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8.7: Chapter Homework

  • Page ID
    6077
  • 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.

    1. Define the random variables \(X\) and \(P^{\prime}\) in words.
    2. Which distribution should you use for this problem? Explain your choice
    3. Calculate \(p^{\prime}\).
    4. Construct a 96% confidence interval for the population proportion of Bam-Bam 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?
    103.

    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.

      • \(\overline x\) = __________
      • \(s_x\) = __________
      • \(n\) = __________
      • \(n – 1\) = __________
    1. Define the random variables \(X\) and \(\overline X\) in words.
    2. Which distribution should you use for this problem? Explain your choice.
    3. Construct a 95% confidence interval for the population mean enrollment at community colleges in the United States.
      • State the confidence interval.
      • Sketch the graph.
    4. What will happen to the error bound and confidence interval if 500 community colleges were surveyed? Why?
    104.

    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.

      • \(\overline x\) = __________
      • \(s_x\) = __________
      • \(n\) = __________
      • \(n – 1\) = __________
    1. Define the random variables \(X\) and \(\overline X\) in words.
    2. Which distribution should you use for this problem? Explain your choice.
    3. Construct a 95% confidence interval for the population mean time wasted.
      • State the confidence interval.
      • Sketch the graph.
    4. Explain in a complete sentence what the confidence interval means.
    105.

    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.

      • \(\overline x\) = __________
      • \(s_x\) = __________
      • \(n\) = __________
      • \(n – 1\) = __________
    1. Define the random variable \(X\) in words.
    2. Define the random variable \(\overline x\) in words.
    3. Which distribution should you use for this problem? Explain your choice.
    4. Construct a 95% confidence interval for the population mean length of time.
      • State the confidence interval.
      • Sketch the graph.
    5. What does it mean to be “95% confident” in this problem?
    106.

    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.

      • \(\overline x\) = __________
      • \(s_x\) = __________
      • \(n\) = __________
      • \(n – 1\) = __________
    1. Define the random variable \(X\) in words.
    2. Define the random variable\(\overline X\) in words.
    3. Which distribution should you use for this problem? Explain your choice.
    4. Construct a 99% confidence interval for the population mean length of time using training wheels.
      • State the confidence interval.
      • Sketch the graph.
    5. Why would the error bound change if the confidence level were lowered to 90%?
    107.

    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 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 \(\PageIndex{3}\)

    \(\overline{x}=\$ 251,854.23\)

    \(s=\$ 521,130.41\)

    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.

    4858516156
    5974635350
    5960605746
    5563574755
    5743616249
    6767555549

    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.

      • \(\overline x\) = __________
      • \(s_x\) = __________
      • \(n\) = __________
      • \(n-1\) = __________
    1. Define the random variables \(X\) and \(\overline X\) in words.
    2. Which distribution should you use for this problem? Explain your choice.
    3. Construct a 92% confidence interval for the population mean number of unoccupied seats per flight.
      • State the confidence interval.
      • Sketch the graph.
    110.

    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 \(\overline x\) in words.
    3. Construct a 95% confidence interval for the population mean cost of a used car.
      • State the confidence interval.
      • Sketch the graph.
    4. Explain what a “95% confidence interval” means for this study.
    111.

    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% confidence interval for the population mean grams of fat per serving of chocolate chip cookies sold in supermarkets.
      • State the confidence interval.
      • Sketch the graph.
    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?
    112.

    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.

      • \(\overline x\) = __________
      • \(s_x\) = __________
      • \(n\) = __________
      • \(n-1\) = __________
    1. Define the random variables \(X\) and \(\overline x\) in words.
    2. Which distribution should you use for this problem? Explain your choice.
    3. Construct a 95% confidence interval for the population mean worth of coupons.
      • State the confidence interval.
      • Sketch the graph.
    4. 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.

    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.

    1. (12.42, 14.18)
    2. (12.32, 14.29)
    3. (12.50, 14.10)
    4. 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.

    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?
    115.

    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.

      • \(x\) = __________
      • \(n\) = __________
      • \(p^{\prime}\) = __________
    1. Define the random variables \(X\) and \(P^{\prime}\), in words.
    2. Which distribution should you use for this problem? Explain your choice.
    3. Construct a 95% confidence interval for the population proportion who claim they always buckle up.
      • State the confidence interval.
      • Sketch the graph.
    4. 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.

    1. Define the random variables \(X\) and \(P^{\prime}\) in words.
    2. Which distribution should you use for this problem? Explain your choice.
    3. 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.

    1. 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.
    2. Which distribution should you use for this problem? Explain your choice.
    3. 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 \(\PageIndex{5}\)
    1. Find the point estimate for the population mean.
    2. Using 95% confidence, calculate the error bound.
    3. Create a 95% confidence interval for the mean total individual contributions.
    4. 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%.

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