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Introduction To The Practice Of Statistics 6th Edition David S. Moore, George P. McCabe, Bruce A. Craig - Solutions
CHALLENGE Fuel efficiency t test. 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 mpg were recorded each time the gas tank was
Two-sided to one-sided P-value. Most software gives P-values for two-sided alternatives. Explain why you cannot always divide these P-values by 2 to obtain P-values for one-sided alternatives.
A final one-sample t test. The one-sample t statistic for testing H0:μ = 20 Ha:μ < 20 based on n = 115 observations has the value t = −1.55.(a) What are the degrees of freedom for this statistic?(b) Between what two values does the P-value of the test fall?(c) If you have software available,
Another one-sample t test. The one-sample t statistic for testing H0:μ = 60 Ha:μ = 60 from a sample of n = 24 observations has the value t = 2.40.(a) What are the degrees of freedom for t?(b) Locate the two critical values t∗ from Table D that bracket t.(c) Between what two values does the
A one-sample t test. The one-sample t statistic for testing H0:μ = 10 Ha:μ > 10 from a sample of n = 20 observations has the value t = 2.10.(a) What are the degrees of freedom for this statistic?(b) Give the two critical values t∗ from Table D that bracket t.(c) Between what two values does the
More on one-sided versus two-sided P-values.Suppose that x = −15.3 in the setting of the previous exercise. Would this change your P-value?Use a sketch of the distribution of the test statistic under the null hypothesis to illustrate and explain your answer.
One-sided versus two-sided P-values. Computer software reports x = 15.3 and P = 0.04 for a t test of H0:μ = 0 versus Ha:μ = 0. Based on prior knowledge, you can justify testing the alternative Ha:μ > 0. What is the P-value for your significance test?
More on the distribution of the t statistic. Repeat the previous exercise for the two situations where the alternative is one-sided.
Distribution of the t statistic. Assume a sample size of n = 20. Draw a picture of the distribution of the t statistic under the null hypothesis. Use Table D and your picture to illustrate the values of the test statistic that would lead to rejection of the null hypothesis at the 5% level for a
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 = 15 observations.(b) A 95% confidence interval
Sign test for energy drink comparison. Exercise 7.8 (page 431) 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.
More on power and the alternative mean μ. Verify your answer to the previous question by doing the calculation for the alternativeμ = 0.75.
Power and the alternative mean μ. If you were to repeat the power calculation in Example 7.9 for a value of μ that is smaller than 1, would you expect the power to be higher or lower than 89%? Why?
Significance test for mounting holes data? Consider data on the distance between mounting holes presented in Figure 1.41 (page 76).Would you feel comfortable applying the t procedures in this case? Explain your answer.
Significance test for CO2 emissions? Consider the CO2 emissions data presented in Figure 1.40 (page 76). 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.Subject 1 2 3 4 5 Drink A 43 79 66 88 78 Drink B 45 78 61 77 70 Is there a difference in preference? State appropriate
Using software. In Example 7.1 (page 421) we calculated the 95%confidence interval for the U.S. average of hours per month spent listening to full-track music on a cell phone. Use software to compute this interval and verify that you obtain the same interval.
Significance test for apartment rents. Recall Exercise 7.1 (page 419). Does this SRS give good reason to believe that the mean rent of all advertised one-bedroom apartments is greater than $550? State the hypotheses, find the t statistic and its P-value, and state your conclusion.
Significance test using the t distribution. A test of a null hypothesis versus a two-sided alternative gives t = 2.35.(a) The sample size is 15. Is the test result significant at the 5% level?Explain how you obtained your answer.(b) The sample size is 6. Is the test result significant at the 5%
90% versus 95% confidence interval. If you were to use 90% confidence, rather than 95% confidence, would the margin of error be larger or smaller? Explain your answer.
More on apartment rents. Recall Exercise 7.1 (page 419). Construct a 95% confidence interval for the mean monthly rent of all advertised one-bedroom apartments.
Finding critical t∗ values. What critical value t∗ from Table D should be used to construct(a) a 95% confidence interval when n = 12?(b) a 99% confidence interval when n = 24?(c) a 90% confidence interval when n = 200?
Apartment rents. You randomly choose 15 unfurnished one-bedroom apartments from a large number of advertisements in your local newspaper. You calculate that their mean monthly rent is $570 and their standard deviation is $105.(a) What is the standard error of the mean?(b) What are the degrees of
CH ALLENGE Find published studies with confidence intervals. Search the Internet or some journals that report research in your field and find two reports that provide an estimate with a margin of error or a confidence interval. For each report:(a) Describe the method used to collect the data.(b)
CHALLENGE Older customer concerns in restaurants.Persons aged 55 and over represented 21.3% of the U.S. population in the year 2000.This group is expected to increase to 30.5% by 2025. In terms of actual numbers of people, the increase is from 58.6 million to 101.4 million.Restauranteurs have found
CHALLENGE Another simulation study of a test of significance. Use the same procedure for generating data as in the previous exercise.Now test the null hypothesis that μ = 23. Explain your results.
C HALLENGE Simulation study of a test of significance.Use a computer to generate n = 12 observations from a Normal distribution with mean 25 and standard deviation 4: N(25, 4). Test the null hypothesis that μ = 25 using a two-sided significance test. Repeat this process 100 times and then count
CHALL ENGE Simulation study of the confidence interval. Use a computer to generate n = 12 observations from a Normal distribution with mean 25 and standard deviation 4: N(25, 4).Find the 95% confidence interval for μ. Repeat this process 100 times and then count the number of times that the
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
More on the meaning of “statistically 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
Meaning of “statistically significant.” When asked to explain the meaning of “statistically significant at the α = 0.01 level,” a student says,“This means there is only probability 0.01 that the null hypothesis is true.” Is this an essentially correct explanation of statistical
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%.
CH ALLENGE Where do you buy? Consumers can purchase nonprescription medications at food stores, mass merchandise stores such as Kmart and Wal-Mart, or pharmacies. About 45% of consumers make such purchases at pharmacies.What accounts for the popularity of pharmacies, which often charge higher
Odor threshold of future wine experts.Many food products contain small quantities of substances that would give an undesirable taste or smell if they are present in large amounts.An example is the “off-odors” caused by sulfur compounds in wine. Oenologists (wine experts)have determined the odor
Cellulose content in alfalfa hay. An agronomist examines the cellulose content of a variety of alfalfa hay. Suppose that the cellulose content in the population has standard deviation σ = 8 milligrams per gram (mg/g). A sample of 15 cuttings has mean cellulose content x = 145 mg/g.(a) Give a 90%
CHALLENGE Blood phosphorus level in dialysis patients. Patients with chronic kidney failure may be treated by dialysis, using a machine that removes toxic wastes from the blood, a function normally performed by the kidneys.Kidney failure and dialysis can cause other changes, such as retention of
CHALLENGE Effect of sample size on significance.You are testing the null hypothesis thatμ = 0 versus the alternative μ > 0 using α = 0.05.Assume σ = 14. Suppose x = 4 and n = 10.Calculate the test statistic and its P-value. Repeat assuming the same value of x but with n = 20. Do the same for
APPLET CHALLENGE Coverage percent of 90%confidence interval. Refer to the previous exercise. Do the simulations and report the results for 90% confidence.
APPLET CHALLENGE Coverage percent of 95%confidence interval. 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
CHALL ENGE 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.24 Here are the
CHALL ENGE Full-time employment and age. A study of late adolescents and early adults reported average months of full-time employment for individuals aged 18 to 26.23 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
A Web-based business. You are in charge of marketing for a Web 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
CH ALLENGE Choose the appropriate distribution. 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 they assign to the values x of X:x 0 1 2 3 4 5 6 p0 0.1 0.1 0.1 0.2 0.1 0.1 0.3 p1 0.3 0.1 0.1 0.2 0.1
Power of the mean SAT score test. Example 6.16(page 385) gives a test of a hypothesis about the SAT scores of California high school students based on an SRS of 500 students. The hypotheses are H0:μ = 450 Ha:μ > 450 Assume that the population standard deviation is σ = 100. The test rejects H0 at
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
APPLET Power of the random east-west distribution of trees test. Refer to the previous exercise. Note that in the east-west direction, the average location was 113.8. Use the Power applet to find the power for the alternativeμ = 110.
APPLET Power of the random north-south distribution of trees test. In Exercise 6.66(page 392) you performed a two-sided significance test of the null hypothesis that the average northsouth location of the longleaf pine trees sampled in the Wade Tract was μ = 100. There were 584 trees in the sample
More on the power for a different alternative. A one-sided test of the null hypothesis μ = 50 versus the alternative μ = 60 has power equal to 0.5. Will the power for the alternative μ = 70 be higher or lower than 0.5? Draw a picture and use this to explain your answer.
Power for a different alternative. The power for a two-sided test of the null hypothesis μ = 0 versus the alternative μ = 5 is 0.82. What is the power versus the alternative μ = −5? 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
CH ALLENGE Significance using the Bonferroni procedure. Refer to the previous problem.A researcher has performed 12 tests of significance and wants to apply the Bonferroni procedure withα = 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,
CH ALLENGE An adjustment for multiple tests. One way to deal with the problem of misleading P-values when performing more than one significance test is to adjust the criterion you use for statistical significance. The Bonferroni procedure does this in a simple way. If you perform 2 tests and want
Interpreting a very small P-value. Assume that you are performing a large number of significance tests. Let n be the number of these tests. How large would n need to be for you to expect about one P-value to be 0.00001 or smaller? Use this information to write an explanation of how to interpret a
More on searching for significance. You perform 1000 significance tests using α = 0.05. Assuming that all null hypotheses are true, about how many of the test results would you expect to be statistically significant? Explain how you obtained your answer.
Searching for significance. Give an example of a situation where searching for significance would lead to misleading conclusions.
CHALLENGE 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
When statistical inference is valid. Give an example of an interesting set of data for which statistical inference is valid. Explain your answer.
When statistical inference is not valid. Give an example of a set of data for which statistical inference is not valid. Explain your answer.
Property damage by tornadoes. Table 1.5 (page 25) gives average property damage per year due to tornadoes for each of the states. Is it appropriate to use the statistical methods we discussed in this chapter for these data? Explain why or why not.
More on coaching for the SAT. Give a 99%confidence interval for the mean SATM score μafter 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
CHALLENGE Find 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
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 to investigate whether 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 calculate the
More on statistical versus practical significance.A study with 14 subjects reported a result that failed to achieve statistical significance at the 5% level.The P-value was 0.052. Write a short summary of how you would interpret these findings.
Statistical versus practical significance. A study with 7500 subjects reported a result that was statistically significant at the 5% level. Explain why this result might not be particularly large or important.
What a test of significance can answer. Explain whether a test of significance can answer each of the following questions.(a) Is the sample or experiment properly designed?(b) Is the observed effect compatible with the null hypothesis?(c) Is the observed effect important?
Interpreting the P-value. A P-value of 0.90 is reported for a significance test for a population mean. Interpret this result.
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
A role as a statistical consultant. You are the statistical expert for a graduate student planning her PhD research. After you carefully present the mechanics of significance testing, she suggests using α = 0.25 for the study because she would be more likely to obtain statistically significant
Home security systems. A recent TV advertisement for home security systems said that homes without an alarm system are 3 times more likely to be broken into. Suppose this conclusion was obtained by examining an SRS of police records of break-ins and determining whether the percent of homes with
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 aboutσ = 100. One hundred students go through a rigorous training program designed to raise their
More on using Table A and Table D to find a P-value. Refer to the previous exercise. Find the P-value for z = −1.37.
Using Table A and Table D to find a P-value.Consider a significance test for a null hypothesis versus a two-sided alternative. Between what values from Table D does the P-value for an outcome z = 1.37 lie? Calculate the P-value using Table A, and verify that it lies between the values you found
More on using Table D to 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.
Using Table D to find a P-value. 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.
Test statistic and levels of significance. Consider a significance test for a null hypothesis versus a two-sided alternative with a z test statistic. Give a value of z that will give a result significant at the 0.5% level but not at the 0.1% level.
More on understanding levels of significance.You are told that a significance test is significant at the 5% level. From this information can you determine whether or not it is significant at the 1%level? Explain your answer.
Understanding levels of significance. Explain in plain language why a significance test that is significant at the 1% level must always be significant at the 5% level.
APPLE T Changes of x on the P-value. The P-Value of a Test of Significance applet illustrates Pvalues of significance tests for Normally distributed data with known standard deviation. Open the applet and keep the default settings for the null(μ = 0) and the alternative (μ > 0) hypotheses, the
APPLE T Changes of α on significance. Repeat the previous exercise with significance levelα = 0.01. How does the choice of α affect the values of x that are far enough away from μ0 to be statistically significant?
APPLE T Changes 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 (μ = 0) and the alternative (μ > 0)
Level of nicotine in 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
Adjusting for changes in the value of the dollar.In Example 6.9 (page 373), we found that the average student debt has risen between 1997 to 2002. In computing the difference, we did not adjust for differing values of the dollar. Using the fact that$1 in 1997 was worth approximately $1.12 in 2002,
Are the mpg measurements similar? Refer to Exercise 6.26 (page 371). In addition to the computer computing mpg, the driver also recorded the mpg by dividing the miles driven by the number of gallons at each fill-up. The following data are the differences between the computer’s and the driver’s
Calcium level in pregnant women in rural Guatemala. The level of calcium in the blood in healthy young adults varies with mean about 9.5 milligrams per deciliter and standard deviation about σ = 0.4. A clinic in rural Guatemala measures the blood calcium level of 160 healthy pregnant women at
Attitudes toward school. 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
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 = 8.9 new words (words not used in the poet’s other works). The standard deviation of the number of new words is σ = 2.5. Now a manuscript
CH ALLENGE Are the pine trees randomly distributed east to west? Answer the questions in the previous exercise for the east-west direction, where the sample mean is 113.8.
CHALLENGE Are the pine trees randomly distributed north to south? In Example 6.1 we looked at the distribution of longleaf pine trees in the Wade Tract. One way to formulate hypotheses about whether or not the trees are randomly distributed in the tract is to examine the average location in the
More on the eighth-grade average science score.The 2005 NAEP report described in the previous exercise states that the average score on their science test for eighth-grade students across the nation was not significantly different from the average score in 2000. A footnote states that
Change in California’s eighth-grade average science score. A report based on the 2005 National Assessment of Educational Progress (NAEP)16 states that the average score on their science test for eighthgrade students in California is significantly higher than in 2000. A footnote states that
Alcohol awareness among college students. A study of alcohol awareness among college students reported a higher awareness for students enrolled in a health and safety class than for those enrolled in a statistics class.15 The difference is described as being statistically significant. Explain what
Symbol of wealth in ancient China? Every society has its own symbols of wealth and prestige. In ancient China, it appears that owning pigs was such a symbol. Evidence comes from examining burial sites. If the skulls of sacrificed pigs tend to appear along with expensive ornaments, that suggests
Purity of a catalyst. A new supplier offers a good price on a catalyst used in your production process.You compare the purity of this catalyst with that of the catalyst offered by your current supplier. The P-value for a test of “no difference” is 0.27. Can you be confident that the purity of
Use of bed nets. A study found that the use of bed nets was associated with a lower prevalence of malarial infections in the Gambia.13 A report of the study states that the significance is P < 0.001.Explain what this means in a way that could be understood by someone who has not studied statistics.
More on a two-sided test and the confidence interval. A 90% confidence interval for a population mean is (23, 28).(a) Can you reject the null hypothesis that μ = 24 at the 10% significance level? Why?(b) Can you reject the null hypothesis that μ = 30 at the 10% significance level? Why?
A two-sided test and the confidence interval. The P-value for a two-sided test of the null hypothesis H0:μ = 30 is 0.04.(a) Does the 95% confidence interval include the value 30? Why?(b) Does the 90% confidence interval include the value 30? Why?
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