Answer: Determine the point estimate of the population

Problem 1AYU A ____ ____ is the value of a statistic that estimates the value of a parameter.

Problem 2AYU The____ ___ _____represents the expected proportion of intervals that will contain the parameter if a large number of different samples is obtained. It is denoted________.

Problem 3AYU True or False: A 95% confidence interval for a population proportion with lower bound 0.45 and upper bound 0.51 means there is a 95% probability the population proportion is between 0.45 and 0.51.

Problem 4AYU The value represents the ____ ____ of the distribution.

As the level of confidence of a confidence interval increases, the margin of error____(increases/decreases).

Problem 6AYU As the sample size used to obtain a confidence interval increases, the margin of error____(increases/decreases).

Problem 7AYU Determine the critical value z?/2 that corresponds to the given level of confidence. 90%

Problem 8AYU Determine the critical value z?/2 that corresponds to the given level of confidence. 99%

In Problems 7-10, determine the critical value \(z_{\alpha / 2}\) that corresponds to the given level of confidence. 98%

In Problems 7-10, determine the critical value \(z_{\alpha / 2}\) that corresponds to the given level of confidence. 92%

Determine the point estimate of the population proportion, the margin of error for each confidence interval, and the number of individuals in the sample with the specified characteristic, x, for the sample size provided. Lower bound: 0.201, upper bound: 0.249, n = 1200

Problem 13AYU Determine the point estimate of the population proportion, the margin of error for each confidence interval, and the number of individuals in the sample with the specified characteristic, x, for the sample size provided. Lower bound: 0.462, upper bound: 0.509, n = 1680

Problem 14AYU Determine the point estimate of the population proportion, the margin of error for each confidence interval, and the number of individuals in the sample with the specified characteristic, x, for the sample size provided. Lower bound: 0.853, upper bound: 0.871, n = 10,732

Construct a confidence interval of the population proportion at the given level of confidence. x = 30, n = 150, 90% confidence

In Problems 15-20, construct a confidence interval of the population proportion at the given level of confidence. x = 80, n = 200, 98% confidence

Construct a confidence interval of the population proportion at the given level of confidence. x = 120, n = 500, 99% confidence

Problem 18AYU Construct a confidence interval of the population proportion at the given level of confidence. x = 400, n = 1200, 95% confidence

Problem 19AYU Construct a confidence interval of the population proportion at the given level of confidence. x = 860, n = 1100, 94% confidence

Problem 20AYU Construct a confidence interval of the population proportion at the given level of confidence. x = 540, n = 900, 96% confidence

Problem 21AYU You Explain It! New Deal Policies In response to the Great Depression, Franklin D. Roosevelt enacted many New Deal policies. One such policy was the enactment of the National Recovery Administration (NRA), which forced business to agree to wages and prices within their particular industry. The thought was that this would encourage higher wages among the working class, thereby spurring consumption. In a Gallup survey conducted in 1933 of 2025 adult Americans, 55% thought that wages paid to workers in industry were too low. The margin of error was 3 percentage points with 95% confidence. Which of the following represents a reasonable interpretation of the survey results? For those that are not reasonable, explain the flaw. (a) We are 95% confident 55% of adult Americans during the Great Depression felt wages paid to workers in industry were too low. ________________ (b) We are 92% to 98% confident 55% of adult Americans during the Great Depression felt wages paid to workers in industry were too low. ________________ (c) We are 95% confident the proportion of adult Americans during the Great Depression who believed wages paid to workers in industry were too low was between 0.52 and 0.58. ________________ (d) In 95% of samples of adult Americans during the Great Depression, the proportion who believed wages paid to workers in industry were too low is between 0.52 and 0.58.

You Explain It! Superstition A USA Today/Gallup poll asked l006 adult Americans how much it would bother them to stay in a room on the 13th floor of a hotel. Interestingly, 13% said it would bother them. The margin of error was 3 percentage points with 95% confidence. Which of the following represents a reasonable interpretation of the survey results? For those not reasonable, explain the flaw. (a) We are 95% confident that the proportion of adult Americans who would be bothered to stay in a room on the 13th floor is between 0.10 and 0.16. (b) We are between 92% and 98% confident that 13% of adult Americans would be bothered to stay in a room on the 13th floor. (c) In 95% of samples of adult Americans, the proportion who would be bothered to stay in a room on the 13th floor is between 0.10 and 0.16. (d) We are 95% confident that 13% of adult Americans would be bothered to stay in a room on the 13th floor.

Problem 23AYU You Explain It! Valentine’s Day A Rasmussen Reports national survey of l000 adult Americans found that 18% dreaded Valentine’s Day. The margin of error for the survey was 4.5 percentage points with 95% confidence. Explain what this means.

Problem 24AYU You Explain It! A Stressful Commute? A Gallup poll of 547 adult Americans employed full or part time asked, “Generally speaking, would you say your commute to work is—very stressful, somewhat stressful, not that stressful, or not stressful at all?” Gallup reported that 24% of American workers said that their commute was “very” or “somewhat” stressful. The margin of error was 4 percentage points with 95% confidence. Explain what this means.

Problem 25AYU Smartphone Purchases In a survey of 3611 adult Americans 18 years and older conducted in July 2010 by SmartRevenue, it was found that 542 have used their smartphone to make a purchase. (a) Obtain a point estimate for the population proportion of adult Americans 18 years and older who have used their smartphone to make a purchase. ________________ (b) Verify that the requirements for constructing a confidence interval about p are satisfied. ________________ (c) Construct a 90% confidence interval for the population proportion of adult Americans who have used their smartphone to make a purchase. ________________ (d) Interpret the interval.

Problem 26AYU Saving for Retirement? A Retirement Confidence Survey of 1153 workers and retirees in the United States 25 years of age and older conducted by Employee Benefit Research Institute in January 2010 found that 496 had less than $10,000 in savings. (a) Obtain a point estimate for the population proportion of workers and retirees in the United States 25 years of age and older who have less than $10,000 in savings. ________________ (b) Verify that the requirements for constructing a confidence interval about p are satisfied. ________________ (c) Construct a 95% confidence interval for the population proportion of workers and retirees in the United States 25 years of age and older who have less than $10,000 in savings. ________________ (d) Interpret the interval.

Luxury or Necessity? A random sample of 1003 adult Americans was asked, "Do you pretty much think televisions are a necessity or a luxury you could do without?" Of the 1003 adults surveyed, 521 indicated that televisions are a luxury they could do without. (a) Obtain a point estimate for the population proportion of adult Americans who believe that televisions are a luxury they could do without. (b) Verify that the requirements for constructing a confidence interval about \(p\) are satisfied. (c) Construct and interpret a 95% confidence interval for the population proportion of adult Americans who believe that televisions are a luxury they could do without. (d) Is it possible that a supermajority (more than 60% ) of adult Americans believe that television is a luxury they could do without? Is it likely? (e) Use the results of part (c) to construct a 95 confidence interval for the population proportion of adult Americans who believe that televisions are a necessity.

Problem 28AYU Family Values In a USA Today/Gallup poll, 768 of 1024 randomly selected adult Americans aged 18 or older stated that a candidate’s positions on the issue of family value are extremely or very important in determining their vote for president. (a) Obtain a point estimate for the proportion of adult Americans aged 18 or older for which the issue of family values is extremely or very important in determining their vote for president. ________________ (b) Verify that the requirements for constructing a confidence interval for p are satisfied. ________________ (c) Construct a 99% confidence interval for the proportion of adult Americans aged 18 or older for which the issue of family values is extremely or very important in determining their vote for president. ________________ (d) Is it possible that the proportion of adult Americans aged 18 or older for which the issue of family values is extremely or very important in determining their vote for president is below 70%? Is this likely? ________________ (e) Use the results of part (c) to construct a 99% confidence interval for the proportion of adult Americans aged 18 or older for which the issue of family values is not extremely or very important in determining their vote for president.

Problem 29AYU Tattoo Attitude A Harris Interactive poll conducted during January 2008 found that 944 of 1748 adult Americans 18 years or older who do not have a tattoo believe that individuals with tattoos are more rebellious. (a) Obtain a point estimate for the proportion of adult Americans without tattoos who believe individuals with tattoos are more rebellious. ________________ (b) Verify that the requirements for constructing a confidence interval for p are satisfied. ________________ (c) Construct a 90% confidence interval for the proportion of adult Americans without tattoos who believe individuals with tattoos are more rebellious. ________________ (d) Construct a 99% confidence interval for the proportion of adult Americans without tattoos who believe individuals with tattoos are more rebellious. ________________ (e) What is the effect of increasing the level of confidence on the width of the interval?

Problem 30AYU Healthcare A Gallup poll conducted in November 2010 found that 493 of 1050 adult Americans believe it is the responsibility of the federal government to make sure all Americans have healthcare coverage. (a) Obtain a point estimate for the proportion of adult Americans who believe it is the responsibility of the federal government to make sure all Americans have healthcare coverage. ________________ (b) Verify that the requirements for constructing a confidence interval for p are satisfied. ________________ (c) Construct a 95% confidence interval for the proportion of adult Americans who believe it is the responsibility of the federal government to make sure all Americans have healthcare coverage. ________________ (d) Construct a 99% confidence interval for the proportion of adult Americans who believe it is the responsibility of the federal government to make sure all Americans have healthcare coverage. ________________ (e) What is the effect of increasing the level of confidence on the width of the interval?

Sullivan Statistics Survey: Deficit Reduction One question on the Sullivan Statistics Survey is “Would you be willing to pay higher taxes if the tax revenue went directly toward deficit reduction?” Treat the survey respondents as a simple random sample of adult Americans. Construct and interpret a 90% confidence interval for the proportion of adult Americans who would be willing to pay higher taxes if the revenue went directly toward deficit reduction.

Sullivan Statistics Survey: Language One question on the Sullivan Statistics Survey is “Is a language other than English the primary language spoken in your home?” Treat the survey respondents as a simple random sample of adult Americans. Construct and interpret a 95% confidence interval for the proportion of adult Americans who speak a language other than English at home.

Problem 33AYU High-Speed Internet Access A researcher wishes to estimate the proportion of households that have broadband Internet access. What size sample should be obtained if she wishes the estimate to be within 0.03 with 99% confidence if (a) she uses a 2009 estimate of 0.635 obtained from the National Telecommunications and Information Administration? ________________ (b) she does not use any prior estimates?

Problem 34AYU Home Ownership An urban economist wishes to estimate the proportion of Americans who own their homes. What size sample should be obtained if he wishes the estimate to be within 0.02 with 90% confidence if (a) he uses a 2010 estimate of 0.669 obtained from the U.S Census Bureau? ________________ (b) he does not use any prior estimates?

Problem 35AYU A Penny for Your Thoughts A researcher for the U.S. Department of the Treasury wishes to estimate the percentage of Americans who support abolishing the penny. What size sample should be obtained if he wishes the estimate to be within 2 percentage points with 98% confidence if (a) he uses a 2006 estimate of 15% obtained from a Coinstar National Currency Poll?

Problem 36AYU Credit-Card Debt A school administrator is concerned about the amount of credit-card debt that college students have. She wishes to conduct a poll to estimate the percentage of fulltime college students who have credit-card debt of $2000 or more. What size sample should be obtained if she wishes the estimate to be within 2.5 percentage points with 94% confidence if (a) a pilot study indicates that the percentage is 34%? ________________ (b) no prior estimates are used?

Problem 40AYU Gun Control In a Gallup Poll conducted October 2010, 44% of the people polled answered “more strict” to the following question: “Do you feel that the laws covering the sale of firearms should be made more strict, less strict, or kept as they are now?” Suppose the margin of error in the poll was 3.5% and the estimate was made with 95% confidence. At least how many people were surveyed?

2004 Presidential Election The Gallup organization conducted a poll of 2014 likely voters just prior to the 2004 presidential election. The results of the survey indicated that George W. Bush would receive 49% of the popular vote and John Kerry would receive 47% of the popular vote. The margin of error was reported to be 3%. The Gallup organization reported that the race was too close to call. Use the concept of a confidence interval to explain what this means.

Simulation - When Model Requirements Fail A Bernoulli random variable is a variable that is either 0 (a failure) or 1 (a success). The probability of success is denoted p. (a) Use StatCrunch, MINITAB, or some other statistical spreadsheet to generate 1000 Bernoulli samples of size n = 20 with p = 0.15. (b) Estimate the population proportion for each of the 1000 Bernoulli samples. (c) Draw a histogram of the 1000 proportions from part (b). What is the shape of the histogram? (d) Construct a 95% confidence interval for each of the 1000 Bernoulli samples using the normal model. (e) What proportion of the intervals do you expect to include the population proportion, p? What proportion of the intervals actually captures the population proportion? Explain any differences.

Problem 43AYU To deal with issues such as the distribution of p not following a normal distribution A. Agresti and B. Coull (Approximate Is Better Than “Exact” for Interval Estimation of Binomial Proportion. American Statistician, 52:119-26, 1998) proposed a modified approach to constructing confidence intervals for a proportion. A (1 – ?) ? 100% confidence interval for p is given by Lower bound: Upper bound: where (x is the number of successes in n trials). Use this result to answer Problem 1-2. Problem 1 Cauliflower? Jane wants to estimate the proportion of students on her campus who eat cauliflower. After surveying 20 students, she finds 2 who eat cauliflower. Obtain and interpret a 95% confidence interval for the proportion of students who eat cauliflower on Jane’s campus using Agresti and Coull’s method. Problem 2 Simulation - When Model Requirements Fail A Bernoulli random variable is a variable that is either 0 (a failure) or 1 (a success). The probability of success is denoted p. (a) Use StatCrunch, MINITAB, or some other statistical spreadsheet to generate 1000 Bernoulli samples of size n = 20 with p = 0.15. ________________ (b) Estimate the population proportion for each of the 1000 Bernoulli samples. ________________ (c) Draw a histogram of the 1000 proportions from part (b). What is the shape of the histogram? ________________ (d) Construct a 95% confidence interval for each of the 1000 Bernoulli samples using the normal model. ________________ (e) What proportion of the intervals do you expect to include the population proportion, p? What proportion of the intervals actually captures the population proportion? Explain any differences.

Problem 44AYU To deal with issues such as the distribution of p not following a normal distribution A. Agresti and B. Coull (Approximate Is Better Than “Exact” for Interval Estimation of Binomial Proportion. American Statistician, 52:119-26, 1998) proposed a modified approach to constructing confidence intervals for a proportion. A (1 – ?) • 100% confidence interval for p is given by Lower bound: Upper bound: where (x is the number of successes in n trials). Use this result to answer Problem 1-2. Problem 1 Walk to Work Alan wants to estimate the proportion of adults who walk to work. In a survey of 10 adults, he finds 0 who walk to work. Explain why a 95% confidence interval using the normal model yields silly results. Then compute and interpret a 95% confidence interval for the proportion of adults who walk to work using Agresti and Coull’s method. Problem 2 Simulation - When Model Requirements Fail A Bernoulli random variable is a variable that is either 0 (a failure) or 1 (a success). The probability of success is denoted p. (a) Use StatCrunch, MINITAB, or some other statistical spreadsheet to generate 1000 Bernoulli samples of size n = 20 with p = 0.15. ________________ (b) Estimate the population proportion for each of the 1000 Bernoulli samples. ________________ (c) Draw a histogram of the 1000 proportions from part (b). What is the shape of the histogram? ________________ (d) Construct a 95% confidence interval for each of the 1000 Bernoulli samples using the normal model. ________________ (e) What proportion of the intervals do you expect to include the population proportion, p? What proportion of the intervals actually captures the population proportion? Explain any differences.

Putting It Together: Hand Washing The American Society for Microbiology (ASM) and the Soap and Detergent Association (SDA) jointly commissioned two separate studies during 2007, both of which were conducted by Harris Interactive. In one of the studies, 1001 adults were interviewed by telephone and asked about their hand-washing habits. In the other study, the hand-washing behavior of 6076 adults was inconspicuously observed within public restrooms in four U.S. cities. (a) In the telephone interviews, 921 of the adults said they always wash their hands in public restrooms. Use this result to obtain a point estimate for the proportion of adults who say they always wash their hands in public restrooms. (b) Verify that the requirements for constructing a confidence interval for p from part (a) are satisfied. (c) Using the results from the telephone interviews, construct a 95% confidence interval for the proportion of adults who say they always wash their hands in public restrooms. (d) Is it possible that the proportion of adults who say they always wash their hands in public restrooms is less than 0.85? Is this likely? (e) In the observational study, 4679 of the 6076 adults were observed washing their hands. Use this result to obtain a point estimate for the proportion of adults who wash their hands in public restrooms. (f) Verify that the requirements for constructing a confidence interval for p from part (e) are satisfied. (g) Using the results from the observational study, construct a 95% confidence interval for the proportion of adults who wash their hands in public restrooms. (h) Is it possible that the proportion of adults who wash their hands in public restrooms is greater than 0.85? Is this likely? (i) Based on your findings in parts (a) through (h), what might you conclude about the proportion of adults who say they always wash their hands versus the proportion of adults who actually wash their hands in public restrooms? Why do you think this is?

Problem 46AYU Explain what “95% confidence” means in a 95% confidence interval.

Problem 47AYU Explain why quadrupling the sample size causes the margin of error to be cut in half.

Problem 48AYU Why do polling companies often survey 1060 individuals when they wish to estimate a population proportion with a margin of error of 3% with 95% confidence?

Katrina wants to estimate the proportion of adult Americans who read at least 10 books last year. To do so, she obtains a simple random sample of 100 adult Americans and constructs a 95% confidence interval. Matthew also wants to estimate the proportion of adult Americans who read at least 10 books last year. He obtains a simple random sample of 400 adult Americans and constructs a 99% confidence interval. Assuming both Katrina and Matthew obtained the same point estimate, whose estimate will have the smaller margin of error? Justify your answer.

Problem 50AYU Two researchers, Jaime and Mariya, are each constructing confidence intervals for the proportion of a population who is left-handed. They find the point estimate is 0.13. Each independently constructed a confidence interval based on the point estimate, but Jaime’s interval has a lower bound of 0.097 and an upper bound of 0.163, while Mariya’s interval has a lower bound of 0.117 and an upper bound of 0.173. Which interval is wrong? Why?

The 112th Congress of the United States of America has 535 members, of which 87 are women. An alien lands near the U.S. Capitol and treats members of Congress as a random sample of the human race. He reports to his superiors that a 95% confidence interval for the proportion of the human race that is female has a lower bound of 0.131 and an upper bound of 0.194. What is wrong with the alien’s approach to estimating the proportion of the human race that is female?