution.

**109**.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.

*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.**113**.Find the 95% Confidence Interval for the true population mean for the amount of soda served.

- (12.42, 14.18)
- (12.32, 14.29)
- (12.50, 14.10)
- Impossible to determine

## 8.3 A Confidence Interval for A Population Proportion

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

- 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?
- 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?

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.

- \(x\) = __________
- \(n\) = __________
- \(p^{\prime}\) = __________

- Define the random variables \(X\) and \(P^{\prime}\), in words.
- Which distribution should you use for this problem? Explain your choice.
- Construct a 95% confidence interval for the population proportion who claim they always buckle up.
- State the confidence interval.
- Sketch the graph.

- If this survey were done by telephone, list three difficulties the companies might have in obtaining random results.

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

- Define the random variables \(X\) and \(P^{\prime}\) in words.
- Which distribution should you use for this problem? Explain your choice.
- Construct a 90% confidence interval for the population proportion of people who feel the president is doing an acceptable job.
- State the confidence interval.
- Sketch the graph.

**117**.An article regarding interracial dating and marriage recently appeared in the Washington Post. Of the 1,709 randomly selected adults, 315 identified themselves as Latinos, 323 identified themselves as blacks, 254 identified themselves as Asians, and 779 identified themselves as whites. In this survey, 86% of blacks said that they would welcome a white person into their families. Among Asians, 77% would welcome a white person into their families, 71% would welcome a Latino, and 66% would welcome a black person.

- We are interested in finding the 95% confidence interval for the percent of all black adults who would welcome a white person into their families. Define the random variables \(X\) and \(P^{\prime}\), in words.
- Which distribution should you use for this problem? Explain your choice.
- Construct a 95% confidence interval.
- State the confidence interval.
- Sketch the graph.

**118**.Refer to the information in Table \(\PageIndex{5}\) 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 \(\sigma\) = $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 \(\PageIndex{5}\) - Find the point estimate for the population mean.
- Using 95% confidence, calculate the error bound.
- Create a 95% confidence interval for the mean total individual contributions.
- Interpret the confidence interval in the context of the problem.

**137**.The American Community Survey (ACS), part of the United States Census Bureau, conducts a yearly census similar to the one taken every ten years, but with a smaller percentage of participants. The most recent survey estimates with 90% confidence that the mean household income in the U.S. 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.

**138**.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?

**139**.If the confidence interval is change to a higher probability, would this cause a lower, or a higher, minimum sample size?

**140**.If the tolerance is reduced by half, how would this affect the minimum sample size?

**141**.If the value of \(p\) is reduced, would this necessarily reduce the sample size needed?

**142**.Is it acceptable to use a higher sample size than the one calculated by \(\frac{z^{2} p q}{e^{2}}\)?

**143**.A company has been running an assembly line with 97.42%% of the products made being acceptable. Then, a critical piece broke down. After the repairs the decision was made to see if the number of defective products made was still close enough to the long standing production quality. Samples of 500 pieces were selected at random, and the defective rate was found to be 0.025%.

- Is this sample size adequate to claim the company is checking within the 90% confidence interval?
- The 95% confidence interval?