#3: Delta Airlines quotes a flight time of 2 hours, 5 minutes for its flights from Cincinnati to Tampa. Suppose we believe that actual flight times are uniformly distributed between 2 hours and 2 hours, 20 minutes.

1. Show the graph of the probability density function for flight time.
2. What is the probability that the flight will be no more than 5 minutes late?
3. What is the probability that the flight will be more than 10 minutes late?
4. What is the expected flight time?

#5: The driving distance for the top 100 golfers on the PGA tour is between 284.7 and 310.6 yards. Assume that the driving distance for these golfers is uniformly distributed over this interval.

1. Give a mathematical expression for the probability density function of driving distance
2. What is the probability the driving distance for one of these golfers is less 290 yards?
3. What is the probability the driving distance for one of these golfers is at? 300 yards?
4. What is the probability the driving distance for one of these golfers is between 290 and 305 yards?
5. How many of these golfers drive the ball at least 290 yards?

#9: A random variable is normally distributed with a mean of \(\mu=50\) and a standard deviation of \(\sigma=5 .\)

1. Sketch a normal curve for the probability density function. Label the horizontal axis with values of 35,40,45,50,55,60, and 65 . Figure 6.4 shows that the normal curve almost touches the horizontal axis at three standard deviations below and at three standard deviations above the mean (in this case at 35 and 65 ).
2. What is the probability the random variable will assume a value between 45 and 55 ?
3. What is the probability the random variable will assume a value between 40 and 60 ?

#17: The average amount parents and children spent per child on back-to-school clothes in Autumn 2001 was $527 (CNBC, September 5, 2001). Assume the standard deviation is $160 and that the amount spent is normally distributed.

1. What is the probability that the amount spent on a randomly selected child is more than $700?
2. What is the probability that the amount spent on a randomly selected child is less than $100?
3. What is the probability that the amount spent on a randomly selected child is between $450 and $700?
4. What is the probability that the amount spent on a randomly selected child is no more than $300?

#23: The time needed to complete a final examination in a particular college course is normally distributed with a mean of 80 minutes and a standard deviation of 10 minutes. Answer the following questions.

1. What is the probability of completing the exam in one hour or less?
2. What is the probability that a student will complete the exam in more than 60 minutes but less than 75 minutes?
3. Assume that the class has 60 students and that the examination period is 90 minutes in length. How many students do you expect will be unable to complete the exam in the allotted time?

#25 : The average ticket price for a Washington Redskins football game was $81.89$ for the 2001 season (USA Today, September 6, 2001). With the additional costs of parking, food, drinks, and souvenirs, the average cost for a family of four to attend a game totaled $442.54. Assume the normal distribution applies and that the standard deviation is $65.

1. What is the probability that a family of four will spend more than $ 400?
2. What is the probability that a family of four will spend $ 300 or less?
3. What is the probability that a family of four will spend between $ 400 and $ 500 ?

GROUP 2:

The weekly incomes of a large group of sales clerks are normally distributed with a mean of $900 and a standard deviation of $110. Use the standard normal distribution to calculate the following:

1. Find the probability that a particular weekly income selected at random is between $750 and $1,500?
2. What is the probability that the income is less than $900.
3. What is the probability that he income is more than $1300?
4. What is the probability that the income is between $1,000$ and $1,400? (Draw a graph to illustrate your results)
5. Find the salary that represents the \(50^{\text {th }}\) percentile.
6. Find the salary that represents the \(90^{\text {th }}\) percentile.
7. \(5 \%\) of the salaries are below what value?
8. The top \(5 \%\) of the salaries are above what value?
9. Between what two values will the middle \(50 \%\) of the data lie?

11. Appliance Magazine provided estimates of the life expectancy of household appliances (USA Today, September 5,2000 ). A simple random sample of 10 VCRs shows the following useful life in years.

1. Develop a point estimate of the population mean life expectancy for VCRs.
2. Develop a point estimate of the population standard deviation for life expectancy of VCRs.

13. A Louis Harris poll used a survey of 1008 adults to learn about how people feel about the economy (BusinessWeek, August 7,2000 ). Responses were as follows:

595 adults The economy is growing.

332 adults The economy is staying about the same.

81 adults The economy is shrinking.

Develop a point estimate of the following population parameters.

1. The proportion of all adults who feel the economy is growing
2. The proportion of all adults who feel the economy is staying about the same
3. The proportion of all adults who feel the economy is shrinking

17. Suppose a simple random sample of size 50 is selected from a population with \(\sigma=10\) Find the value of the standard error of the mean in each of the following cases (use the finite population correction factor if appropriate).

1. The population size is infinite.
2. The population size is \(N=50,000\).
3. The population size is \(N=5000\).
4. The population size is \(N=500\).

21. The College Board American College Testing Program reported a population mean SAT score of \(\mu=1020\) (The World Almanac 2003 ). Assume that the population standard deviation is \(\sigma=100\).

1. What is the probability that a random sample of 75 students will provide a sample mean SAT score within 10 of the population mean?
2. What is the probability a random sample of 75 students will provide a sample mean SAT score within 20 of the population mean?

23. BusinessWeek conducted a survey of graduates from 30 top MBA programs (BusinessWeek, September 22, 2003). On the basis of the survey, assume that the mean annual salary for male and female graduates 10 years after graduation is $168,000 and $117,000, respectively. Assume the standard deviation for the male graduates is $40,000, and for the female graduates it is $25,000.

1. What is the probability that a simple random sample of 40 male graduates will provide a sample mean within $10,000 of the population mean, $168,00?
2. What is the probability that a simple random sample of 40 female graduates will provide a sample mean within $10,000 of the population mean, $117,000 ?
3. In which of the preceding two cases, part (a) or part (b), do we have a higher probability of obtaining a sample estimate within $10,000 of the population mean? Why?
4. What is the probability that a simple random sample of 100 male graduates will provide a sample mean more than $4000 below the population mean?

29. Assume that the population proportion is .55. Compute the standard error of the proportion, \(\sigma_{\bar{p}}\), for sample sizes of 100, 200, 500, and 1000 . What can you say about the size the standard error of the proportion as the sample size is increased?

# 33: Time/CNN voter polls monitored public opinion for the presidential candidates during the 2000 presidential election campaign. One Time/CNN poll conducted by Yankelovich Partners, Inc., used a sample of 589 likely voters (Time, June 26,2000 ). Assume the population proportion for a presidential candidate is \(p=.50\). Let \(\bar{p}\) be the sample proportion of likely voters favoring the presidential candidate.

1. Show the sampling distribution of \(\bar{p}\).
2. What is the probability the Time/CNN poll will provide a sample proportion within \(\pm .04\) of the population proportion?
3. What is the probability the Time/CNN poll will provide a sample proportion within \(\pm .03\) of the population proportion?
4. What is the probability the Time/CNN poll will provide a sample proportion within \(\pm .02\) of the population proportion?

# 35: The Democrat and Chronicle reported that \(25 \%\) of the flights arriving at the San Diego airport during the first five months of 2001 were late (Democrat and Chronicle, July 23 , 2001). Assume the population proportion is \(p=.25\).

1. Show the sampling distribution of \(\bar{p}\), the proportion of late flights in a sample of 1000 flights.
2. What is the probability that the sample proportion will be within \(\pm .03\) of the population proportion if a sample of size 1000 is selected?
3. Answer part (b) for a sample of 500 flights.

41. The mean television viewing time for Americans is 15 hours per week (Money, November 2003). Suppose a sample of 60 Americans is taken to further investigate viewing habits. Assume the population standard deviation for weekly viewing time is \(\sigma=4\) hours.

1. What is the probability the sample mean will be within 1 hour of the population mean?
2. What is the probability the sample mean will be within 45 minutes of the population mean?

Sample Methods Worksheet

Sample Methods Worksheet

In each of the following situations determine the type of sampling (simple random sample, cluster sample, stratified sample, convenient sample or systematic sample) that was used. Be able to give an explanation.

1. A marketing research firm is trying to determine the effectiveness of an advertising campaign. They have sponsored a series of television ads for a new cleaning product which have run in a major city. They plan a survey by dividing the city according to the squares on a transportation road map. They take a simple random sample of the squares on the map. For each square chosen they send an employee out to go to each house in the square with their questionnaire. They are able to complete a total of 345 questionnaires. One of the questions asked, "Have you heard of Wonder detergent?" and another asked "How many times during the week do you visit the grocery store?"
2. Zooms produces computer games. The quality control department of Zooms has a procedure to inspect its games. During the first hour of production the manager picks a random value between 1 and 60 . If the number is 35 then at exactly thirty five minutes past the hour, for that hour and for every other hour after that during that day, she sends her assistant to pull a copy of the software game program from the assembly line. The programs pulled will get put through a complete testing process.
3. The Franklin University Student Advisory Committee is trying to determine how much students spend on average for books each term. Each of the six members of the committee randomly chooses five friends and asks the friend to determine how much was spent on books for that term.
4. A small company was trying to determine the amount of the average order for parts. It had received over 1000 orders for parts. The office manager had the computer select 35 of those orders randomly. He determined that the mean amount of these 35 orders was $389.45. One of the orders was for $350 in miscellaneous cables. He informed his boss that he was \(95 \%\) certain that the average order that they had received was between $337.97 and $441.03

1. A simple random sample of 40 items resulted in a sample mean of 25 . The population standard deviation is \(\sigma=5\).

1. What is the standard error of the mean, \(\sigma_{\bar{x}}\) ?
2. At \(95 \%\) confidence, what is the margin of error?

3. A simple random sample of 60 items resulted in a sample mean of 80 . The population standard deviation is \(\sigma=15\).

1. Compute the \(95 \%\) confidence interval for the population mean.
2. Assume that the same sample mean was obtained from a sample of 120 items. Provide a \(95 \%\) confidence interval for the population mean.
3. What is the effect of a larger sample size on the interval estimate?

11. For a \(t\) distribution with 16 degrees of freedom, find the area, or probability, in each region

1. To the right of 2.120
2. To the left of 1.337
3. To the left of \(-1.746\)
4. To the right of 2.583
5. Between \(-2.120\) and 2.120
6. Between \(-1.746\) and 1.746

17. The International Air Transport Association surveys business travelers to develop quality ratings for transatlantic gateway airports. The maximum possible rating is 10 . Suppose a simple random sample of 50 business travelers is selected and each traveler is asked to provide a rating for the Miami International Airport. The ratings obtained from the sample of 50 business travelers follow.

19. A National Retail Foundation survey found households intended to spend an average of $649 during the December holiday season (The Wall Street Journal, December 2, 2002). Assume that the survey included 600 households and that the sample standard deviation was $175.

1. With \(95 \%\) confidence, what is the margin of error?
2. What is the \(95 \%\) confidence interval estimate of the population mean?
3. The prior year, the population mean expenditure per household was $632. Discuss the change in holiday season expenditures over the one-year period.

21. Complaints about rising prescription drug prices caused the U.S. Congress to consider laws that would force pharmaceutical companies to offer prescription discounts to senior citizens without drug benefits. The House Government Reform Committee provided data on the prescription cost for some of the most widely used drugs (Newsweek, May 8, 2000) Assume the following data show a sample of the prescription cost in dollars for Zocor drug used to lower cholesterol.

Given a normal population, what is the \(95 \%\) confidence interval estimate of the population mean cost for a prescription of Zocor?

23. How large a sample should be selected to provide a \(95 \%\) confidence interval with a margin of error of 10 ? Assume that the population standard deviation is 40 .

27. Annual starting salaries for college graduates with degrees in business administration are generally expected to be between $30,000 and $45,000. Assume that a \(95 \%\) confidence interval estimate of the population mean annual starting salary is desired. What is the planning value for the population standard deviation? How large a sample should be taken if the desired margin of error is

1. $500?
2. $200?
3. $100?
4. Would you recommend trying to obtain the $100 margin of error? Explain.

29. The travel-to-work time for residents of the 15 largest cities in the United States is reported in the 2003 Information Please Almanac. Suppose that a preliminary simple random sample of residents of San Francisco is used to develop a planning value of 6.25 minutes for the population standard deviation.

1. If we want to estimate the population mean travel-to-work time for San Francisco residents with a margin of error of 2 minutes, what sample size should be used? Assume 95% confidence.
2. If we want to estimate the population mean travel-to-work time for San Francisco residents with a margin of error of 1 minute, what sample size should be used? Assume 95% confidence.

33. In a survey, the planning value for the population proportion is \(p^{*}=.35\). How large a sample should be taken to provide a \(95 \%\) confidence interval with a margin of error of. 05 ?

41. An American Express retail survey found that \(16 \%\) of U.S. consumers used the Internet to buy gifts during the holiday season (USA Today, January 18, 2000). If 1285 customers participated in the survey, what is the margin of error and what is the interval estimate of the population proportion of customers using the Internet to buy gifts? Use \(95 \%\) confidence.

43. A Phoenix Wealth Management/Harris Interactive survey of 1500 individuals with worth of $1 million or more provided a variety of statistics on wealthy people (Businesses Week, September 22,2003 ). The previous three-year period had been bad for the stock market, which motivated some of the questions asked.

1. The survey reported that \(53 \%\) of the respondents lost \(25 \%\) or more of their portfolio value over the past three years. Develop a \(95 \%\) confidence interval for the proportion of wealthy people who lost \(25 \%\) or more of their portfolio value over the past three years.
2. The survey reported that \(31 \%\) of the respondents feel they have to save more for retirement to make up for what they lost. Develop a \(95 \%\) confidence interval for the population proportion.
3. Five percent of the respondents gave $25,000 or more to charity over the previous year Develop a \(95 \%\) confidence interval for the proportion who gave $25,000 or more to charity.
4. Compare the margin of error for the interval estimates in parts (a), (b), and (c), How is the margin of error related to \(\bar{p}\) ? When the same sample is being used to estimate variety of proportions, which of the proportions should be used to choose the planning value \({{p}^{*}}\) Why do you think \(p^{*}=.50\) is often used in these cases?

CONFIDENCE INTERVALS AND SAMPLE SIZE

Team TWO

Part I.

Suppose the coach of the football team wants to estimate the proportion of the population of fans who support his current starter lineup. The coach wants the estimate to be .04 of the true proportion. Assume a 90 percent level of confidence. The coach estimated the proportion supporting the current starter lineup to be .60.

Answer the following questions:

1. Construct a \(90 \%\) confidence interval using a sample size of 50 , then of 100 , then of 1,000.
2. How did changing the sample size affect the size of the interval?
3. What is the error of the estimate for each of these sample sizes?
4. How large of a sample is required for the error of the estimate to be within \(\pm .04\) of the population proportion?
5. How large would the sample have to be if the coach of the team were not available?

Show your work. Be prepared to present your analysis it to the class.

Part II.

A study is being done to determine how many hours college students study before taking an exam. A pilot study indicated that the mean time during the week of the exam is 4 hours, with a standard deviation of 1 hour. It is desired to estimate the mean study time within \(1 / 2\) hour. The 80 percent degree of confidence is to be used.

Answer the following question:

How many students should be surveyed?

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