Critical Thinking An article in a newspaper states that

Statistical Literacy To use the normal distribution to test a proportion p, the conditions np  5 and nq  5 must be satisfied. Does the value of p come from H0, or is it estimated by using p from the sample?

Statistical Literacy Consider a binomial experiment with n trials and r successes. To construct a test for a proportion p, what value do we use for the sample test statistic?

Critical Thinking In general, if sample data are such that the null hypothesis is rejected at the a  1% level of significance based on a two-tailed test, is H0 also rejected at the a  1% level of significance for a corresponding one-tailed test? Explain

Critical Thinking An article in a newspaper states that the proportion of traffic accidents involving road rage is higher this year than it was last year, when it was 15%. Reconstruct the information of the study in terms of a hypothesis test. Discuss possible hypotheses, possible issues about the sample, possible levels of significance, and the absolute truth of the conclusion. For Problems 519, please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) What sampling distribution will you use? Do you think the sample size is sufficiently large? Explain. What is the value of the sample test statistic? (c) Find the P-value of the test statistic. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? (e) Interpret your conclusion in the context of the application.

Focus Problem: Benfords Law Please read the Focus Problem at the beginning of this chapter. Recall that Benfords Law claims that numbers chosen from very large data files tend to have 1 as the first nonzero digit disproportionately often. In fact, research has shown that if you randomly draw a number from a very large data file, the probability of getting a number with 1 as the leading digit is about 0.301 (see the reference in this chapters Focus Problem). Now suppose you are an auditor for a very large corporation. The revenue report involves millions of numbers in a large computer file. Let us say you took a random sample of numerical entries from the file and of the entries had a first nonzero digit of 1. Let p represent the population proportion of all numbers in the corporate file that have a first nonzero digit of 1. i. Test the claim that p is less than 0.301. Use ii. If p is in fact less than 0.301, would it make you suspect that there are not enough numbers in the data file with leading 1s? Could this indicate that the books have been cooked by pumping up or inflating the numbers? Comment from the viewpoint of a stockholder. Comment from the perspective of the Federal Bureau of Investigation as it looks for money laundering in the form of false profits. iii. Comment on the following statement: If we reject the null hypothesis at level of significance , we have not proved to be false. We can say that the probability is that we made a mistake in rejecting Based on the outcome of the test, would you recommend further investigation before accusing the company of fraud?

Focus Problem: Benfords Law Again suppose you are the auditor for a very large corporation. The revenue file contains millions of numbers in a large computer data bank (see Problem 5). You draw a random sample of numbers from this file and have a first nonzero digit of 1. Let p represent the population proportion of all numbers in the computer file that have a leading digit of 1. i. Test the claim that p is more than 0.301. Use ii. If p is in fact larger than 0.301, it would seem there are too many numbers in the file with leading 1s. Could this indicate that the books have been cooked by artificially lowering numbers in the file? Comment from the point of view of the Internal Revenue Service. Comment from the perspective of the Federal Bureau of Investigation as it looks for profit skimming by unscrupulous employees. iii. Comment on the following statement: If we reject the null hypothesis at level of significance , we have not proved to be false. We can say that the probability is that we made a mistake in rejecting Based on the outcome of the test, would you recommend further investigation before accusing the company of fraud?

Sociology: Crime Rate Is the national crime rate really going down? Some sociologists say yes! They say that the reason for the decline in crime rates in the 1980s and 1990s is demographics. It seems that the population is aging, and older people commit fewer crimes. According to the FBI and the Justice Department, 70% of all arrests are of males aged 15 to 34 years. (Source: True Odds, by J. Walsh, Merritt Publishing.) Suppose you are a sociologist in Rock Springs, Wyoming, and a random sample of police files showed that of 32 arrests last month, 24 were of males aged 15 to 34 years. Use a 1% level of significance to test the claim that the population proportion of such arrests in Rock Springs is different from 70%.

College Athletics: Graduation Rate Women athletes at the University of Colorado, Boulder, have a long-term graduation rate of 67% (Source: The Chronicle of Higher Education). Over the past several years, a random sample of 38 women athletes at the school showed that 21 eventually graduated. Does this indicate that the population proportion of women athletes who graduate from the University of Colorado, Boulder, is now less than 67%? Use a 5% level of significance.

Highway Accidents: DUI The U.S. Department of Transportation, National Highway Traffic Safety Administration, reported that 77% of all fatally injured automobile drivers were intoxicated. A random sample of 27 records of automobile driver fatalities in Kit Carson County, Colorado, showed that 15 involved an intoxicated driver. Do these data indicate that the population proportion of driver fatalities related to alcohol is less than 77% in Kit Carson County? Use \(\alpha = 0.01\).

Preference: Color What is your favorite color? A large survey of countries, including the United States, China, Russia, France, Turkey, Kenya, and others, indicated that most people prefer the color blue. In fact, about 24% of the population claim blue as their favorite color. (Reference: Study by J. Bunge and A. Freeman-Gallant, Statistics Center, Cornell University.) Suppose a random sample of college students were surveyed and of them said that blue is their favorite color. Does this information imply that the color preference of all college stud

Wildlife: Wolves The following is based on information from The Wolf in the Southwest: The Making of an Endangered Species, by David E. Brown (University of Arizona Press). Before 1918, the proportion of female wolves in the general population of all southwestern wolves was about 50%. However, after 1918, southwestern cattle ranchers began a widespread effort to destroy wolves. In a recent sample of 34 wolves, there were only 10 females. One theory is that male wolves tend to return sooner than females to their old territories, where their predecessors were exterminated. Do these data indicate that the population proportion of female wolves is now less than 50% in the region? Use a  0.01.

Fishing: Northern Pike Athabasca Fishing Lodge is located on Lake Athabasca in northern Canada. In one of its recent brochures, the lodge advertises that 75% of its guests catch northern pike over 20 pounds. Suppose that last summer 64 out of a random sample of 83 guests did, in fact, catch northern pike weighing over 20 pounds. Does this indicate that the population proportion of guests who catch pike over 20 pounds is different from 75% (either higher or lower)? Use a  0.05.

Platos Republic: Syllable Patterns Prose rhythm is characterized as the occurrence of five-syllable sequences in long passages of text. This characterization may be used to assess the similarity among passages of text and sometimes the identity of authors. The following information is based on an article by D. Wishart and S. V. Leach appearing in Computer Studies of the Humanities and Verbal Behavior (Vol. 3, pp. 9099). Syllables were categorized as long or short. On analyzing Platos Republic, Wishart and Leach found that about 26.1% of the five-syllable sequences are of the type in which two are short and three are long. Suppose that Greek archaeologists have found an ancient manuscript dating back to Platos time (about 427347 B.C.). A random sample of 317 five-syllable sequences from the newly discovered manuscript showed that 61 are of the type two short and three long. Do the data indicate that the population proportion of this type of fivesyllable sequence is different (either way) from the text of Platos Republic? Use a  0.01.

Plato’s Dialogues: Prose Rhythm Symposium is part of a larger work referred to as Plato’s Dialogues. Wishart and Leach (see source in Problem 13) found that about 21.4% of five-syllable sequences in Symposium are of the type in which four are short and one is long. Suppose an antiquities store in Athens has a very old manuscript that the owner claims is part of Plato’s Dialogues. A random sample of 493 five-syllable sequences from this manuscript showed that 136 were of the type four short and one long. Do the data indicate that the population proportion of this type of five-syllable sequence is higher than that found in Plato’s Symposium? Use \(\alpha = 0.01\).

. Consumers: Product Loyalty USA Today reported that about 47% of the general consumer population in the United States is loyal to the automobile manufacturer of their choice. Suppose Chevrolet did a study of a random sample of 1006 Chevrolet owners and found that 490 said they would buy another Chevrolet. Does this indicate that the population proportion of consumers loyal to Chevrolet is more than 47%? Use a  0.01.

Supermarket: Prices Harper’s Index reported that 80% of all supermarket prices end in the digit 9 or 5. Suppose you check a random sample of 115 items in a supermarket and find that 88 have prices that end in 9 or 5. Does this indicate that less than 80% of the prices in the store end in the digits 9 or 5? Use \(\alpha = 0.05\).

Medical: Hypertension This problem is based on information taken from The Merck Manual (a reference manual used in most medical and nursing schools). Hypertension is defined as a blood pressure reading over 140 mm Hg systolic and/or over 90 mm Hg diastolic. Hypertension, if not corrected, can cause long-term health problems. In the college-age population (1824 years), about 9.2% have hypertension. Suppose that a blood donor program is taking place in a college dormitory this week (final exams week). Before each student gives blood, the nurse takes a blood pressure reading. Of 196 donors, it is found that 29 have hypertension. Do these data indicate that the population proportion of students with hypertension during final exams week is higher than 9.2%? Use a 5% level of significance.

Medical: Hypertension Diltiazem is a commonly prescribed drug for hypertension (see source in Problem 17). However, diltiazem causes headaches in about 12% of patients using the drug. It is hypothesized that regular exercise might help reduce the headaches. If a random sample of 209 patients using diltiazem exercised regularly and only 16 had headaches, would this indicate a reduction in the population proportion of patients having headaches? Use a 1% level of significance.

Myers-Briggs: Extroverts Are most student government leaders extroverts? According to Myers-Briggs estimates, about 82% of college student government leaders are extroverts. (Source: Myers-Briggs Type Indicator Atlas of Type Tables.) Suppose that a Myers-Briggs personality preference test was given to a random sample of 73 student government leaders attending a large national leadership conference and that 56 were found to be extroverts. Does this indicate that the population proportion of extroverts among college student government leaders is different (either way) from 82%? Use a  0.01.

Critical Region Method: Testing Proportions Solve Problem 7 using the critical region method of testing. Since the sampling distribution of is the normal p 440 Chapter 9 HYPOTHESIS TESTING distribution, you can use critical values from the standard normal distribution as shown in Figure 9-8 or part (c) of Table 5, Appendix II. Compare your conclusions with the conclusions obtained by using the P-value method. Are they the same?

Critical Region Method: Testing Proportions Solve Problem 9 using the critical region method of testing. Hint: See Problem 20. Compare your conclusions with the conclusions obtained by using the P-value method. Are they the same?

Critical Region Method: Testing Proportions Solve Problem 15 using the critical region method of testing. Hint: See Problem 20. Compare your conclusions with the conclusions obtained by using the P-value method. Are they the same?