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The Practice Of Statistics For Business And Economics 4th Edition Layth C. Alwan, Bruce A. Craig - Solutions
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. Now you want 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 5 6.9%, and the standard deviation of the increases was s 5 55%.
Using software with summary measures.Most statistical software packages provide an option of find confidence interval limits by inputting the sample mean, sample size, population standard deviation, and desired confidence level.• JMP users: Do the following pull-down sequence:Help S Sample Data
Using software on a data set. Refer to Exercise 6.125 and the DMS odor threshold data. As noted in the exercise, assume s 5 7 mg/l. Read the data into statistical software, and obtain the 95%confidence interval for the mean DMS. Standard Excel does not provide an option for confidence intervals for
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?A study
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
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
Supply chain practices. In a Stanford University study of supply chain practices, researchers gathered data on numerous companies and computed the correlations between various managerial practices and metrics on social responsibility.25 In the report, the researchers only report correlations that
Really small P-value. For Example 6.21(page 329), we noted that the P-value for testing the null hypothesis of m 5 0 is 2PsZ $ 25.33d. Without calculation, we further noted that the P-value is obviously much less than 0.001.(a) Just how small is the P-value? Excel will actually report very small
Job satisfaction. A study of job satisfaction of Croatian employees was conducted on a research sample of 40001 employees.24 The researcher developed a metric for overall job satisfaction based on the rating of numerous factors, including nature of work, top management, promotion, pay, status,
Change the confidence level. Refer to Example 6.21 (page 329) and construct a 95%confidence interval for the mean initial return for the population of Chinese IPO firms.
Coverage percent of 90% confidence interval. Refer to the previous exercise. Do the simulations and report the results for 90% confidence.
Coverage percent of 95% confidence interval. For this exercise, 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 (that is, percent of intervals including the population
Change in number insured. The Wall Street Journal reported a Rand study on the estimated change in insured Americans from September 2013 to March 2014.23 Here is an excerpt:Á a net gain of 9.3 million people with coverage.That number came with a wide margin of error(3.5 million people), was driven
Computer-assisted career guidance systems. A wide variety of computer-assisted career guidance systems have been developed over the past decade. These programs use factors such as student interests, aptitude, skills, personality, and family history to recommend a career path. For simplicity,
Choose the appropriate distribution. You must decide which of two discrete distributions a random variable X has. We 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
Power of the mean blood pressure.Example 6.20 (pages 327–328) gives a test of a hypothesis about systolic blood pressure of company executives based on a sample size of 72. The hypotheses are H0: m 5 128 Ha: m . 128 Assume that the population standard deviation is s 5 15.Consider the test at the
Sample size determination.Example 6.26 (page 343) considers the test of H0: m 5 473 against H0: m Þ 473, where m is the mean fill amount. The population standard deviation is given to be s 5 2. Suppose that the testing is performed at a 5% significance level. Without use of software, determine the
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
Effect of changing the alternative m on power. The Statistical Power applet illustrates the power calculation similar to that in Figure 6.18 (page 345). Open the applet and keep the default settings for the null (m 5 0) and the alternative(m . 0) hypotheses, the sample size (n 5 10), the standard
Power versus a different alternative. A one-sided test of the null hypothesis m 5 60 versus the alternative m 5 50 has power equal to 0.5. Will the power for the alternative m 5 40 be higher or lower than 0.5? Draw a picture and use this to explain your answer.
Power versus a different alternative. The power for a two-sided test of the null hypothesis m 5 0 versus the alternative m 5 5 is 0.73. What is the power versus the alternative m 5 25? 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
False-positives. Refer to the setting of the previous problem. Define X as the number of false-positives occurring among the 80 correlation tests.(a) What is the distribution of the number X of tests that are significant?(b) Find the probability that two or more of the tests are significant.
False-positive rate. With the big data movement, companies are searching through thousands of variables to find patterns in the data to make better predictions on key business variables. For example, Walmart found that sales of strawberry Pop-Tarts increased significantly when the surrounding
More than one test and critical value.Suppose that you are performing 12 two-sided tests of significance using the Bonferroni procedure with a 5 0.05.(a) If you were to perform the testing procedure using a critical value z*, what would be z*?(b) As the number of test increases, what will happen to
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 a 5 0.05. The calculated P-values are 0.039, 0.549, 0.003, 0.316, 0.001, 0.006, 0.251, 0.031, 0.778, 0.012, 0.002, and ,0.001. 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 a then means that if all the null hypotheses are true, the probability is a that any of the tests rejects its null
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
Turning insignificance in significance. 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. Consider the following randomly generated digits used
Do you agree? 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 larger than 0.05, the null hypothesis is true.(b) Practical significance is not the same as statistical significance.(c) We can
How far do rich parents take us? How much education children get is strongly associated with the wealth and social status of their parents, termed“socioeconomic status,” or SES. The SES of parents, however, has little influence on whether children who have graduated from college continue their
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
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 a 5 0.20 for the study because you would be more likely to obtain statistically
Student satisfaction. Each year Forbes publishes its rankings of 650 American colleges. The category of student satisfaction carries a weight of 25%toward the overall score of a college. The major component of the student satisfaction measure is based on student evaluations from RateMyProfessor for
Is it significant? More than 200,000 people worldwide take the GMAT examination each year when they apply for MBA programs. Their scores vary Normally with mean about m 5 525 and standard deviation about s 5 100. One hundred students go through a rigorous training program designed to raise their
Finding a P-value. You have performed a one-sided test of significance for greater-than alternative and obtained a value of z 5 20.382.(a) Use Table A to find the approximate P-value for this test.(b) Use software to find the P-value even more accurately.
Test statistic and levels of significance.Consider a significance test for a null hypothesis versus a two-sided alternative. Give a value of z that will give a result significant at the 1% level but not at the 0.5% level.
Finding a P-value. You have performed a two-sided test of significance and obtained a value of z 5 3.1.(a) Use Table A to find the P-value for this test.(b) Use software to find the P-value even more accurately.
Other changes and the P-value. Refer to the previous exercise.(a) What happens to the P-values when you change the significance level a to 0.01? Explain the result.(b) What happens to the P-values when you change the sample size n from 10 to 40? Explain the result.6.89 Why is it significant at the
Changing to a two-sided alternative, continued. Repeat the previous exercise but with the two-sided alternative hypothesis. How does this change affect the P-values associated with each x?Explain why the P-values change in this way.
Impact of x on the P-value. We can also study the P-value using the Statistical Significance applet. Reset the applet to the default settings for the null (m 5 0) and the alternative(m . 0) hypotheses, the sample size (n 5 10), the standard deviation (s 5 1), and the significance level(a 5 0.05).
Changing the sample size. Refer to Exercise 6.82. Suppose that you increase the sample size n from 10 to 40. Again make a table giving x and the results of the significance tests at the 0.05 significance level. What do you conclude?
Changing to a two-sided alternative.Repeat the previous exercise but with the two-sided alternative hypothesis. How does this change affect which values of x are far enough away from m0 to be statistically significant at the 0.01 level?
Effect of changing a on significance.Repeat the previous exercise with significance level a 5 0.01. How does the choice of a affect which values of x are far enough away from m0 to be statistically significant?
Impact of x on significance. The Statistical Significance applet illustrates statistical tests with a fixed level of significance for Normally distributed data with known standard deviation. Open the applet and keep the default settings for the null(m 5 0) and the alternative (m . 0) hypotheses,
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 the
E-cigarette use among the youth. E-cigarettes are battery operated devices that aim to mimic standard cigarettes. They don’t contain tobacco but operate by heating nicotine into a vapor that is inhaled. Here is an excerpt from a 2014 UK public health report in which the use of e-cigarettes among
Corn yield. The 10-year historical average yield of corn in the United States is about 160 bushels per acre. A survey of 50 farmers this year gives a sample mean yield of x 5 158.4 bushels per acre. We want to know whether this is good evidence that the national mean this year is not 160 bushels
Study habits. The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures the motivation, attitude toward school, and study habits of students. Scores range from 0 to 200. The mean score for U.S. college students is about 115, and the standard deviation is about 30. A
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 m 5 6.9 new words (words not used in the poet’s other works). The standard deviation of the number of new words is s 5 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 5 0.038) was found between the sexes, with men earning more on the average. No difference (P 5
Exercise and statistics exams. A study examined whether exercise affects how students 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
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 household
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
Hypotheses. In each of the following situations, a significance test for a population mean m 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 m.State the appropriate null hypothesis H0 and alternative hypothesis Ha in each case.(a) David’s car averages 28 miles per gallon on the highway. He now switches to a new motor oil that is
Interpreting P-value. The reporting of P-values is standard practice in statistics. Unfortunately, misinterpretations of P-values by producers and readers of statistical reports are common. The previous two exercises dealt with a few incorrect applications of the P-value. This exercise explores the
What’s wrong? Here are several situations in which there is an incorrect application of the ideas presented in this section. Write a short explanation of what is wrong in each situation and why it is wrong.(a) The z statistic had a value of 22.3 for a two-sided test. The null hypothesis is not
What’s wrong? Here are several situations in which there is an incorrect application of the ideas presented in this section. Write a short explanation of what is wrong in each situation and why it is wrong.(a) A report says that the alternative hypothesis is rejected because the P-value is
What’s wrong? Here are several situations in which there is an incorrect application of the ideas presented in this section. Write a short explanation of what is wrong in each situation and why it is wrong.(a) A manager wants to test the null hypothesis that average weekly demand is not equal to
More on P-value and significance level. The P-value for a significance test is 0.079.(a) Do you reject the null hypothesis at level a 5 0.05?(b) Do you reject the null hypothesis at level a 5 0.01?(c) Explain how you determined your answers in parts (a) and (b).
P-value and significance level. The P-value for a significance test is 0.023.(a) Do you reject the null hypothesis at level a 5 0.05?(b) Do you reject the null hypothesis at level a 5 0.01?(c) Explain how you determined your answers in parts (a) and (b).
Can you reject the null hypothesis? A 95% confidence interval for a population mean is s42,51d.(a) Can you reject the null hypothesis that m 5 53 at the 5% significance level?Why?(b) Can you reject the null hypothesis that m 5 44 at the 5% significance level?Why?
Does the confidence interval include m0? The P-value for a two-sided test of the null hypothesis H0: m 5 20 is 0.037.(a) Does the 95% confidence interval include the value 20? Explain.(b) Does the 99% confidence interval include the value 20? Explain.
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 0.31. Can you be confident that the purity of the new product is the same
Computing the test statistic and P-value. You will perform a significance test of H0: m 5 19 based on an SRS of n 5 25. Assume that s 5 13.(a) If x 5 23, what is the test statistic z?(b) What is the P-value if Ha: m . 19?(c) What is the P-value if Ha: m Þ 19?
Testing a random number generator. Statistical software has a “random number generator” that is supposed to produce numbers uniformly distributed between 0 and 1. If this is true, the numbers generated come from a population with m 5 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* 5 2 or 3 are generally convincing statistical evidence. For a two-sided test, what significance level corresponds to z* 5 2? To z*
Significance. You are testing H0: m 5 0 against Ha: m . 0 based on an SRS of 30 observations from a Normal population. What values of the z statistic are statistically significant at the a 5 0.01 level?
Significance. You are testing H0: m 5 0 against Ha: m Þ 0 based on an SRS of 30 observations from a Normal population. What values of the z statistic are statistically significant at the a 5 0.01 level?
Finding significant z-scores. Consider a two-sided significance test for a population mean.(a) Sketch a Normal curve similar to that shown in Figure 6.13 (page 323), but find the value z such that P 5 0.05.(b) Based on your curve from part (a), what values of the z statistic are statistically
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 5 28.6%, less than the national 31%, their analyst tests H0: m 5 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 its area. It interviews a sample of 40 households in the Cleveland metropolitan area to learn
Laboratory quality control. Hospital laboratories routinely check their diagnostic equipment to ensure that patient lab test results are accurate. To check if the equipment is well calibrated, lab technicians make several measurements on a control substance known to have a certain quantity of the
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
Sample size determination. Refer to Example 6.3 (page 293) to find the standard deviation of the delay departures for Delta Airlines is given by s 5 25.83.(a) Use the sample size formula (page 311) to determine what sample size you need to have a margin of error equal to two minutes with 90%
Satisfied with your job? The Gallup-Healthways Well-Being Index is a single metric on a 0 to 100 percentage scale based on six domains of well-being, including life evaluation, emotional health, work environment, physical health, healthy behaviors, and basic access. In 2013, the estimate for the
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 plan to use several such intervals, however, our
What is the cost? In Exercise 6.44, you found an estimate with a margin of error for the fuel efficiency expressed in miles per gallon. Suppose that fuel costs $3.80 per gallon. Find the estimate and margin of error for fuel efficiency in terms of miles per dollar. To convert miles per gallon to
Confidence intervals for average annual income. Based on a 2012 survey, the National Statistics Office of the Republic of the Philippines released a report on various estimates related to family income and expenditures in Philippine pesos. With respect to annual family income, we would find the
Fuel efficiency in metric units. In the previous exercise, you found an estimate with a margin of error for the average miles per gallon. Convert your estimate and margin of error to the metric units kilometers per liter (kpl). To change mpg to kpl, use the fact that 1 mile = 1.609 kilometers and 1
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). For one vehicle equipped in this way, the car was set to 60 miles per hour by cruise control, and the
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
Average starting salary. The University of Texas at Austin McCombs School of Business performs and reports an annual survey of starting salaries for recent bachelor’s in business administration graduates.12 For 2013, there were a total of 430 respondents.(a) Respondents who were supply chain
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 m? Explain why such an interval is of no practical use.(b) What would be a 0% confidence
More confidence interval mistakes and misunderstandings. Suppose that 100 randomly selected members of the Karaoke Channel were asked how much time they typically spend on the site during the week.11 The sample mean x was found to be 3.8 hours. Assume that the population standard deviation is known
Confidence interval mistakes and misunderstandings. Suppose that 500 randomly selected alumni of the University of Okoboji were asked to rate the university’s academic advising services on a 1 to 10 scale.The sample mean x was found to be 8.6. Assume that the population standard deviation is
Reporting margins of error. A U.S. News &World Report article of July 17, 2014, reported Commerce Department estimates of changes in the construction industry:Construction fell 9.3 percent last month to a seasonally adjusted annual rate of 893,000 homes, the Commerce Department said Thursday.If we
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 of 100 sophomore students in your major at your college or university. For each of these samples
Change the confidence. Consider the setting of the previous two exercises. Suppose that the sample mean is still 82, the sample size is 30, and the population standard deviation is 7. Make a diagram similar to Figure 6.11 (page 310) that illustrates the effect of the confidence level on the width
Change the sample size. Consider the setting of the previous exercise. Suppose that the sample mean is again 82 and the population standard deviation is 7. Make a diagram similar to Figure 6.10(page 309) that illustrates the effect of sample size on the width of a 95% interval. Use the following
Margin of error and the confidence interval. A study based on a sample of size 30 reported a mean of 82 with a margin of error of 7 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
Nonresponse in a survey. Let’s revisit Example 6.10 (pages 307–308). Of the 1601 participants in the survey, only 532 reported the undergraduate’s outstanding credit card balance. For that example, we proceeded as if we had a random sample and calculated a margin of error at 95% confidence of
Changes in sample size. Suppose that, in the setting of the previous exercise, you have the resources to contact 500 recent graduates. If all respond, will your margin of error be larger or smaller than $1000? What if only 50% respond?Verify your answers by performing the calculations.
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