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probability statistics
The Practice Of Statistics For Business And Economics 3rd Edition David S. Moore, George P. McCabe, Layth C. Alwan, Bruce A. Craig, William M. Duckworth - Solutions
Vitamin C and colds. In a study of the suggestion that taking vitamin C will prevent colds, 400 subjects are assigned at random to one of two groups. The experimental group takes a vitamin C tablet daily, while the control group takes a placebo. At the end of the experiment, the researchers
Significance tests. Which of the following questions does a test of significance answer?(a) Is the sample or experiment properly designed?(b) Is the observed effect due to chance?(c) Is the observed effect important?
Discuss these views. State whether or not you agree with each of the following statements, and provide a short summary of the reasons for your answers.(a) If the P-value is not less than 0.05, the null hypothesis is proven.(b) Practical significance is not the same as statistical significance.(c)
Find some journal articles. Find two journal articles that report results with statistical analyses. For each article, summarize how the results are reported and write a critique of the presentation.Be sure to include details regarding use of significance testing at a particular level of
What do you know? A research report described two results that both achieved statistical significance at the 5% level.The P-value for the first is 0.049; for the second it is 0.00002. Do the P-values add any useful information beyond that conveyed by the statement that both results are
Your role on a team. You are the statistical expert on a team that is planning a study. After you have made a careful presentation of the mechanics of significance testing, one of the team members suggests using α = 0.50 for the study because you would be more likely to obtain statistically
Predicting success of trainees. What distinguishes managerial trainees who eventually become executives from those who, after expensive training, don’t succeed and leave the company? We have abundant data on past trainees—data on their personalities and goals, their college preparation and
Give an example. Give an example of a set of data for which statistical inference is not valid.
How far do rich parents take us? How much education children get is strongly associated with the wealth and social status of their parents. In social science jargon, this is “socioeconomic status,” or SES. But the SES of parents has little influence on whether children who have graduated from
Is it significant? More than 200,000 people worldwide take the GMAT examination each year as they apply for MBA programs. Their scores vary Normally with mean about μ = 525 and standard deviation about σ = 100. One hundred students go through a rigorous training program designed to raise their
Market pioneers. Market pioneers, companies that are among the first to develop a new product or service, tend to have higher market shares than latecomers to the market. What accounts for this advantage? Here is an excerpt from the conclusions of a study of a sample of 1209 manufacturers of
Cockroaches. Your company is developing a better means to eliminate cockroaches from buildings. In the process, your research and development team studies the absorption of sugar by these insects.10 They feed cockroaches a diet containing measured amounts of a particular sugar. After 10 hours, the
Clothing for runners. Your company sells exercise clothing and equipment on the Internet. To design the clothing, you collect data on the physical characteristics of your different types of customers. Here are the weights for a sample of 24 male runners.Assume that these runners can be viewed as a
Radon. Radon is a colorless, odorless gas that is naturally released by rocks and soils and may concentrate in tightly closed houses. Because radon is slightly radioactive, there is some concern that it may be a health hazard. Radon detectors are sold to homeowners worried about this risk, but the
Check the P-value. Between what critical values from Table D does the P-value for the outcome z = −1.27 in Exercise 6.71 lie? Calculate the P-value using Table A, and verify that it lies between the values you found from Table D.
What values will lead to rejection of H0? You will perform a significance test of H0: μ = 0 versus Ha: μ > 0.(a) What values of z would lead you to reject H0 at the 5% level?(b) If the alternative hypothesis was Ha: μ = 0, what values of z would lead you to reject H0 at the 5% level?(c) Explain
Find a P-value. You have performed a one-sided test of significance and obtained a value of z = 0.35. Use Table D to find the approximate P-value for this test.
Use Table D. You have performed a two-sided test of significance and obtained a value of z = 3.1. Use Table D to find the approximate P-value for this test.
Why is it significant at the 5% level? Explain in plain language why a significance test that is significant at the 1% level must always be significant at the 5% level.
Academic probation and TV watching. There are other z statistics that we have not yet met. We can use Table D to assess the significance of any z statistic. A study compares the habits of students who are on academic probation with students whose grades are satisfactory. One variable measured is
Cigarettes. According to data from the Tobacco Institute Testing Laboratory, Camel Lights King Size cigarettes contain an average of 1.4 milligrams of nicotine. An advocacy group commissions an independent test to see if the mean nicotine content is higher than the industry laboratory claims.(a)
Corn yield. The mean yield of corn in the United States is about 135 bushels per acre. A survey of 50 farmers this year gives a sample mean yield of x = 138.4 bushels per acre. We want to know whether this is good evidence that the national mean this year is not 135 bushels per acre. Assume that
Who is the author? Statistics can help decide the authorship of literary works. Sonnets by a certain Elizabethan poet are known to contain an average of μ = 6.9 new words (words not used in the poet’s other works). The standard deviation of the number of new words is σ = 2.7. Now a manuscript
Financial aid. The financial aid office of a university asks a sample of students about their employment and earnings. The report says that “for academic year earnings, a significant difference(P = 0.038) was found between the sexes, with men earning more on the average. No difference (P = 0.476)
Exercise and statistics exams. A study examined whether exercise affects howstudents perform on their final exam in statistics.The P-value was given as 0.68.(a) State null and alternative hypotheses that could be used for this study. (Note that there is more than one correct answer.)(b) Do you
Handshakes and getting a job. When building business relationships or forming new friendships, first impressions are regarded as crucial. One of those first-impression opportunities is the handshake. In the sales arena, it is conventional wisdom that a firm handshake makes for a good first
Responding to customer complaints. Maritz is a leading marketing research firm (www.maritz.com) that conducts marketing analysis for various industries worldwide. In a 2004 study, Maritz conducted a survey of new-vehicle buyers. The focus of the study was on customer complaint resolution and
Hypotheses. Translate each of the following research questions into appropriate H0 and Ha.(a) Census Bureau data show that the mean household income in the area served by a shopping mall is $62,500 per year. A market research firm questions shoppers at the mall to find out whether the mean
Hypotheses. In each of the following situations, state an appropriate null hypothesis H0 and alternative hypothesis Ha. Be sure to identify the parameters that you use to state the hypotheses.(We have not yet learned how to test these hypotheses.)(a) A sociologist asks a large sample of high school
Hypotheses. In each of the following situations, a significance test for a population mean μ is called for. State the null hypothesis H0 and the alternative hypothesis Ha in each case.(a) A university gives credit in French language courses to students who pass a placement test. The language
Hypotheses. Each of the following situations requires a significance test about a population mean μ. State the appropriate null hypothesis H0 and alternative hypothesis Ha in each case.(a) Larry’s car averages 31 miles per gallon on the highway. He now switches to a new motor oil that is
Compare student credit card debt for different age groups. The credit card study in the previous exercise examined the amount of debt held by different age groups. It was found that the mean credit card debt for graduate students in the United States aged 22 to 29 years was $6479, while the debt
Compare student credit card debt for different regions.Nellie Mae, a subsidiary of Sallie Mae, is a national student loan company providing loans to undergraduate and graduate students.Nellie Mae conducts statistical studies to estimate various types of debt in the student population. One such
What’s wrong? Here are several situations where there is an incorrect application of the ideas presented in this section.Write a short paragraph explaining what is wrong in each situation and why it is wrong.(a) A change is made that should improve student satisfaction with the way grades are
What’s wrong? Here are several situations where there is an incorrect application of the ideas presented in this section.Write a short paragraph explaining what is wrong in each situation and why it is wrong.(a) A fleet manager for a trucking company wants to test the null hypothesis that the
Suppose the P-value is a little higher? The P-value for a significance test is 0.051.(a) Do you reject the null hypothesis at level α = 0.05?(b) Do you reject the null hypothesis at level α = 0.01?(c) Explain your answers.
Do you reject the null hypothesis? The P-value for a significance test is 0.049.(a) Do you reject the null hypothesis at level α = 0.05?(b) Do you reject the null hypothesis at level α = 0.01?(c) Explain your answers.
Can you reject the null hypothesis? A95% confidence interval for a population mean is (16, 32).(a) Can you reject the null hypothesis that μ = 34 at the 5% significance level? Why?(b) Can you reject the null hypothesis that μ = 17 at the 5% significance level? Why?
Does the confidence interval include μ0? The P-value for a two-sided test of the null hypothesis H0: μ = 15 is 0.07.(a) Does the 95% confidence interval include the value 15? Why?(b) Does the 90% confidence interval include the value 15? Why?
A new supplier. A new supplier offers a good price on a catalyst used in your production process. You compare the purity of this catalyst with that from your current supplier. The P-value for a test of “no difference” is 0.27. Can you be confident that the purity of the new product is the same
The effects of different alternative hypotheses. A test of the null hypothesis H0: μ = μ0 gives test statistic z = 1.9.(a) What is the P-value if the alternative is Ha: μ > μ0?(b) What is the P-value if the alternative is Ha: μ < μ0?(c) What is the P-value if the alternative is Ha: μ = μ0?
Testing a random number generator. Statistical software has a “random number generator” that is supposed to produce numbers uniformly distributed between 0 to 1. If this is true, the numbers generated come from a population with μ = 0.5.A command to generate 100 random numbers gives outcomes
The Supreme Court speaks. Court cases in such areas as employment discrimination often involve statistical evidence. The Supreme Court has said that z-scores beyond z∗ = 2 or 3 are generally convincing statistical evidence. For a two-sided test, what significance level corresponds to z∗ = 2? To
Significance. You are testing H0: μ = 0 against Ha: μ > 0 based on an SRS of 30 observations from a Normal population. What values of the z statistic are statistically significant at the α = 0.005 level?
Significance. You are testing H0: μ = 0 against Ha: μ = 0 based on an SRS of 30 observations from a Normal population. What values of the z statistic are statistically significant at the α = 0.005 level?
How to show that you are rich. Every society has its own marks of wealth and prestige. In ancient China, it appears that owning pigswas such a mark. Evidence comes from examining burial sites. If the skulls of sacrificed pigs tend to appear along with expensive ornaments, that suggests that the
Why is this wrong? The homebuilders wonder if the national finding applies in the Cleveland area. They have no idea whether Cleveland residents spend more or less than the national average. Because their interviews find that x = 28.6%, less than the national 31%, their analyst tests H0: μ = 31%Ha:
State null and alternative hypotheses. In the setting of the previous exercise, suppose that the Cleveland homebuilders were convinced, before interviewing their sample, that residents of Cleveland spend less than the national average on housing.Do the interviews support their conviction? State
Spending on housing. The Census Bureau reports that households spend an average of 31% of their total spending on housing. A homebuilders association in Cleveland wonders if the national finding applies in their area. They interview a sample of 40 households in the Cleveland metropolitan area to
DEXA scanners. A dual-energy X-ray absorptiometry (DEXA) scanner is used to measure bone mineral density for people who may be at risk for osteoporosis. Customers want assurance that your company’s latest-model DEXA scanner is accurate.You therefore supply an object called a “phantom” that
Customer feedback. Feedback from your customers shows that many think it takes too long to fill out the online order form for your products. You redesign the form and plan a survey of customers to determine whether or not they think that the new form is actually an improvement. Sampled customers
An outlier. Exercise 6.24 gives the weights of a sample of 24 male runners. Suppose that the sample actually contained 25 runners. The extra runner claimed to weigh 92.3 kg.(a) Compute the 95% confidence interval for the mean with this observation included.(b)Would you report the interval you
Talk show poll. A radio talk show invites listeners to enter a dispute about a proposed pay increase for city council members. “What yearly pay do you think council members should get? Call us with your number.” In all, 958 people call.The station calculates the 95% confidence interval for the
Manager trainee wages. A newspaper ad for a manager trainee position contained the statement “Our manager trainees have a first-year earnings average of $20,000 to $30,000.” Do you think that the ad is describing a confidence interval? Explain your answer.
An election poll. A newspaper headline describing a poll of registered voters taken two weeks before a recent election reads “Ringel leads with 52%.” The accompanying article says that the margin of error is 3% with 95% confidence.(a) Explain in plain language to someone who knows no statistics
More than one confidence interval. As we prepare to take a sample and compute a 95% confidence interval, we know that the probability that the interval we compute will cover the parameter is 0.95. That’s the meaning of 95% confidence. If we use several such intervals, however, our confidence that
Other variables. Refer to the previous two exercises.Gallup polls such as this often present the results for subgroups of respondents such as females and males. Suggest subgroups for this job satisfaction question. Give reasons for your answer.
Like or love your job? Refer to the previous exercise. In the same poll, 32% of the respondents said that they loved their job.(a) Combine the respondents who like their job with those who love their job. Give the percent who either love their job or like their job.(b) Assuming that the same margin
Like your job? A Gallup Poll asked 1001 adult workers about their job satisfaction. One question was “All in all, which best describes how you feel about your job?” The possible answers were “love job,” “like job,” “dislike job,” and “hate job.”Fifty-nine percent of the sample
Hotel managers. How large a sample of the hotel managers in Exercise 6.27 would be needed to estimate the mean μwithin ±1 year with 99% confidence?
Supermarket shoppers. Suppose that youwant to perform a study similar to the survey of supermarket shoppers described in Exercise 6.28. If you want the margin of error to be about $5.00, howmany shopperswould you need in your sample? (Use $22.00 for the standard deviation in your calculations.)
Supermarket shoppers. Amarketing consultant observed 40 consecutive shoppers at a supermarket. One variable of interest was how much each shopper spent in the store. Here are the data (in dollars), arranged in increasing order: DATADATA DATADATADATA DATADATADATADATADATA DATADATADATADATADATA
Hotel managers. In a study of the career paths of hotel general managers, questionnaires were sent to an SRS of 160 hotels belonging to major U.S. hotel chains. There were 114 responses.The average time these 114 general managers had spent with their current company was 11.8 years. Give a 99%
99% versus 95% confidence interval. Find a 99% confidence interval for the mean weight μ of the population of male runners in Exercise 6.24. Is the 99% confidence interval wider or narrower than the 95% interval found in Exercise 6.24? Explain in plain language why this is true.
Pounds versus kilograms. Suppose that the weights of the runners in Exercise 6.24 were recorded in pounds rather than kilograms. Use your answers to Exercise 6.24, and the fact that 1 kilogram equals 2.2 pounds, to answer these questions.(a) What is the mean weight of these runners?(b) What is the
Clothing for runners. Your company sells exercise clothing and equipment on the Internet. To design the clothing, you collect data on the physical characteristics of your different types of customers. Here are the weights (in kilograms) for a sample of 24 male runners. Assume that these runners can
Study habits. Aquestionnaire about study habitswas given to a random sample of students taking a large introductory statistics class. The sample of 35 students reported that they spent an average of 115 minutes per week studying statistics. Assume that the standard deviation is 40 minutes.(a) Give
Apartment rental rates. You want to rent an unfurnished one-bedroom apartment for next semester. The mean monthly rent for a random sample of 10 apartments advertised in the local newspaper is $640. Assume that the standard deviation is $90.Find a 95% confidence interval for the mean monthly rent
What is the cost? In Exercise 6.16 you found an estimate with a margin of error for the fuel efficiency expressed in miles per gallon. Suppose that fuel costs $3.10 per gallon. Find the estimate and margin of error for fuel efficiency in terms of miles per dollar. To convert miles per gallon to
Convert to metric. In Exercise 6.16 you found an estimate with a margin of error for the average fuel efficiency expressed in miles per gallon (mpg). Convert your estimate and margin of error to kilometers per liter (kpl). To change mpg to kpl, multiply by 0.4251.
Survey response and margin of error. Suppose that a business conducts a marketing survey. As is often done, the survey is conducted by telephone. As it turns out, the business was only able to illicit responses from less than 10% of the randomly chosen customers. The low response rate is
Confidence intervals for informal sector. The International Labor Organization (an agency of the United Nations)coined the term “informal sector” to refer to economic activities that are not regulated by labor or taxation laws and are not included in gross domestic product estimates. This
Wanting another confidence interval. Suppose you were told that the 90% confidence interval for the mean μ based on some known σ is (329.87, 356.46). You, however, want a 95%confidence interval.With only the information provided here determine the 95% confidence interval.
Fuel efficiency. Computers in some vehicles calculate various quantities related to performance. One of these is the fuel efficiency, or gas mileage, usually expressed as miles per gallon(mpg). One of the authors of this book conducted an experiment with his 2006 Toyota Highlander Hybrid by
In the extremes. As suggested in our discussions, 90%, 95%, and 99% are probably the most common confidence levels chosen in practice.(a) In general, what would be a 100% confidence interval for the mean μ? Explain why such an interval is of no practical use.(b) What would be a 0% confidence
Reporting margins of error. A Wall Street Journal article(“Home Construction at Record Slow Pace,” January 23, 2009) reported Commerce Department estimates of changes in the construction industry:Construction of new single- and multifamily homes sank 15.5% in December from the previous month to
Populations sampled and margins of error. Consider the following two scenarios. (A) Take a simple random sample of 100 sophomore students at your college or university. (B) Take a simple random sample 100 sophomore students in your major at your college or university. For each of these samples you
Change the confidence. A study with 25 observations gave a mean of 70. Assume that the standard deviation is 15.Make a diagram similar to Figure 6.8 (page 343) that illustrates the effect of the confidence level on the width of the interval.Use 80%, 90%, 95%, and 99%. Summarize what the diagram
Change the sample size. Suppose that the sample mean is 50 and the standard deviation is assumed to be 5. Make a diagram similar to Figure 6.7 (page 343) that illustrates the effect of sample size on the width of a 95% interval. Use the following sample sizes: 10, 20, 40, and 100. Summarize what
Margin of error and the confidence interval. A study based on a sample of size 25 reported a mean of 76 with a margin of error of 12 for 95% confidence.(a) Give the 95% confidence interval.(b) If you wanted 99% confidence for the same study, would your margin of error be greater than, equal to, or
Confidence interval mistakes and misunderstandings. Consider the following scenario. Suppose 100 randomly selected customers were asked to rate a particular service they recently had on a 0 to 10 scale. The sample mean (x) was found to be 7.3.Assume that the population standard deviation is known
Will the sample size needed be larger or smaller? Suppose that in the setting of the previous exercise you are willing to settle for a margin of error of $1000. Will the required sample size be larger or smaller? Verify your answer by performing the calculations.
Starting salaries. You are planning a survey of starting salaries for recent business major graduates from your college. From a pilot study you estimate that the standard deviation is about $9000. What sample size do you need to have a margin of error equal to $500 with 95% confidence?
Is the margin of error larger or smaller? In the setting of the previous exercise, would the margin of error for 99% confidence be larger or smaller? Verify your answer by performing the calculations.
Health insurance. For the 453 large firms in the Kaiser health coverage survey(Example 6.5), the mean monthly cost of the premium for single-coverage HMO plans perworkerwas $405.02. The sample standard deviation of premium costswas $112.08.Assume that the sample standard deviation can be used in
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
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 μ. Fill in the blank.
Company invoices. The mean amount μ 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 $220. If you take a random sample of 100 invoices, what is the value of the standard
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. In a USA Today analysis of airline statistics, it was found that
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
Life satisfaction. Refer to Example 5.19 (page 313).We learned that the probability that at least 610 out of 1010 randomly selected adults would be “very satisfied” with their lives is 0.4119.(a) In Excel, fill the first column of the spreadsheet with the numbers 610, . . . , 1010. In the
Poisson distribution? Suppose you find in your spam folder an average of 2 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 model
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
Who is driving? A sociology professor asks her class to observe cars having a man and a woman in the front seat and record which of the two is the driver.(a) Explain why it is reasonable to use the binomial distribution for the number of male drivers in n cars if all observations are made in the
Is this coin balanced? While he was a prisoner of the Germans duringWorldWar II, John Kerrich tossed a coin 10,000 times. He got 5067 heads.TakeKerrich’s tosses to be anSRSfrom the population of all possible tosses of his coin. If the coin is perfectly balanced, p = 0.5. Is there reason to think
Computer training. Macintosh users make up about 5%of all computer users. A computer training school that wants to attract Macintosh users mails an advertising flyer to 25,000 computer users.(a) If the mailing list can be considered a random sample of the population, what is the mean number of
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
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