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The Practice Of Statistics For Business And Economics 4th Edition Layth C. Alwan, Bruce A. Craig - Solutions

Starting salaries. You are planning a survey of starting salaries for recent business majors. In the latest survey by the National Association of Colleges and Employers, the average starting salary was reported to be $55,144.10 If you assume that the standard deviation is $11,000, what sample size
Changing the confidence level. In the setting of Exercise 6.28, would the margin of error for 99% confidence be larger or smaller? Verify your answer by performing the calculations.
Changing the sample size. In the setting of the previous exercise, would the margin of error for 95% confidence be roughly doubled or halved if the sample size were raised to n 5 6400? Verify your answer by performing the calculations.
Average amount paid for college. Refer to Example 6.10 (pages 307–308).The average annual amount the n 5 1601 families paid for college was $20,902.9 If the population standard deviation is $7500, give the 95% confidence interval for m, the average amount a family pays for a college undergraduate.
80% confidence intervals. The idea of an 80% confidence interval is that the interval captures the true parameter value in 80% of all samples. That’s not high enough confidence for practical use, but 80% hits and 20% misses make it easy to see how a confidence interval behaves in repeated samples
Generating a single confidence interval. Using the default settings in the Confidence Interval applet (95% confidence level and n 5 20), click “Sample”to choose an SRS and display its confidence interval.(a) Is the spread in the data, shown as yellow dots below the confidence interval, larger
An interval for 95% of the sample means. In the setting of the previous two exercises, about 95% of all samples will capture the true mean of all the invoices in the interval x plus or minus . Fill in the blank.
Use the 68–95–99.7 rule. In the setting of the previous exercise, the 68–95–99.7 rule says that the probability is about 0.95 that x is within of the population mean m. Fill in the blank.
Company invoices. The mean amount m for all the invoices for your company last month is not known. Based on your past experience, you are willing to assume that the standard deviation of invoice amounts is about $260. If you take a random sample of 100 invoices, what is the value of the standard
Supplier delivery times. Supplier on-time delivery performance is critical to enabling the buyer’s organization to meet its customer service commitments. Therefore, monitoring supplier delivery times is critical. Based on a great deal of historical data, a manufacturer of personal computers finds
Increasing sample size. Heights of adults are well approximated by the Normal distribution.Suppose that the population of adult U.S. males has mean of 69 inches and standard deviation of 2.8 inches.(a) What is the probability that a randomly chosen male adult is taller than 6 feet?(b) What is the
Grades in a math course. Indiana University posts the grade distributions for its courses online.6 Students in one section of Math 118 in the fall 2012 semester received 33% A’s, 33% B’s, 20% C’s, 12% D’s, and 2% F’s.(a) Using the common scale A 5 4, B 5 3, C 5 2, D 5 1, F = 0, take X to
Safe flying weight. In response to the increasing weight of airline passengers, the Federal Aviation Administration told airlines to assume that passengers average 190 pounds in the summer, including clothing and carry-on baggage. But passengers vary: the FAA gave a mean but not a standard
ACT scores of high school seniors. The scores of your state’s high school seniors on the ACT college entrance examination in a recent year had mean m 5 22.3 and standard deviation s 5 6.2. The distribution of scores is only roughly Normal.(a) What is the approximate probability that a single
Generating a sampling distribution. Let’s illustrate the idea of a sampling distribution in the case of a very small sample from a very small population.The population is the sizes of 10 medium-sized businesses, where size is measured in terms of the number of employees. For convenience, the 10
Number of friends on Facebook. Facebook recently examined all active Facebook users (more than 10% of the global population) and determined that the average user has 190 friends. This distribution takes only integer values, so it is certainly not Normal.It is also highly skewed to the right, with a
Determining sample size. Refer to the previous exercise. Now you want to use a sample size such that about 95% of the averages fall within 6 10 minutes(0.17 hour) of the true mean m 5 6.78.(a) Based on your answer to part (b) in Exercise 6.14, should the sample size be larger or smaller than 150?
Total sleep time of college students. In Example 6.1 (page 289), the total sleep time per night among college students was approximately Normally distributed with mean m 5 6.78 hours and standard deviation s 5 1.24 hours. You plan to take an SRS of size n 5 150 and compute the average total sleep
Why the difference? Refer to the previous exercise. In Exercise 6.1 (page 289), a survey by AppsFire reported a median of 108 apps per device. This is very different from the average reported in the previous exercise.(a) Do you think that the two populations are comparable? Explain your answer.(b)
Number of apps on a smartphone. At a recent Appnation conference, Nielsen reported an average of 41 apps per smartphone among U.S. smartphone subscribers.4 State the population for this survey, the statistic, and some likely values from the population distribution.
Business employees. There are more than 7 million businesses in the United States with paid employees. The mean number of employees in these businesses is about 16. A university selects a random sample of 100 businesses in Colorado and finds that they average about 11 employees. Is each of the bold
What is wrong? Explain what is wrong in each of the following statements.(a) The central limit theorem states that for large n, the population mean m is approximately Normal.(b) For large n, the distribution of observed values will be approximately Normal.(c) For sufficiently large n, the
What is wrong? Explain what is wrong in each of the following statements.(a) If the population standard deviation is 20, then the standard deviation of x for an SRS of 10 observations will be 20y10 5 2.(b) When taking SRSs from a large population, larger sample sizes will result in larger standard
Find a probability. Refer to Example 6.8. Find the probability that the mean time between text messages is less than 16 minutes. The exact probability is 0.6944. Compare your answer with the exact one.
Use the Central Limit Theorem applet again. Refer to the previous exercise.In the setting of Example 6.6, let’s approximate the sampling distribution for samples of size n 5 2, 10, and 25 observations.(a) For each sample size, compute the mean and standard deviation of x.(b) For each sample size,
Use the Central Limit Theorem applet. Let’s consider the uniform distribution between 0 and 10. For this distribution, all intervals of the same length between 0 and 10 are equally likely. This distribution has a mean of 5 and standard deviation of 2.89.(a) Approximate the population distribution
The effect of increasing the sample size. In the setting of Exercise 6.4, suppose that we increase the sample size to 1225. Use the 95 part of the 68–95–99.7 rule to describe the variability of this sample mean. Compare your results with those you found in Exercise 6.4.
Use the 68–95–99.7 rule. You take an SRS of size 49 from a population with mean 185 and standard deviation 70. According to the central limit theorem, what is the approximate sampling distribution of the sample mean? Use the 95 part of the 68–95–99.7 rule to describe the variability of x.
The effect of increasing the sample size. In the setting of the previous exercise, repeat the calculations for a sample size of 441. Explain the effect of the sample size increase on the mean and standard deviation of the sampling distribution.
Find the mean and the standard deviation of the sampling distribution.Compute the mean and standard deviation of the sampling distribution of the sample mean when you plan to take an SRS of size 49 from a population with mean 420 and standard deviation 21.
Number of apps on a smartphone. AppsFire is a service that shares the names of the apps on an iOS device with everyone else using the service. This, in a sense, creates an iOS device app recommendation system. Recently, the service drew a sample of 1000 AppsFire users and reported a median of 108
More about inventory control. Refer to the previous exercise. In practice, the amount of inventory held on the shelf during the lead time is known as the reorder point. Firms use the term service level to indicate the percentage of the time that the amount of inventory is sufficient to meet demand
Inventory control. OfficeShop experiences a one-week order time to restock its HP printer cartridges. During this reorder time, also known as lead time, OfficeShop wants to ensure a high level of customer service by not running out of cartridges. Suppose the average lead time demand for a
Airline overbooking. Airlines regularly overbook flights to compensate for no-show passengers. In doing so, airlines are balancing the risk of having to compensate bumped passengers against lost revenue associated with empty seats. Historically, no-show rates in the airline industry range from 10
Poisson distribution? Suppose you find in your spam folder an average of two spam emails every 10 minutes. Furthermore, you find that the rate of spam mail from midnight to 6 a.m. is twice the rate during other parts of the day. Explain whether or not the Poisson distribution is an appropriate
Binomial distribution? Suppose a manufacturing colleague tells you that 1% of items produced in first shift are defective, while 1.5% in second shift are defective and 2% in third shift are defective. He notes that the number of items produced is approximately the same from shift to shift, which
Six Sigma. Six Sigma is a quality improvement strategy that strives to identify and remove the causes of defects. Processes that operate with Six-Sigma quality produce defects at a level of 3.4 defects per million. Suppose 10,000 independent items are produced from a Six-Sigma process. What is the
Is this coin balanced? While he was a prisoner of the Germans during World War II, John Kerrich tossed a coin 10,000 times. He got 5067 heads. Take Kerrich’s tosses to be an SRS from the population of all possible tosses of his coin. If the coin is perfectly balanced, p = 0.5. Is there reason to
Leaking gas tanks. Leakage from underground gasoline tanks at service stations can damage the environment. It is estimated that 25% of these tanks leak. You examine 15 tanks chosen at random, independently of each other.(a) What is the mean number of leaking tanks in such samples of 15?(b) What is
Environmental credits. An opinion poll asks an SRS of 500 adults whether they favor tax credits for companies that demonstrate a commitment to preserving the environment. Suppose that, in fact, 45%of the population favor this idea. What is the probability that more than half of the sample are in
Benford’s law, continued Benford’s law suggests that the proportion of legitimate invoices with a first digit of 1, 2, or 3 is much greater than if the digits were distributed as equally likely outcomes. As a fraud investigator, you would be suspicious of some potential wrongdoing if the count
Wi-fi interruptions. Refer to Example 5.16(page 268) in which we were told that the mean number of wi-fi interruptions per day is 0.9. We also found in Example 5.16 that the probability of no interruptions on a given day is 0.4066.(a) Treating each day as a trial in a binomial setting, use the
Benford’s law. We learned in Chapter 4 that there is a striking fact that the first digits of numbers in legitimate records often follow a distribution known as Benford’s law. Here it is:First digit 1 2 3 4 5 6 7 8 9 Proportion 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046(a) What is
Bias and variability. Figure 5.15 shows histograms of four sampling distributions of statistics intended to estimate the same parameter. Label each distribution relative to the others as high or low bias and as high or low variability.
Describe the population and the sample. For each of the following situations, describe the population and the sample.(a) A survey of 17,096 students in U.S. four-year colleges reported that 19.4% were binge drinkers.(b) In a study of work stress, 100 restaurant workers were asked about the impact
What’s wrong? State what is wrong in each of the following scenarios.(a) A parameter describes a sample.(b) Bias and variability are two names for the same thing.(c) Large samples are always better than small samples.(d) A sampling distribution is something generated by a computer.
The effect of the sample size. Refer to Exercise 5.63, where you simulated the sampling distribution of the mean of two uniform variables, and Exercise 5.65, where you simulated the sampling distribution of the mean of 12 uniform variables.(a) Based on what you know about the effect of the sample
Is it unbiased? A statistic has a sampling distribution that is somewhat skewed. The median is 5 and the quartiles are 2 and 10. The mean is 8.(a) If the population parameter is 5, is the estimator unbiased?(b) If the population parameter is 10, is the estimator unbiased?(c) If the population
Normal distributions. Many software packages generate standard Normal variables by taking the sum of 12 uniform variables and subtracting 6.(a) Simulate 1000 random values using this method.(b) Use numerical and graphical summaries to assess how well the distribution of the 1000 values approximates
Increase the number of simulations. Refer to the previous exercise and to Exercise 5.64. Use 500 simulations to study the sampling distribution of the mean of a sample of size 12 from a uniform distribution. Write a summary of what you have found.
Change the sample size to 12. Refer to Exercise 5.63. Change the sample size to 12 and answer parts (a) through (c) of that exercise. Note that the theoretical mean of the sampling distribution is still 0.5 but the standard deviation is the square root of 1•144 or, simply, 1•12. Investigate how
What is the effect of increasing the number of simulations? Refer to the previous exercise.Increase the number of simulations from 100 to 500.Compare your results with those you found in the previous exercise. Write a report summarizing your findings. Include a comparison with the results from the
Simulate a sampling distribution. In Exercise 1.72(page 41) and Example 4.26 (pages 213–214), you examined the density curve for a uniform distribution ranging from 0 to 1. The population mean for this uniform distribution is 0.5 and the population variance is 1•12. Let’s simulate taking
Simulate a sampling distribution for p⁄ . In the previous exercise, you were asked to use statistical software’s capability to generate Poisson counts. Here, you will use software to generate binomial counts from the B(n, p) distribution. We can use this fact to simulate the sampling
Simulating Poisson counts. Most statistical software packages can randomly generate Poisson counts for a given m. In this exercise, you will generate 1000 Poisson counts for m = 9.• JMP users: With a new data table, right-click on header of Column 1 and choose Column Info. In the dragdown dialog
What population and sample? Twenty fourthyear students from your college who are majoring in English are randomly selected to be on a committee to evaluate changes in the statistics requirement for the major. There are 76 fourth-year English majors at your college. The current rules say that a
How much less spread? Refer to Example 5.24 in which we showed the sampling distributions of p⁄for n = 100 and n = 2500 with p = 0.60 in both cases.(a) In terms of a multiple, how much larger is the standard deviation of the sampling distribution for n = 100 versus n = 2500 when p = 0.60?(b) Show
Web polls. If you connect to the website boston.cbslocal.com/wbz-daily-poll, you will be given the opportunity to give your opinion about a different question of public interest each day. Can you apply the ideas about populations and samples that we have just discussed to this poll? Explain why or
Sexual harassment of college students. A recent survey of undergraduate college students reports that 62% of female college students and 61% of male college students say they have encountered some type of sexual harassment at their college.16 Describe the samples and the populations for the survey.
Baseball runs scored. We found in Example 5.20 (pages 271–272) that, in soccer, goal scoring is well described by the Poisson model. It will be interesting to investigate if that phenomenon carries over to other sports. Consider data on the number of runs scored per game by the Washington
Calculator convenience. Suppose that X follows a Poisson distribution with mean m.(a) Show that P(X = 0) = e−m.(b) Show that P(X = k) =m kP(X = k − 1) for any whole number k ≥ 1.(c) Suppose m = 3. Use part (a) to compute P(X = 0).(d) Part (b) gives us a nice calculator convenience that allows
Mishandled baggage. In the airline industry, the term “mishandled baggage’’ refers to baggage that was lost, delayed, damaged, or stolen. In 2013, American Airlines had an average of 3.02 mishandled baggage per 1000 passengers.14 Consider an incoming American Airlines flight carrying 400
Website hits, continued. Refer to the previous exercise to determine the number of website hits in one hour. Use the Normal distribution to find the range in which we would expect 99.7% of the hits to fall.
Website hits. A “hit’’ for a website is a request for a file from the website’s server computer. Some popular websites have thousands of hits per minute.One popular website boasts an average of 6500 hits per minute between the hours of 9 a.m. and 6 p.m. Assume that the hits per hour are
How many zeroes expected? Refer to Example 5.21 (page 272). We would find 1099 of the observed counts to have a value of 0. Based on the provided information in the example, how many more observed zeroes are there in the data set than what the best-fitting Poisson model would expect?
Initial public offerings. The number of companies making their initial public offering of stock(IPO) can be modeled by a Poisson distribution with a mean of 15 per month.(a) What is the probability of three or fewer IPOs in a month?(b) What is the probability of 10 or fewer in a two-month
Email, continued. Refer to Exercise 5.46, where we learned that a particular employee at your company receives an average of five emails per hour.(a) What is the distribution of the number of emails over the course of an eight-hour day?(b) What is the probability of receiving 50 or more emails
Flaws in carpets. Flaws in carpet material follow the Poisson model with mean 0.8 flaw per square yard. Suppose an inspector examines a sample of carpeting measuring 1.25 yards by 1.5 yards.(a) What is the distribution for the number of flaws in the sample carpeting?(b) What is the probability that
Traffic model. The number of vehicles passing a particular mile marker during 15-minute units of time can be modeled as a Poisson random variable. Counting devices show that the average number of vehicles passing the mile marker every 15 minutes is 48.7.(a) What is the probability of 50 or more
Email. Suppose the average number of emails received by a particular employee at your company is five emails per hour. Suppose the count of emails received can be adequately modeled as a Poisson random variable. Compute the following probabilities without the aid of software.(a) What is the
EPL goals. Refer to Example 5.20(pages 271–272) in which we found that the total number of goals scored in a game is well modeled by the Poisson distribution. Compute the following probabilities without the aid of software.(a) What is the probability that a game will end in a 0 – 0 tie?(b) What
How many calls? Calls to the customer service department of a cable TV provider are made randomly and independently at a rate of 11 per minute. The company has a staff of 20 customer service specialists who handle all the calls. Assume that none of the specialists are on a call at this moment and
A safety initiative. This year, a “safety culture change’’ initiative attempts to reduce the number of accidents at the plant described in the previous exercise.There are 60 reportable accidents during the year. Suppose that the Poisson distribution of the previous exercise continues to
Industrial accidents. A large manufacturing plant has averaged seven“reportable accidents’’ per month. Suppose that accident counts over time follow a Poisson distribution with mean seven per month.(a) What is the probability of exactly seven accidents in a month?(b) What is the probability
Number of wi-fi interruptions. Refer to Example 5.16. What is the probability of having at least one wi-fi interruption on any given day?
ATM customers. Refer to Example 5.17. Use the Poisson model to compute the probability that four or fewer customers will use the ATM machine during any given hour between 9 a.m. and 5 p.m.
The geometric distribution. Generalize your work in Exercise 5.38. You have independent trials, each resulting in a success or a failure. The probability of a success is p on each trial. The binomial distribution describes the count of successes in a fixed number of trials. Now, the number of
Tossing a die. You are tossing a balanced die that has probability 1/6 of coming up 1 on each toss.Tosses are independent. We are interested in how long we must wait to get the first 1.(a) The probability of a 1 on the first toss is 1/6. What is the probability that the first toss is not a 1 and
Does your vote matter? Consider a common situation in which a vote takes place among a group of people and the winning result is associated with having one vote greater than the losing result. For example, if a management board of 11 members votes Yes or No on a particular issue, then minimally a
Show that these facts are true. Use the definition of binomial coefficients to show that each of the following facts is true. Then restate each fact in words in terms of the number of ways that k successes can be distributed among n observations.(a) 1n n2 5 1 for any whole number n $ 1.(b) 1n 02 5
Checking for survey errors. One way of checking the effect of undercoverage, nonresponse, and other sources of error in a sample survey is to compare the sample with known facts about the population.About 13% of American adults are black. The number X of blacks in a random sample of 1500 adults
Are we shipping on time? Your mail-order company advertises that it ships 90% of its orders within three working days. You select an SRS of 100 of the 5000 orders received in the past week for an audit. The audit reveals that 86 of these orders were shipped on time.(a) If the company really ships
Multiple-choice tests. Here is a simple probability model for multiple-choice tests. Suppose that each student has probability p of correctly answering a question chosen at random from a universe of possible questions. (A strong student has a higher p than a weak student.) The correctness of an
Finding P(X 5 k). In Example 5.5, we found P(X 5 10) 5 0.106959 when X has a B(150, 0.08)distribution. Suppose we wish to find P(X 5 10)using the Normal approximation.(a) What is the value for P(X 5 10) if the Normal approximation is used without continuity correction?(b) What is the value for P(X
Shooting free throws. Since the mid-1960s, the overall free throw percent at all college levels, for both men and women, has remained pretty consistent. For men, players have been successful on roughly 69% of these free throws, with the season percent never falling below 67% or above 70%.10 Assume
Online learning. Recently, the U.S. Department of Education released a report on online learning stating that blended instruction, a combination of conventional face-to-face and online instruction, appears more effective in terms of student performance than conventional teaching.9 You decide to
Random digits. Each entry in a table of random digits like Table B has probability 0.1 of being a 0, and digits are independent of each other.(a) Suppose you want to determine the probability of getting at least one 0 in a group of five digits. Explain what is wrong with the logic of computing it
Internet video postings. Suppose (as is roughly true) about 30% of all adult Internet users have posted videos online. A sample survey interviews an SRS of 1555 Internet users.(a) What is the actual distribution of the number X in the sample who have posted videos online?(b) Use software to find
More on paying for music downloads.Consider the settings of Exercises 5.24 and 5.26.(a) Using the 75% rate of the Canadian teenagers, what is the smallest number m out of n 5 15 U.S. teenagers such that P(X # m) is no larger than 0.05? You might consider m or fewer students as evidence that the
Paying for music downloads, continued. Refer to Exercise 5.24. Suppose that only 60% of the U.S.teenagers used a fee-based website to download music.(a) If you interview 15 U.S. teenagers at random, what is the mean of the count X who used a fee-based website to download music? What is the mean of
Getting to work. Many U.S. cities are investing and encouraging a shift of commuters toward the use of public transportation or other modes of non-auto commuting. Among the 10 largest U.S. cities, New York City and Philadelphia have the two highest percentages of non-auto commuters at 73% and 41%,
Paying for music downloads. A survey of Canadian teens aged 12 to 17 years reported that roughly 75% of them used a fee-based website to download music.7 You decide to interview a random sample of 15 U.S. teenagers. For now, assume that they behave similarly to the Canadian teenagers.(a) What is
Random stock prices. As noted in Example 5.4(a)(page 246), the S&P 500 index has a probability 0.56 of increasing in any week. Moreover, the change in the index in any given week is not influenced by whether it rose or fell in earlier weeks. Let X be the number of weeks among the next five weeks in
Checking smartphone. A 2014 Bank of America survey of U.S. adults who own smartphones found that 35% of the respondents check their phones at least once an hour for each hour during the waking hours.6 Such smartphone owners are classified as “constant checkers.’’ Suppose you were to draw a
Should you use the binomial distribution?In each of the following situations, is it reasonable to use a binomial distribution for the random variable X?Give reasons for your answer in each case. If a binomial distribution applies, give the values of n and p.(a) A poll of 200 college students asks
Should you use the binomial distribution? In each of the following situations, is it reasonable to use a binomial distribution for the random variable X?Give reasons for your answer in each case. If a binomial distribution applies, give the values of n and p.(a) In a random sample of 20 students in
What is wrong? Explain what is wrong in each of the following scenarios.(a) If you toss a fair coin four times and a head appears each time, then the next toss is more likely to be a tail than a head.(b) If you toss a fair coin four times and observe the pattern HTHT, then the next toss is more
What is wrong? Explain what is wrong in each of the following scenarios.(a) In the binomial setting, X is a proportion.(b) The variance for a binomial count is Ïps1 2 pdyn.(c) The Normal approximation to the binomial distribution is always accurate when n is greater than 1000.(d) We can use the
The effect of sample size. The SRS of size 200 described in the previous exercise finds that 100 of the 200 respondents are concerned about nutrition. We wonder if this is reason to conclude that the percent in your area is higher than the national 40%.(a) Find the probability that X is 100 or
Restaurant survey. Return to the survey described in Exercise 5.8 (page 250).You plan to use random digit dialing to contact an SRS of 200 households by telephone rather than just 20.(a) What are the mean and standard deviation of the number of nutritionconscious people in your sample if p 5 0.4 is

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