8.11: Homework
- Page ID
- 6077
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.
- Define the random variables X and P′ in words.
- Which distribution should you use for this problem? Explain your choice
- Calculate p′.
- Construct a 96% confidence interval for the population proportion of Bam-Bam snack pieces per bag.
- State the confidence interval.
- Sketch the graph.
- Calculate the error bound.
- Do you think that six packages of fruit snacks yield enough data to give accurate results? Why or why not?
-
- x−𝑥- = __________
- sx𝑠𝑥 = __________
- n = __________
- n – 1 = __________
- Define the random variables X𝑋 and X−𝑋- in words.
- Which distribution should you use for this problem? Explain your choice.
- Construct a 95% confidence interval for the population mean enrollment at community colleges in the United States.
- State the confidence interval.
- Sketch the graph.
- What will happen to the error bound and confidence interval if 500 community colleges were surveyed? Why?
-
- x−𝑥- = __________
- sx𝑠𝑥 = __________
- n = __________
- n – 1 = __________
- Define the random variables X𝑋 and X−𝑋- in words.
- Which distribution should you use for this problem? Explain your choice.
- Construct a 95% confidence interval for the population mean time wasted.
- State the confidence interval.
- Sketch the graph.
- Explain in a complete sentence what the confidence interval means.
-
- x¯𝑥¯ = __________
- sx𝑠𝑥 = __________
- n = __________
- n – 1 = __________
- Define the random variable X𝑋 in words.
- Define the random variable X−𝑋- in words.
- Which distribution should you use for this problem? Explain your choice.
- Construct a 95% confidence interval for the population mean length of time.
- State the confidence interval.
- Sketch the graph.
- What does it mean to be “95% confident” in this problem?
-
- x−𝑥- = __________
- sx𝑠𝑥 = __________
- n = __________
- n – 1 = __________
- Define the random variable X𝑋 in words.
- Define the random variableX−𝑋- in words.
- Which distribution should you use for this problem? Explain your choice.
- Construct a 99% confidence interval for the population mean length of time using training wheels.
- State the confidence interval.
- Sketch the graph.
- 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 |
\(x_\bar\)=$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 \(\PageIndex{2}\) 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 |
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.
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.
-
- x¯𝑥¯ = __________
- sx𝑠𝑥 = __________
- n = __________
- n-1 = __________
- Define the random variables X𝑋 and X¯¯¯𝑋¯ in words.
- Which distribution should you use for this problem? Explain your choice.
- Construct a 92% confidence interval for the population mean number of unoccupied seats per flight.
- State the confidence interval.
- Sketch the graph.
- Which distribution should you use for this problem? Explain your choice.
- Define the random variable X−𝑋- in words.
- Construct a 95% confidence interval for the population mean cost of a used car.
- State the confidence interval.
- Sketch the graph.
- Explain what a “95% confidence interval” means for this study.
- 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.
- 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?
- Go to the store and record the grams of fat per serving of six brands of chocolate chip cookies.
- Calculate the mean.
- Is the mean within the interval you calculated in part a? Did you expect it to be? Why or why not?
-
- x−𝑥- = __________
- sx𝑠𝑥 = __________
- n = __________
- n-1 = __________
- Define the random variables X𝑋 and X−𝑋- in words.
- Which distribution should you use for this problem? Explain your choice.
- Construct a 95% confidence interval for the population mean worth of coupons.
- State the confidence interval.
- Sketch the graph.
- 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.
- (12.42, 14.18)
- (12.32, 14.29)
- (12.50, 14.10)
- Impossible to determine
8.4 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.
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.
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.
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.
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.
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.
- Define the random variables X and P′ in words.
- Which distribution should you use for this problem? Explain your choice.
- Construct a 97% confidence interval for the population proportion of people over 50 who ran and died in the same eight–year period.
- State the confidence interval.
- Sketch the graph.
- Explain what a “97% confidence interval” means for this study.
- Define the random variables X and P′ in words.
- Which distribution should you use for this problem? Explain your choice.
- Construct a 95% confidence interval for the population proportion of adult Americans who feel that crime is the main problem.
- State the confidence interval.
- Sketch the graph.
- Suppose we want to lower the sampling error. What is one way to accomplish that?
- The sampling error given by Yankelovich Partners, Inc. (which conducted the poll) is ±3%. In one to three complete sentences, explain what the ±3% represents.
- Define the random variables X and P′ in words.
- Which distribution should you use for this problem? Explain your choice.
- Construct a 95% confidence interval for the population proportion of adult Americans who are worried a lot about the quality of education in our schools.
- State the confidence interval.
- Sketch the graph.
- The sampling error given by Yankelovich Partners, Inc. (which conducted the poll) is ±3%. In one to three complete sentences, explain what the ±3% represents.
Use the following information to answer the next three exercises: 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.
- 0.90
- 1.27
- 0.79
- 400
- (0.761, 0.820)
- (0.125, 0.188)
- (0.755, 0.826)
- (0.130, 0.183)
Use the following information to answer the next two exercises: Five hundred and eleven (511) homes in a certain southern California community are randomly surveyed to determine if they meet minimal earthquake preparedness recommendations. One hundred seventy-three (173) of the homes surveyed met the minimum recommendations for earthquake preparedness, and 338 did not.
- (0.2975, 0.3796)
- (0.6270, 0.6959)
- (0.3041, 0.3730)
- (0.6204, 0.7025)
- 0.6614
- 0.3386
- 173
- 338
- Determine the estimated proportion from the sample.
- Determine the sample size.
- Identify CL and α.
- Calculate the error bound based on the information provided.
- Compare the error bound in part d to the margin of error reported by Gallup. Explain any differences between the values.
- Create a confidence interval for the results of this study.
- A reporter is covering the release of this study for a local news station. How should she explain the confidence interval to her audience?
- Find the point estimate and the error bound for this confidence interval.
- Can we (with 95% confidence) conclude that more than half of all American adults believe this?
- Use the point estimate from part a and n = 1,000 to calculate a 75% confidence interval for the proportion of American adults that believe that major college sports programs corrupt higher education.
- Can we (with 75% confidence) conclude that at least half of all American adults believe this?
- Create a 99% confidence interval for the true proportion of American adults who have illegally downloaded music.
- 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.
- Without performing any calculations, describe how the confidence interval would change if the confidence level changed from 99% to 90%.
8.5 Calculating the Sample Size n: Continuous and Binary Random Variables
-
- x−𝑥- =________
- σ =________
- n =________
- In words, define the random variables X and X−𝑋-.
- Which distribution should you use for this problem? Explain your choice.
- Construct a 95% confidence interval for the population mean height of male Swedes.
- State the confidence interval.
- Sketch the graph.
- What will happen to the level of confidence obtained if 1,000 male Swedes are surveyed instead of 48? Why?
- In words, define the random variables X and X−𝑋-.
- Which distribution should you use for this problem? Explain your choice.
- Construct a 95% confidence interval for the population mean length of engineering conferences.
- State the confidence interval.
- Sketch the graph.
-
- x−𝑥- =________
- σ =________
- n =________
- In words, define the random variables X and X−𝑋-.
- Which distribution should you use for this problem? Explain your choice.
- Construct a 90% confidence interval for the population mean time to complete the tax forms.
- State the confidence interval.
- Sketch the graph.
- 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?
- 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?
- 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?
-
- x−𝑥- =________
- σ =________
- sx =________
- In words, define the random variable X.
- In words, define the random variable X−𝑋-.
- Which distribution should you use for this problem? Explain your choice.
- Construct a 90% confidence interval for the population mean weight of the candies.
- State the confidence interval.
- Sketch the graph.
- Construct a 98% confidence interval for the population mean weight of the candies.
- State the confidence interval.
- Sketch the graph.
- Calculate the error bound.
- In complete sentences, explain why the confidence interval in part f is larger than the confidence interval in part e.
- In complete sentences, give an interpretation of what the interval in part f means.
-
- x−𝑥- =________
- σ =________
- n =________
- Define the random variables X and X−𝑋- in words.
- Which distribution should you use for this problem? Explain your choice.
- Construct a 90% confidence interval for the population mean number of letters campers send home.
- State the confidence interval.
- Sketch the graph.
- What will happen to the error bound and confidence interval if 500 campers are surveyed? Why?
- If we took repeated samples, approximately 90% of the samples would produce the same confidence interval.
- If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the sample mean.
- If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the true value of the population mean.
- If we took repeated samples, the sample mean would equal the population mean in approximately 90% of the samples.
| $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 |
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%.