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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
Wheat prices revisited. Example 7.13 (page 424) gives summary statistics for prices received by wheat producers in September and July. The two sample standard deviations are very similar, so we may be willing to assume equal population standard deviations. Calculate the pooled t test statistic and
Can you do better? Design your own output for a two-sample t analysis and illustrate it with the results given in Figure 7.8. Pay particular attention to the use of labels and how numbers are rounded.
Compare the outputs. Figure 7.8 gives the outputs from five software systems for the two-sample t procedures. Make tables to compare the outputs, giving the numerical values reported by the software.(a) How do they report the means?(b) How do they report variability for each of the groups?(c) How
Calculating the Satterthwaite df. The SAS output in Figure 7.8 gives df = 6.6.Use the sample standard deviations in that output and the Satterthwaite formula to verify this value.
Why randomize? A coworker suggested that you give the new monitors to the next 10 employees who need new computers and the standard monitor to the following 10. Explain why your randomized design in the previous exercise is better.
New computer monitors? The purchasing department has suggested that all new computer monitors for your company should be a new style. You want data to be assured that employees will like these new monitors. You designate the next 20 employees needing a new computer to participate in an experiment
Changing the experiment. Suppose you know that online sales differ across days of the week. To compare twoWeb page designs, you choose two successive weeks in the middle of a month. You flip a coin to assign one Monday to the first design and the other Monday to the second. You repeat this for each
Comparison of twoWeb designs. Consider the previous problem. If the t statistic for comparing the daily sales were 1.93, what P-value would you report? What would you conclude using α = 0.05?
Selling on the Internet. You want to compare the daily sales for two different designs ofWeb pages for your Internet business. You assign the next 90 days to either Design A or Design B, 45 days to each.(a) Would you use a one-sided or two-sided significance test for this problem?Explain your
Sign test for fuel efficiency comparison. Use the sign test to assess whether the computer calculates a higher mpg than the driver in Exercise 7.45. State the hypotheses, give the P-value using the binomial table (Table C), and report your conclusion.
Sign test for potential insurance fraud. The differences in the repair estimates in Exercise 7.44 can also be analyzed using a sign test. Set up the appropriate null and alternative hypotheses, carry out the test, and summarize the results.How do these results compare with those that you obtained
Learning Spanish. Use the sign test to assess whether the intensive language training of Exercise 7.46 improves Spanish listening skills. State the hypotheses, give the P-value using the binomial table (Table C), and report your conclusion
Design of controls. Apply the sign test to the data in Exercise 7.42 to assess whether the subjects can complete a task with a right-hand thread significantly faster than with a left-hand thread.(a) State the hypotheses two ways, in terms of a population median and in terms of the probability of
A field trial. The tomato experts who carried out the field trial described in Exercise 7.47 suspect that the relative lack of significance is due to low power. They would like to be able to detect a mean difference in yields of 0.6 pound per plant at the 0.05 significance level. Based on the
Credit card fees. The bank in Exercise 7.40 tested a new idea on a sample of 100 customers. Suppose that the bank wanted to be quite certain of detecting a mean increase ofμ = $300 in the credit card amount charged, at the α = 0.01 significance level.Perhaps a sample of only n = 60 customers
Areas of continents. While browsing the Internet, you find a site that gives the land area (in square kilometers) for Africa, Antarctica, Asia, Australia, Europe, North America, and South America. You feed the data into your statistical software package and the output produced includes a confidence
Which design? The following situations all require inference about a mean or means. Identify each as (1) a single sample,(2) matched pairs, or (3) two independent samples. The procedures of this section apply to the first two situations. We will learn procedures for the third in the next section.
Which design? The following situations all require inference about a mean or means. Identify each as (1) a single sample,(2) matched pairs, or (3) two independent samples. The procedures of this section apply to the first two situations. We will learn procedures for the third in the next section.
A field trial. An agricultural field trial compares the yield of two varieties of tomatoes for commercial use. The researchers divide in half each of 10 small plots of land in different locations and plant each tomato variety on one half of each plot. After harvest, they compare the yields in
Executives learn Spanish. The table below gives the pretest and posttest scores on the MLA listening test in Spanish for 20 executives who received intensive training in Spanish.12 The setting is identical to the one described in Example 7.7(page 405). DATADATA DATADATADATA DATADATADATADATADATA
Fuel efficiency comparison t test. One of the authors of this book records the mpg of his car each time he fills the tank.He does this by dividing the miles driven since the last fill-up by the amount of gallons at fill-up. He wants to determine if these calculations differ from what his car’s
Potential insurance fraud? Insurance adjusters are concerned about the high estimates they are receiving from Jocko’s Garage. To see if the estimates are unreasonably high, each of 10 damaged cars was taken to Jocko’s and to a “trusted” garage and the estimates recorded. Here are the
Is the difference important? Give a 90% confidence interval for the mean time advantage of right-hand over left-hand threads in the setting of the previous exercise. Do you think that the time saved would be of practical importance if the task were performed many times—for example, by an
Design of controls. The design of controls and instruments has a large effect on how easily people can use them. A student project investigated this effect by asking 25 right-handed students to turn a knob (with their right hands) that moved an indicator by screw action. There were two identical
Truth in advertising. A leading agricultural company advertises that their bags of seeds contain at least a certain number of seeds. An SRS of bags from one warehouse is chosen, and these bags are opened and the number of seeds in each bag is counted. The actual counts are compared with the
Credit card fees. A bank wonders whether omitting the annual credit card fee for customers who charge at least $4000 in a year would increase the amount charged on its credit card. The bank makes this offer to an SRS of 100 of its existing credit card customers. It then compares how much these
A customer satisfaction survey. Many organizations do surveys to determine the satisfaction of their customers. Attitudes toward various aspects of campus life were the subject of one such study conducted at Purdue University. Each question was rated on a 1 to 5 scale, with 5 being the highest
Health care costs. The cost of health care is the subject of many studies that use statistical methods. One such study estimated that the average length of service for home health care among people aged 65 and over who use this type of service is 304 days with a standard error of 25.2 days.
Corn prices. The U.S. Department of Agriculture (USDA)uses sample surveys to obtain important economic estimates. One USDApilot study estimated the price received by farmers for corn sold in January from a sample of 20 farms. The mean price was reported as $3.64 per bushel with a standard error of
The influence of big shoppers. Eliminate the three largest observations and redo parts (a), (b), and (c) of the previous exercise. Do these observations have a large influence on the results?
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
Is all the output helpful? When the data for Exercise 7.32 were fed to a statistical package, the output included a t statistic under the label “Test for Location: Mu0=0.” Explain why this test is meaningless for the IBI water quality data.
to determine a confidence interval for the proportion of Ozark Highland ecoregion streams that have very poor or poor IBI scores? Explain your answer.
Classifying poor-quality streams. Refer to the previous exercise. Streams with IBI scores of less than 20 are classified as“very poor,” while those with scores greater than 20 but less than 40 are classified as “poor.”(a) Count the number of streams in the sample that are classified in
Plant capacity. A leading company chemically treats its product before packaging. The company monitors the weight of product per hour that each machine treats. An SRS of 90 hours of production data for a particular machine is collected. The measured variable is in pounds. DATADATA DATADATADATA
ReRuns R Fun. The largest nonprofit childrenís consignment sale in America is called ReRuns R Fun.7 Consignors to the sale keep 35% of the profits generated by the sale of their items, and the rest goes to a variety of charities. A random sample of n = 4 consignor gross sales records (in dollars)
Significance test for the average number of seeds. Refer to the previous two exercises.(a) Do these data provide evidence that the average number of seeds in a one-pound scoop is greater than 1550? Using a significance level of 5%, state your hypotheses, the P-value, and your conclusion.(b) Do
How many seeds on average? Refer to the previous exercise.(a) Find the mean, the standard deviation, and the standard error of the mean for this sample.(b) If you were to calculate the margin of error for the average number of seeds at 90% and 95% confidence, which would be smaller? Briefly explain
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 Toyota Highlander Hybrid by randomly
Critical values. What critical value t∗ would you use to construct(a) a 90% confidence interval when n = 15?(b) a 95% confidence interval when n = 26?(c) a 99% confidence interval when n = 150?
A final one-sample t test. The one-sample t statistic for testing H0: μ = 20 Ha:μ < 20 based on n = 150 observations has the value t = −3.83.(a) What are the degrees of freedom for this statistic?(b) How would you report the P-value based on Table D?(c) If you have software available, find the
Another one-sample t test. The one-sample t statistic for testing H0: μ = 60 Ha: μ = 60 from a sample of n = 28 observations has the value t = 0.75.(a) What are the degrees of freedom for t?(b) Locate the two critical values t∗ from Table D that bracket t.What are the right-tail probabilities
A one-sample t test. The one-sample t statistic for testing H0: μ = 10 Ha:μ > 10 from a sample of n = 18 observations has the value t = 1.83.(a) What are the degrees of freedom for this statistic?(b) Give the two critical values t∗ from Table D that bracket t.(c) What are the right-tail
Finding critical t∗-values. What critical value t∗ from Table D should be used to calculate the margin of error for a confidence interval for the mean of the population in each of the following situations?(a) A 90% confidence interval based on n = 9 observations.(b) A 90% confidence interval
Finding the critical value t∗. What critical value t∗ from Table D should be used to calculate the margin of error for a confidence interval for the mean of the population in each of the following situations?(a) A 95% confidence interval based on n = 13 observations.(b) A 95% confidence
Sign test for the energy drink comparison. Exercise 7.10 (page 406) gives data on the appeal of two popular energy drinks. Is there evidence that the medians are different? State the hypotheses, carry out the sign test, and report your conclusion.
Another power calculation. Verify your answer to the previous exercise by doing the calculation for the alternative μ = 2.4.
Power for other values of μ. If you repeat the calculation in Example 7.9 for values of μ that are larger than 2, would you expect the power to be higher or lower than 0.8944? Why?
Significance test for the average T-bill interest rate? Consider data on the T-bill interest rate presented in Figure 1.29 (page 56). Would you feel comfortable applying the t procedures in this case? Explain your answer.
Significance test for the average time to start a business? Consider the sample of time data presented in Figure 1.30 (page 57). Would you feel comfortable applying the t procedures in this case? Explain your answer.
95% confidence interval for the difference in energy drinks. For the companies producing these drinks, the real question is how much difference there is between the two preferences. Use the data above to give a 95% confidence interval for the difference in preference between Drink A and Drink B.
Comparison of two energy drinks. Consider the following study to compare two popular energy drinks. Each drink was rated on a 0 to 100 scale, with 100 being the highest rating. For each subject, a coin was tossed to see which drink would be tried first. The drinks were served in identical Styrofoam
Using software to perform a significance test. In Example 7.2(page 400) we tested whether the average household expenditure for landline telephone service in your communitywas different from the U.S. average. Use software to perform this test and verify that you obtain the same P-value.
Using software to compute a confidence interval. In Example 7.1(page 398) we calculated the 95% confidence interval for the average household expenditure for landline telephone service in your community. Use software to compute this interval and verify that you obtain the same interval.
Have sales changed? You will have complete sales information for last month in a week, but right now you have data from a random sample of 40 stores. The mean change in sales in the sample is −3.2%, and the standard deviation of the changes is 12%. Are average sales for all stores different from
Significant? A test of a null hypothesis versus a two-sided alternative gives t = 2.18.(a) The sample size is 16. Is the test result significant at the 5% level? Explain how you obtained your answer.(b) The sample size is 10. Is the test result significant at the 5% level?(c) Sketch the two t
Apartment rents. Refer to Exercise 7.1 (page 397). Do these data give good reason to believe that the average rent for all advertised one-bedroom apartments is greater than $550 per month? Make sure to state the hypotheses, find the t statistic and its P-value, and state your conclusion using the
90% versus 95% confidence interval. If you chose 90%, rather than 95%, confidence in the previous exercise, would your margin of error be larger or smaller?Explain your answer.
More on apartment rents. Refer to Exercise 7.1 (page 397). Construct a 95%confidence interval for the mean monthly rent of all advertised one-bedroom apartments.
Increasing the sample size. Refer to the previous exercise. Suppose that instead of an SRS of 16, you had sampled 25 advertisements. Would the standard error of the mean be larger or smaller in this case? Verify your answer by doing the calculation.
Apartment rents. Your local newspaper contains a large number of advertisements for unfurnished one-bedroom apartments. You choose 16 at random and calculate that their mean monthly rent is $508 and that the standard deviation of their rents is $78.(a) What is the standard error of the mean?(b)
Simulation. In the previous exercise you simulated the attorney fees of 25 SRSs of 100 Chapter 7 consumer bankruptcy cases. Now use these samples to demonstrate the behavior of a significance test. We know that the population standard deviation is σ = 433, and we are willing to assume that the
Simulation. Figure 6.5 (page 339) demonstrates the behavior of a confidence interval in repeated sampling by showing the results of 25 samples from the same population.Now you will do a similar demonstration, though in an artificial setting. Suppose that the attorney fees for Chapter 7 consumer
Simulation. Use a computer to generate n = 5 observations from the Normal distribution N(20, 5). Test H0 : μ = 20 versus Ha : μ = 20 at the α = 0.05 significance level. Repeat this process 100 times and then count the number of times that you reject H0. Explain your results.6.130 Simulation.
Simulation. Use a computer to generate n = 5 observations from the Normal distribution N(20, 5). Find the 95%confidence interval for μ. Repeat this process 100 times and then count the number of times that the confidence interval includes the value μ = 20. Explain your results.
Welfare reform. A study compares two groups of mothers with young children who were on welfare two years ago.One group attended a voluntary training program offered free of charge at a local vocational school and advertised in the local news media. The other group did not choose to attend the
Significant. Another student, when asked why statistical significance appears so often in research reports, says, “Because saying that results are significant tells us that they cannot easily be explained by chance variation alone.” Do you think that this statement is essentially correct?
Significant. When asked to explain the meaning of “statistically significant at the α = 0.05 level,” a student says, “This means there is only probability 0.05 that the null hypothesis is true.” Is this a correct explanation of statistical significance? Explain your answer.
Roulette. A roulette wheel has 18 red slots among its 38 slots. You observe many spins and record the number of times that red occurs. Nowyouwant to use these data to test whether the probability of a red has the value that is correct for a fair roulette wheel. State the hypotheses H0 and Ha that
Large samples. Statisticians prefer large samples. Describe briefly the effect of increasing the size of a sample (or the number of subjects in an experiment) on each of the following.(a) The width of a level C confidence interval.(b) The P-value of a test, when H0 is false and all facts about the
CEO pay. A study of the pay of corporate chief executive officers (CEOs) examined the increase in cash compensation of the CEOs of 104 companies, adjusted for inflation, in a recent year. The mean increase in real compensation was x = 6.9%, and the standard deviation of the increases was s = 55%.
Where do you buy? Consumers can purchase nonprescription medications at food stores, mass merchandise stores such as Kmart and Walmart, or pharmacies. About 45% of consumers make such purchases at pharmacies. What accounts for the popularity of pharmacies, which often charge higher prices?Astudy
Too much cellulose to be profitable? Excess cellulose in alfalfa reduces the “relative feed value” of the product that will be fed to dairy cows. If the cellulose content is too high, the price will be lower and the producer will have less profit. An agronomist examines the cellulose content of
Annual household income. Refer to the previous exercise.Give a 90% confidence interval for the 2007 median weekly household income. Use 52.14 (= 365/7) as the number of weeks in a year.
Annual household income. A government report gives a 90% confidence interval for the 2007 median annual household income as $50, 233 ± $230. This result was calculated by advanced methods from the Current Population Survey, a multistage random sample of about 50,000 households.(a) Would a 95%
Wine. Many food products contain small quantities of substances that would give an undesirable taste or smell if they were present in large amounts. An example is the “off-odors”caused by sulfur compounds in wine. Oenologists (wine experts)have determined the odor threshold, the lowest
Inheriting control and company performance. “The only reason I was on the payroll is because I was the son of the boss,” testified John Tyson, chief executive officer (CEO) of Tyson Food, Inc., in a legal proceeding.16 Promotion of one’s kin in the businessworld is quite common. Showing
Change the confidence level. Refer to Example 6.19(page 364) and construct a 95% confidence interval for the mean change in profitability for the population of IPO firms.
Change the confidence level. Refer to the previous exercise.Dothe simulations and report the results for90%confidence.
Generate some confidence intervals. For this exercise you will use the Confidence Interval applet. Set the confidence level at 95% and click the “Sample” button 10 times to simulate 10 confidence intervals. Record the percent hit.Simulate another 10 intervals by clicking another 10 times (do
Workers’ perceptions about safety. The Safety Climate Index (SCI) measures workers’ perceptions about the safety of their work environment. A study of safe work practices of industrial workers reported mean SCI scores for workers classified by workplace size.15 Here are the means:Workplace
Full-time employment and age. A study reported average months of full-time employment for individuals aged 18 to 26.14 Here are the means:Age: 18 19 20 21 22 23 24 25 26 Months employed: 2.9 4.2 5.0 5.3 6.4 7.4 8.5 8.9 9.3 Assume that the standard deviation for each of these means is 4.5 months and
Inspecting shipments. In Example 6.27 (page 383), we considered the situation where a sample from a shipment of many parts serves as a basis for rejecting or accepting the shipment.Suppose that a sampling test has probability 0.05 of rejecting a good shipment of manufactured parts and probability
A Web-based business. You are in charge of marketing for aWeb site that offers automated medical diagnoses. The program will scan the results of routine medical tests (pulse rate, blood pressure, urinalysis, etc.) and either clear the patient or refer the case to a doctor. You are marketing the
Decide. You must decide which of two discrete distributions a random variable X has. We will call the distributions p0 and p1. Here are the probabilities that the distributions assign to the values x of X:x: 0 1 2 3 4 5 6 p0: 0.1 0.1 0.1 0.1 0.2 0.1 0.3 p1: 0.2 0.1 0.1 0.2 0.2 0.1 0.1 You have a
Decreasing population standard deviation. Improved measurement systems, better technology, and changes to standard operating procedures are among various strategies to reduce population variability in manufacturing and service applications.Suppose variation reduction strategies are implemented and
Profitability of IPO firms. Use the result of Exercise 6.102 to give the probabilities of Type I and Type II errors for the test discussed there.
DecreasingType II error probability. What are four distinct ways to decrease the probability of a Type II error? Explain why these ways will decrease the error probability.
Mail-order catalog sales. You want to see if a redesign of the cover of a mail-order catalog will increase sales. A very large number of customers will receive the original catalog, and a random sample of customers will receive the one with the new cover. For planning purposes, you are willing to
Sample size determination. Example 6.23 (page 377)discusses a test of H0 : μ = 0 against H0 : μ > 0 where μ is the mean percent change in total body bone mineral content of young women. The population standard deviation is given to beσ = 2, and the testing was performed at a 5% significance
Profitability of IPO firms. In Example 6.19 (page 364), we ask if the mean of the profitability of IPO firms has changed during the initial three-year period since becoming public. If μis the mean percent change for the population of IPO firms, the hypotheses are H0: μ = 0 and Ha: μ = 0. The
Power versus a different alternative. A one-sided test of the null hypothesis μ = 50 versus the alternative μ = 70 has power equal to 0.5. Will the power for the alternative μ = 80 be higher or lower than 0.5? Draw a picture and use this to explain your answer.
Explain power and sample size. Two studies are identical in all respects except for the sample sizes. Consider the power versus a particular sample size. Will the study with the larger sample size have more power or less power than the one with the smaller sample size? Explain your answer in terms
Make a recommendation. Your manager has asked you to review a research proposal that includes a section on sample size justification. A careful reading of this section indicates that the power is 20% for detecting an effect that you would consider important. Write a short report for your manager
More than one test. Refer to the previous exercise. A researcher has performed 12 tests of significance and wants to apply the Bonferroni procedure with α = 0.05. The calculated P-values are 0.041, 0.569, 0.050, 0.416, 0.001, 0.004, 0.256, 0.041, 0.888, 0.010, 0.002, and 0.433. Which of these
More than one test. A P-value based on a single test is misleading if you perform several tests. The Bonferroni procedure gives a significance level for several tests together. Level αthen means that if all the null hypotheses are true, the probability is α that any of the tests rejects its null
Extrasensory perception. A researcher looking for evidence of extrasensory perception (ESP) tests 500 subjects. Four of these subjects do significantly better (P < 0.01) than random guessing.(a) Is it proper to conclude that these four people have ESP?Explain your answer.(b) What should the
Student loan poll. A local television station announces a question for a call-in opinion poll on the six o’clock news and then gives the response on the eleven o’clock news. Today’s question concerns a proposed increase in funds for student loans.Of the 2372 calls received, 1921 oppose the
Coaching for the SAT. Give a 99% confidence interval for the mean SATM score μ after coaching in each part of the previous exercise. For large samples, the confidence interval says,“Yes, the mean score is higher after coaching, but only by a small amount.”
Coaching for the SAT. Every user of statistics should understand the distinction between statistical significance and practical importance. A sufficiently large sample will declare very small effects statistically significant. Let us suppose that SAT Mathematics (SATM) scores in the absence of
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