Much concern has been expressed in recent years regarding

Explain why the statement is not a legitimate hypothesis.

For the following pairs, indicate which do not comply with the rules for setting up hypotheses, and explain why:5 H0: m
123, Ha: m
125 H0: m
123, Ha: m 123 H0: p
.4, Ha: p  .6 H0:

To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of H0: p
.1, Ha: p
125 H0: m
123, Ha: m
125 H0: m
123, Ha: m 123 H0: p
.4, Ha: p .6 H0: m
15, Ha: m
15 x
50 welds is selected and tests are conducted on each weld in the sample. Weld strength is measured as the force required to break the weld. Suppose that the specifications state that the mean strength of welds should exceed 100 lb/in.2 . The inspection team decides to test versus . Explain why this alternative hypothesis was chosen rather than .

Do state laws that allow private citizens to carry concealed weapons result in a reduced crime rate? The author of a study carried out by the Brookings Institution is reported as saying, The strongest thing I could say is that I dont see any strong evidence that they are reducing crime (San Luis Obispo Tribune, January 23, 2003).a. Is this conclusion consistent with testing H0: concealed weapons laws reduce crime versus Ha: concealed weapons laws do not reduce crime or with testing H0: concealed weapons laws do not reduce crime versus Ha: concealed weapons laws reduce crime Explain. b. Does the stated conclusion indicate that the null hypothesis was rejected or not rejected? Explain.

Consider the following quote from the article Review Finds No Link Between Vaccine and Autism (San Luis Obispo Tribune, October 19, 2005): We found no evidence that giving MMR causes Crohns disease and/or autism in the children that get the MMR, said Tom Jefferson, one of the authors of The Cochrane Review. That does not mean it doesnt cause it. It means we could find no evidence of it. (MMR is a measles-mumps-rubella vaccine.) In the context of a hypothesis test with the null hypothesis being that MMR does not cause autism, explain why the author could not just conclude that the MMR vaccine does not cause autism.

A certain university has decided to introduce the use of plus and minus with letter grades, as long as there is evidence that more than 60% of the faculty favor the change. A random sample of faculty will be selected, and the resulting data will be used to test the relevant hypotheses. If p represents the true proportion of all faculty that favor a change to plusminus grading, which of the following pair of hypotheses should the administration test:

A certain television station has been providing live coverage of a particularly sensational criminal trial. The stations program director wishes to know whether more than half the potential viewers prefer a return to regular daytime programming. A survey of randomly selected viewers is conducted. Let p represent the true proportion of viewers who prefer regular daytime programming. What hypotheses should the program director test to answer the question of interest?

Researchers have postulated that because of differences in diet, Japanese children have a lower mean blood cholesterol level than U.S. children do. Suppose that the mean level for U.S. children is known to be 170. Let m represent the true mean blood cholesterol level for Japanese children. What hypotheses should the researchers test?

A county commissioner must vote on a resolution that would commit substantial resources to the construction of a sewer in an outlying residential area. Her fiscal decisions have been criticized in the past, so she decides to take a survey of constituents to find out whether they favor spending money for a sewer system. She will vote to appropriate funds only if she can be fairly certain that a majority of the people in her district favor the measure. What hypotheses should she test?

The mean length of long-distance telephone calls placed with a particular phone company was known to be 7.3 min under an old rate structure. In an attempt to be more competitive with other long-distance carriers, the phone company lowered long-distance rates, thinking that its customers would be encouraged to make longer calls and thus that there would not be a big loss in revenue. Let m denote the true mean length of long-distance calls after the rate reduction. What hypotheses should the phone company test to determine whether the mean length of long-distance calls increased with the lower rates?

Many older homes have electrical systems that use fuses rather than circuit breakers. A manufacturer of 40-amp fuses wants to make sure that the mean amperage at which its fuses burn out is in fact 40. If the mean amperage is lower than 40, customers will complain because the fuses require replacement too often. If the mean amperage is higher than 40, the manufacturer might be liable for damage to an electrical system as a result of fuse malfunction. To verify the mean amperage of the fuses, a sample of fuses is selected and tested. If a hypothesis test is performed using the resulting data, what null and alternative hypotheses would be of interest to the manufacturer?

University analyzed records of breast cancer screening and diagnostic evaluations (Mammogram Cancer Scares More Frequent than Thought, USA Today, April 16, 1998). Discussing the benefits and downsides of the screening process, the article states that, although the rate of false-positives is higher than previously thought, if radiologists were less aggressive in following up on suspicious tests, the rate of false-positives would fall but the rate of missed cancers would rise. Suppose that such a screening test is used to decide between a null hypothesis of H0: no cancer is present and an alternative hypothesis of Ha: cancer is present. (Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) a. Would a false-positive (thinking that cancer is present when in fact it is not) be a Type I error or a Type II error? b. Describe a Type I error in the context of this problem, and discuss the consequences of making a Type I error. c. Describe a Type II error in the context of this problem, and discuss the consequences of making a Type II error. d. What aspect of the relationship between the probability of Type I and Type II errors is being described by the statement in the article that if radiologists were less ag gressive in following up on suspicious tests, the rate of false-positives would fall but the rate of missed cancers would rise?

Medical personnel are required to report suspected cases of child abuse. Because some diseases have symptoms that mimic those of child abuse, doctors who see a child with these symptoms must decide between two competing hypotheses: H0: symptoms are due to child abuse Ha: symptoms are due to disease (Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) The article Blurred Line Between Illness, Abuse Creates Problem for Authorities (Macon Telegraph, February 28, 2000) included the following quote from a doctor in Atlanta regarding the consequences of making an incorrect decision: If its disease, the worst you have is an angry family. If it is abuse, the other kids (in the family) are in deadly danger. a. For the given hypotheses, describe Type I and Type II errors. b. Based on the quote regarding consequences of the two kinds of error, which type of error does the doctor quoted consider more serious? Explain.

Ann Landers, in her advice column of October 24, 1994 (San Luis Obispo Telegram-Tribune), described the reliability of DNA paternity testing as follows: To get a completely accurate result, you would have to be tested, and so would (the man) and your mother. The test is 100 percent accurate if the man is not the father and 99.9 percent accurate if he is. a. Consider using the results of DNA paternity testing to decide between the following two hypotheses: H0: a particular man is the father Ha: a particular man is not the father In the context of this problem, describe Type I and Type II errors. (Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) b. Based on the information given, what are the values of a, the probability of Type I error, and b, the probability of Type II error? c. Ann Landers also stated, If the mother is not tested, there is a 0.8 percent chance of a false positive. For the hypotheses given in Part (a), what are the values of a and b if the decision is based on DNA testing in which the mother is not tested?

Pizza Hut, after test-marketing a new product called the Bigfoot Pizza, concluded that introduction of the Bigfoot nationwide would increase its sales by more than 14% (USA Today, April 2, 1993). This conclusion was based on recording sales information for a random sample of Pizza Hut restaurants selected for the marketing trial. With m denoting the mean percentage increase in sales for all Pizza Hut restaurants, consider using the sample data to decide between H0: m
14 and Ha: m 14. a. Is Pizza Huts conclusion consistent with a decision to reject H0 or to fail to reject H0? b. If Pizza Hut is incorrect in its conclusion, is the company making a Type I or a Type II error?

A television manufacturer claims that (at least) 90% of its TV sets will need no service during the first 3 years of operation. A consumer agency wishes to check this claim, so it obtains a random sample of n
100 purchasers and asks each whether the set purchased needed repair during the first 3 years after purchase. Let p be the sample proportion of responses indicating no repair (so that no repair is identified with a success). Let p denote the true proportion of successes for all sets made by this manufacturer. The agency does not want to claim false advertising unless sample evidence strongly suggests that p .9. The appropriate hypotheses are then H0: p
.9 versus Ha: p .9. a. In the context of this problem, describe Type I and Type II errors, and discuss the possible consequences of each. b. Would you recommend a test procedure that uses a
.10 or one that uses a
.01? Explain.

A manufacturer of hand-held calculators receives large shipments of printed circuits from a supplier. It is too costly and time-consuming to inspect all incoming circuits, so when each shipment arrives, a sample is selected for inspection. Information from the sample is then used to test H0: p
.05 versus Ha: p .05, where p is the true proportion of defective circuits in the shipment. If the null hypothesis is not rejected, the shipment is accepted, and the circuits are used in the production of calculators. If the null hypothesis is rejected, the entire shipment is returned to the supplier because of inferior quality. (A shipment is defined to be of inferior quality if it contains more than 5% defective circuits.)a. In this context, define Type I and Type II errors. b. From the calculator manufacturers point of view, which type of error is considered more serious? c. From the printed circuit suppliers point of view, which type of error is considered more serious?

Water samples are taken from water used for cooling as it is being discharged from a power plant into a river. It has been determined that as long as the mean temperature of the discharged water is at most 150F, there will be no negative effects on the rivers ecosystem. To investigate whether the plant is in compliance with regulations that prohibit a mean discharge water temperature above 150F, researchers will take 50 water samples at randomly selected times and record the temperature of each sample. The resulting data will be used to test the hypotheses H0: m
150F versus Ha: m 150F. In the context of this example, describe Type I and Type II errors. Which type of error would you consider more serious? Explain.

Occasionally, warning flares of the type contained in most automobile emergency kits fail to ignite. A consumer advocacy group wants to investigate a claim against a manufacturer of flares brought by a person who claims that the proportion of defective flares is much higher than the value of .1 claimed by the manufacturer. A large number of flares will be tested, and the results will be used to decide between H0: p
.1 and Ha: p .1, where p represents the true proportion of defective flares made by this manufacturer. If H0 is rejected, charges of false advertising will be filed against the manufacturer. a. Explain why the alternative hypothesis was chosen to be Ha: p .1. b. In this context, describe Type I and Type II errors, and discuss the consequences of each.

Suppose that you are an inspector for the Fish and Game Department and that you are given the task of determining whether to prohibit fishing along part of the Oregon coast. You will close an area to fishing if it is determined that fish in that region have an unacceptably high mercury content. a. Assuming that a mercury concentration of 5 ppm is considered the maximum safe concentration, which of the following pairs of hypotheses would you test: H0: m
5 versus Ha: m 5 or Give the reasons for your choice. b. Would you prefer a significance level of .1 or .01 for your test? Explain.

The National Cancer Institute conducted a 2-year study to determine whether cancer death rates for areas near nuclear power plants are higher than for areas without nuclear facilities (San Luis Obispo Telegram-Tribune, September 17, 1990). A spokesperson for the Cancer Institute said, From the data at hand, there was no convincing evidence of any increased risk of death from any of the cancers surveyed due to living near nuclear facilities. However, no study can prove the absence of an effect. a. Let p denote the true proportion of the population in areas near nuclear power plants who die of cancer during a given year. The researchers at the Cancer Institute might have considered the two rival hypotheses of the form H0: p
value for areas without nuclear facilities Ha: p value for areas without nuclear facilities Did the researchers reject H0 or fail to reject H0? b. If the Cancer Institute researchers were incorrect in their conclusion that there is no increased cancer risk associated with living near a nuclear power plant, are they making a Type I or a Type II error? Explain. c. Comment on the spokespersons last statement that no study can prove the absence of an effect. Do you agree with this statement?

An automobile manufacturer is considering using robots for part of its assembly process. Converting to robots is an expensive process, so it will be undertaken only if there is strong evidence that the proportion of defective installations is lower for the robots than for human assemblers. Let p denote the true proportion of defective installations for the robots. It is known that human assemblers have a defect proportion of .02. a. Which of the following pairs of hypotheses should the manufacturer test: or Explain your answer.

Use the definition of the P-value to explain the following: a. Why H0 would certainly be rejected if P-value
.0003 b. Why H0 would definitely not be rejected if P-value
.350

For which of the following P-values will the null hypothesis be rejected when performing a level .05 test: a. .001 d. .047 b. .021 e. .148 c. .078

Pairs of P-values and significance levels, a, are given. For each pair, state whether the observed P-value leads to rejection of H0 at the given significance level. a. P-value
.084, a
.05 b. P-value
.003, a
.001 c. P-value
.498, a
.05 d. P-value
.084, a
.10 e. P-value
.039, a
.01 f. P-value
.218, a
.10

Let p denote the proportion of grocery store customers that use the stores club card. For a large sample z test of H0: p
.5 versus Ha: p .5, find the P-value associated with each of the given values of the test statistic: a. 1.40 d. 2.45 b. 0.93 e. 0.17 c. 1.96

Assuming a random sample from a large population, for which of the following null hypotheses and sample sizes n is the large-sample z test appropriate:

The article Poll Finds Most Oppose Return to Draft, Wouldnt Encourage Children to Enlist (Associated Press, December 18, 2005) reports that in a random sample of 1000 American adults, 700 indicated that they H0: p
.05, n
75 H0: p
.9, n
100 H0: p
.6, n
210 H0: p
.2, n
25 oppose the reinstatement of a military draft. Is there convincing evidence that the proportion of American adults who oppose reinstatement of the draft is greater than twothirds? Use a significance level of .05.

The poll referenced in the previous exercise (Military Draft Study, AP-Ipsos, June 2005) also included the following question: If the military draft were reinstated, would you favor or oppose drafting women as well as men? Forty-three percent of the 1000 people responding said that they would favor drafting women if the draft were reinstated. Using a .05 significance level, carry out a test to determine if there is convincing evidence that fewer than half of adult Americans would favor the drafting of women.

The article Irritated by Spam? Get Ready for Spit (USA Today, November 10, 2004) predicts that spit, spam that is delivered via Internet phone lines and cell phones, will be a growing problem as more people turn to web-based phone services. In a 2004 poll of 5500 cell phone users conducted by the Yankee Group, 20% indicated that they had received commercial messages and ads on their cell phones. Is there sufficient evidence that the proportion of cell phone users who have received commercial messages or ads in 2004 was greater than the proportion of .13 reported for the previous year?

In a survey conducted by Yahoo Small Business, 1432 of 1813 adults surveyed said that they would alter their shopping habits if gas prices remain high (Associated Press, November 30, 2005). The article did not say how the sample was selected, but for purposes of this exercise, assume that it is reasonable to regard this sample as representative of adult Americans. Based on these survey data, is it reasonable to conclude that more than three-quarters of adult Americans plan to alter their shopping habits if gas prices remain high?

According to a Washington Post-ABC News poll, 331 of 502 randomly selected U.S. adults interviewed said they would not be bothered if the National Security Agency collected records of personal telephone calls they had made. Is there sufficient evidence to conclude that a majority of U.S. adults feel this way? Test the appropriate hypotheses using a .01 significance level.

According to a survey of 1000 adult Americans conducted by Opinion Research Corporation, 210 of those surveyed said playing the lottery would be the most practical way for them to accumulate $200,000 in net wealth in their lifetime (One in Five Believe Path to Riches Is the Lottery, San Luis Obispo Tribune, January 11, 2006). Although the article does not describe how the sample was selected, for purposes of this exercise, assume that the sample can be regarded as a random sample of adult Americans. Is there convincing evidence that more than 20% of adult Americans believe that playing the lottery is the best strategy for accumulating $200,000 in net wealth?

The article Theaters Losing Out to Living Rooms (San Luis Obispo Tribune, June 17, 2005) states that movie attendance declined in 2005. The Associated Press found that 730 of 1000 randomly selected adult Americans preferred to watch movies at home rather than at a movie theater. Is there convincing evidence that the majority of adult Americans prefer to watch movies at home? Test the relevant hypotheses using a .05 significance level.

The article referenced in Exercise 10.34 also reported that 470 of 1000 randomly selected adult Americans thought that the quality of movies being produced was getting worse. a. Is there convincing evidence that fewer than half of adult Americans believe that movie quality is getting worse? Use a significance level of .05. b. Suppose that the sample size had been 100 instead of 1000, and that 47 thought that the movie quality was getting worse (so that the sample proportion is still .47). Based on this sample of 100, is there convincing evidence that fewer than half of adult Americans believe that movie quality is getting worse? Use a significance level of .05. c. Write a few sentences explaining why different conclusions were reached in the hypothesis tests of Parts (a) and (b).

The report 2005 Electronic Monitoring & Surveillance Survey: Many Companies Monitoring, Recording, Videotapingand FiringEmployees (American Management Association, 2005) summarized the results of a survey of 526 U.S. businesses. Four hundred of these companies indicated that they monitor employees web site visits. For purposes of this exercise, assume that it is reasonable to regard this sample as representative of businesses in the United States. a. Is there sufficient evidence to conclude that more than 75% of U.S. businesses monitor employees web site visits? Test the appropriate hypotheses using a significance level of .01. b. Is there sufficient evidence to conclude that a majority of U.S. businesses monitor employees web site visits? Test the appropriate hypotheses using a significance level of .01.

In an AP-AOL sports poll (Associated Press, December 18, 2005), 272 of 394 randomly selected baseball fans stated that they thought the designated hitter rule should either be expanded to both baseball leagues or eliminated. Based on the given information, is there suf- ficient evidence to conclude that a majority of baseball fans feel this way?

In a representative sample of 1000 adult Americans, only 430 could name at least one justice who is currently serving on the U.S. Supreme Court (Ipsos, January 10, 2006). Using a significance level of .01, carry out a hypothesis test to determine if there is convincing evidence to support the claim that fewer than half of adult Americans can name at least one justice currently serving on the Supreme Court.

In a national survey of 2013 adults, 1590 responded that lack of respect and courtesy in American society is a serious problem, and 1283 indicated that they believe that rudeness is a more serious problem than in past years (Associated Press, April 3, 2002). Is there convincing evidence that more than three-quarters of U.S. adults believe that rudeness is a worsening problem? Test the relevant hypotheses using a significance level of .05.

The success of the U.S. census depends on people filling out and returning census forms. Despite extensive advertising, many Americans are skeptical about claims that the Census Bureau will guard the information it collects from other government agencies. In a USA Today poll (March 13, 2000), only 432 of 1004 adults surveyed said that they believe the Census Bureau when it says the information you give about yourself is kept confidential. Is there convincing evidence that, despite the advertising campaign, fewer than half of U.S. adults believe the Cen- sus Bureau will keep information confidential? Use a significance level of .01.

The article Americans Seek Spiritual Guidance on Web (San Luis Obispo Tribune, October 12, 2002) reported that 68% of the general population belong to a religious community. In a survey on Internet use, 84% of religion surfers (defined as those who seek spiritual help online or who have used the web to search for prayer and devotional resources) belong to a religious community. Suppose that this result was based on a sample of 512 religion surfers. Is there convincing evidence that the proportion of religion surfers who belong to a religious community is different from .68, the proportion for the general population? Use a
.05.

Teenagers (age 15 to 20) make up 7% of the driving population. The article More States Demand Teens Pass Rigorous Driving Tests (San Luis Obispo Tribune, January 27, 2000) described a study of auto accidents conducted by the Insurance Institute for Highway Safety. The Institute found that 14% of the accidents studied involved teenage drivers. Suppose that this percentage was based on examining records from 500 randomly selected accidents. Does the study provide convincing evidence that the proportion of accidents involving teenage drivers differs from .07, the proportion of teens in the driving population?

Students at the Akademia Podlaka conducted an experiment to determine whether the Belgium-minted Euro coin was equally likely to land heads up or tails up. Coins were spun on a smooth surface, and in 250 spins, 140 landed with the heads side up (New Scientist, January 4, 2002). Should the students interpret this result as convincing evidence that the proportion of the time the coin would land heads up is not .5? Test the relevant hypotheses using a
.01. Would your conclusion be different if a signifi- cance level of .05 had been used? Explain.

The article Fewer Parolees Land Back Behind Bars (Associated Press, April 11, 2006) includes the following statement: Just over 38 percent of all felons who were released from prison in 2003 landed back behind bars by the end of the following year, the lowest rate since 1979. Explain why it would not be necessary to carry out a hypothesis test to determine if the proportion of felons released in 2003 was less than .40.

Newly purchased automobile tires of a certain type are supposed to be filled to a pressure of 30 psi. Let m denote the true average pressure. Find the P-value associated with each of the following given z statistic values for testing H0: m
30 versus Ha: m  30 when s is known: a. 2.10 d. 1.44 b. 1.75 e. 5.00 c. 0.58

The desired percentage of silicon dioxide in a certain type of cement is 5.0%. A random sample of n
36 specimens gave a sample average percentage of . Let m be the true average percentage of silicon dioxide in this type of cement, and suppose that s is known to be 0.38. Test H0: m
5 versus Ha: m  5 using a significance level of .01.

Give as much information as you can about the P-value of a t test in each of the following situations: a. Upper-tailed test, df
8, t
2.0 b. Upper-tailed test, n
14, t
3.2 c. Lower-tailed test, df
10, t
2.4 d. Lower-tailed test, n
22, t
4.2 e. Two-tailed test, df
15, t
1.6 f. Two-tailed test, n
16, t
1.6 g. Two-tailed test, n
16, t
6.3

Give as much information as you can about the P-value of a t test in each of the following situations: a. Two-tailed test, df
9, t
0.73 b. Upper-tailed test, df
10, t
0.5 c. Lower-tailed test, n
20, t
2.1 d. Lower-tailed test, n
20, t
5.1 e. Two-tailed test, n
40, t
1.7

Paint used to paint lines on roads must reflect enough light to be clearly visible at night. Let m denote the true average reflectometer reading for a new type of paint under consideration. A test of H0: m
20 versus Ha: m 20 based on a sample of 15 observations gave t
3.2. What conclusion is appropriate at each of the following significance levels? a. a
.05 c. a
.001 b. a
.01

A certain pen has been designed so that true average writing lifetime under controlled conditions (involving x
5.21 the use of a writing machine) is at least 10 hr. A random sample of 18 pens is selected, the writing lifetime of each is determined, and a normal probability plot of the resulting data support the use of a one-sample t test. The relevant hypotheses are H0: m
10 versus Ha: m 10. a. If t
2.3 and a
.05 is selected, what conclusion is appropriate? b. If t
1.83 and a
.01 is selected, what conclusion is appropriate? c. If t
0.47, what conclusion is appropriate?

The true average diameter of ball bearings of a certain type is supposed to be 0.5 in. What conclusion is appropriate when testing H0: m
0.5 versus Ha: m  0.5 in each of the following situations: a. n
13, t
1.6, a
.05 b. n
13, t
1.6, a
.05 c. n
25, t
2.6, a
.01 d. n
25, t
3.6

A credit bureau analysis of undergraduate students credit records found that the average number of credit cards in an undergraduates wallet was 4.09 (Undergraduate Students and Credit Cards in 2004, Nellie Mae, May 2005). It was also reported that in a random sample of 132 undergraduates, the sample mean number of credit cards carried was 2.6. The sample standard deviation was not reported, but for purposes of this exercise, suppose that it was 1.2. Is there convincing evidence that the mean number of credit cards that undergraduates report carrying is less than the credit bureaus figure of 4.09?

Medical research has shown that repeated wrist extension beyond 20 degrees increases the risk of wrist and hand injuries. Each of 24 students at Cornell University used a proposed new mouse design, and while using the mouse their wrist extension was recorded for each one. Data consistent with summary values given in the paper Comparative Study of Two Computer Mouse Designs (Cornell Human Factors Laboratory Technical Report RP7992) are given. Use these data to test the hypothesis that the mean wrist extension for people using this new mouse design is greater than 20 degrees. Are any assumptions required in order for it to be appropriate to generalize the results of your test to the population of Cornell students? To the population of all university students? (data on next page)

The international polling organization Ipsos reported data from a survey of 2000 randomly selected Canadians who carry debit cards (Canadian Account Habits Survey, July 24, 2006). Participants in this survey were asked what they considered the minimum purchase amount for which it would be acceptable to use a debit card. Suppose that the sample mean and standard deviation were $9.15 and $7.60 respectively. (These values are consistent with a histogram of the sample data that appears in the report.) Do these data provide convincing evidence that the mean minimum purchase amount for which Canadians consider the use of a debit card to be appropriate is less than $10? Carry out a hypothesis test with a significance level of .01

A comprehensive study conducted by the National Institute of Child Health and Human Development tracked more than 1000 children from an early age through elementary school (New York Times, November 1, 2005). The study concluded that children who spent more than 30 hours a week in child care before entering school tended to score higher in math and reading when they were in the third grade. The researchers cautioned that the findings should not be a cause for alarm because the effects of child care were found to be small. Explain how the difference between the mean math score for third graders who spent long hours in child care and the overall mean for thirdgraders could be small but the researchers could still reach the conclusion that the mean for the child care group is significantly higher than the overall mean for third-graders.

In a study of computer use, 1000 randomly selected Canadian Internet users were asked how much time they spend using the Internet in a typical week (Ipsos Reid, August 9, 2005). The mean of the 1000 resulting observations was 12.7 hours. a. The sample standard deviation was not reported, but suppose that it was 5 hours. Carry out a hypothesis test with a significance level of .05 to decide if there is convincing evidence that the mean time spent using the Internet by Canadians is greater than 12.5 hours. b. Now suppose that the sample standard deviation was 2 hours. Carry out a hypothesis test with a significance level of .05 to decide if there is convincing evidence that the mean time spent using the Internet by Canadians is greater than 12.5 hours.c. Explain why the null hypothesis was rejected in the test of Part (b) but not in the test of Part (a).

A survey of teenagers and parents in Canada conducted by the polling organization Ipsos (Untangling the Web: The Facts About Kids and the Internet, January 25, 2006) included questions about Internet use. It was reported that for a sample of 534 randomly selected teens, the mean number of hours per week spent online was 14.6 and the standard deviation was 11.6. a. What does the large standard deviation, 11.6 hours, tell you about the distribution of online times for this sample of teens? b. Do the sample data provide convincing evidence that the mean number of hours that teens spend online is greater than 10 hours per week?

The same survey referenced in the previous exercise reported that for a random sample of 676 parents of Canadian teens, the mean number of hours parents thought their teens spent online was 6.5 and the sample standard deviation was 8.6. a. Do the sample data provide convincing evidence that the mean number of hours that parents think their teens spend online is less than 10 hours per week? b. Write a few sentences commenting on the results of the test in Part (a) and of the test in Part (b) of the previous exercise.

The paper titled Music for Pain Relief (The Cochrane Database of Systematic Reviews, April 19, 2006) concluded, based on a review of 51 studies of the effect of music on pain intensity, that Listening to music reduces pain intensity levels . . . However, the magnitude of these positive effects is small, the clinical relevance of music for pain relief in clinical practice is unclear. Are the authors of this paper claiming that the pain reduction attributable to listening to music is not statistically signifi- cant, not practically significant, or neither statistically nor practically significant? Explain.

Typically, only very brave students are willing to speak out in a college classroom. Student participation may be especially difficult if the individual is from a different culture or country. The article An Assessment of Class Participation by International Graduate Students (Journal of College Student Development [1995]: 132 140) considered a numerical speaking-up scale, with possible values from 3 to 15 (a low value means that a student rarely speaks). For a random sample of 64 males from Asian countries where English is not the official language, the sample mean and sample standard deviation were 8.75 and 2.57, respectively. Suppose that the mean for the population of all males having English as their native language is 10.0 (suggested by data in the article). Does it appear that the population mean for males from non-English-speaking Asian countries is smaller than 10.0?

A well-designed and safe workplace can contribute greatly to increasing productivity. It is especially important that workers not be asked to perform tasks, such as lifting, that exceed their capabilities. The following data on maximum weight of lift (MWOL, in kilograms) for a frequency of 4 lifts per minute were reported in the article The Effects of Speed, Frequency, and Load on Measured Hand Forces for a Floor-to-Knuckle Lifting Task (Ergonomics [1992]: 833843): 25.8 36.6 26.3 21.8 27.2 Suppose that it is reasonable to regard the sample as a random sample from the population of healthy males, age 1830. Do the data suggest that the population mean MWOL exceeds 25? Carry out a test of the relevant hypotheses using a .05 significance level.

An article titled Teen Boys Forget Whatever It Was appeared in the Australian newspaper The Mercury (April 21, 1997). It described a study of academic performance and attention span and reported that the mean time to distraction for teenage boys working on an independent task was 4 min. Although the sample size was not given in the article, suppose that this mean was based on a random sample of 50 teenage Australian boys and that the sample standard deviation was 1.4 min. Is there convincing evidence that the average attention span for teenage boys is less than 5 min? Test the relevant hypotheses using a
.01.

Many consumers pay careful attention to stated nutritional contents on packaged foods when making purchases. It is therefore important that the information on packages be accurate. A random sample of n
12 frozen dinners of a certain type was selected from production during a particular period, and the calorie content of each one was determined. (This determination entails destroying the product, so a census would certainly not be desirable!) Here are the resulting observations, along with a boxplot and normal probability plot: 255 244 239 242 265 245 259 248 225 226 251 233 265 245 259 248 a. Is it reasonable to test hypotheses about true average calorie content m by using a t test? b. The stated calorie content is 240. Does the boxplot suggest that true average content differs from the stated value? Explain your reasoning. c. Carry out a formal test of the hypotheses suggested in Part (b).

Much concern has been expressed in recent years regarding the practice of using nitrates as meat preservatives. In one study involving possible effects of these chemicals, bacteria cultures were grown in a medium containing nitrates. The rate of uptake of radio-labeled amino acid was then determined for each culture, yielding the following observations: 7251 6871 9632 6866 9094 5849 8957 7978 7064 7494 7883 8178 7523 8724 7468 0000 Suppose that it is known that the true average uptake for cultures without nitrates is 8000. Do the data suggest that the addition of nitrates results in a decrease in the true average uptake? Test the appropriate hypotheses using a significance level of .10.

The power of a test is influenced by the sample size and the choice of significance level. a. Explain how increasing the sample size affects the power (when significance level is held fixed). b. Explain how increasing the significance level affects the power (when sample size is held fixed).

is being discharged from a power plant into a river. It has been determined that as long as the mean temperature of the discharged water is at most 150F, there will be no negative effects on the river ecosystem. To investigate whether the plant is in compliance with regulations that prohibit a mean discharge water temperature above 150F, a scientist will take 50 water samples at randomly selected times and will record the water temperature of each sample. She will then use a z statistic z
x  150 s 1n to decide between the hypotheses H0: m
150 and Ha: m 150, where m is the true mean temperature of discharged water. Assume that s is known to be 10. a. Explain why use of the z statistic is appropriate in this setting. b. Describe Type I and Type II errors in this context. c. The rejection of H0 when z 1.8 corresponds to what value of a? (That is, what is the area under the z curve to the right of 1.8?) d. Suppose that the true value for m is 153 and that H0 is to be rejected if z 1.8. Draw a sketch (similar to that of Figure 10.5) of the sampling distribution of , and shade the region that would represent b, the probability of making a Type II error. e. For the hypotheses and test procedure described, compute the value of b when m
153. f. For the hypotheses and test procedure described, what is the value of b if m
160? g. If H0 is rejected when z 1.8 and
152.8, what is the appropriate conclusion? What type of error might have been made in reaching this conclusion?

Let m denote the true average lifetime for a certain type of pen under controlled laboratory conditions. A test of H0: m
10 versus Ha: m 10 will be based on a sample of size 36. Suppose that s is known to be 0.6, from which The appropriate test statistic is then a. What is a for the test procedure that rejects H0 if z  1.28? b. If the test procedure of Part (a) is used, calculate b when m
9.8, and interpret this error probability. c. Without doing any calculation, explain how b when m
9.5 compares to b when m
9.8. Then check your assertion by computing b when m
9.5. d. What is the power of the test when m
9.8? when m
9.5?

The city council in a large city has become concerned about the trend toward exclusion of renters with children in apartments within the city. The housing coordinator has decided to select a random sample of 125 apartments and determine for each whether children are permitted. Let p be the true proportion of apartments that prohibit children. If p exceeds .75, the city council will consider appropriate legislation. a. If 102 of the 125 sampled apartments exclude renters with children, would a level .05 test lead you to the conclusion that more than 75% of all apartments exclude children? b. What is the power of the test when p
.8 and a
.05?

The amount of shaft wear after a fixed mileage was determined for each of 7 randomly selected internal combustion engines, resulting in a mean of 0.0372 in. and a standard deviation of 0.0125 in. z
x  10 0.1 sx
0.1. a. Assuming that the distribution of shaft wear is normal, test at level .05 the hypotheses H0: m
.035 versus Ha: m .035. b. Using s
0.0125, a
.05, and Appendix Table 5, what is the approximate value of b, the probability of a Type II error, when m
.04? c. What is the approximate power of the test when m
.04 and a
.05?

Optical fibers are used in telecommunications to transmit light. Current technology allows production of fi- bers that transmit light about 50 km (Research at Rensselaer, 1984). Researchers are trying to develop a new type of glass fiber that will increase this distance. In evaluating a new fiber, it is of interest to test H0: m
50 versus Ha: m 50, with m denoting the true average transmission distance for the new optical fiber. a. Assuming s
10 and n
10, use Appendix Table 5 to find b, the probability of a Type II error, for each of the given alternative values of m when a level .05 test is employed: i. 52 ii. 55 iii. 60 iv. 70 b. What happens to b in each of the cases in Part (a) if s is actually larger than 10? Explain your reasoning.

Let m denote the true average diameter for bearings of a certain type. A test of H0: m
0.5 versus Ha: m  0.5 will be based on a sample of n bearings. The diameter distribution is believed to be normal. Determine the value of b in each of the following cases: a. n
15, a
.05, s
0.02, m
0.52 b. n
15, a
.05, s
0.02, m
0.48 c. n
15, a
.01, s
0.02, m
0.52 d. n
15, a
.05, s
0.02, m
0.54 e. n
15, a
.05, s
0.04, m
0.54 f. n
20, a
.05, s
0.04, m
0.54 g. Is the way in which b changes as n, a, s, and m vary consistent with your intuition? Explain.

Generate 200 random samples of size 5 from a normal population with mean 100 and standard deviation 10. Using MINTAB, go to the Calc Menu. Then Calc S Random Data S Normal In the Generate box, enter 200 In the Store in columns box, enter c1-c5 In the mean box, enter 100 In the standard deviation box, enter 10 Click on OK You should now see 200 rows of data in each of the first 5 columns of the MINITAB worksheet.

Each row contains five values that have been randomly selected from a normal population with mean 100 and standard deviation 10. Viewing each row as a sample of size 5 from this population, calculate the mean and standard deviation for each of the 200 samples (the 200 rows) by using MINITABs row statistics functions, which can also be found under the Calc menu: Calc S Row statistics Choose the Mean button In the Input Variables box, enter c1-c5 In the Store result in box, enter c7 Click on OK You should now see the 200 sample means in column 7 of the MINITAB worksheet. Name this column x-bar, by typing the name in the gray box at the top of c7. Now follow a similar process to compute the 200 sample standard deviations, and store them in c8. Name c8 s.

Next, calculate the value of the z statistic for each of the 200 samples. We can calculate z in this example because we know that the samples were selected from a population for which s
10. Use the calculator function of MINITAB to compute as follows: Calc S Calculator In the Store results in box, enter c10 In the Expression box type in the following: (c7-100)/(10/sqrt(5)) Click on OK You should now see the z values for the 200 samples in c10. Name c10 z.

Now calculate the value of the t statistic for each of the 200 samples. Use the calculator function of MINITAB to compute as follows: t
x  m 1s/ 1n2
x  100 1s/ 152 Calc S Calculator In the Store results in box, enter c11 In the Expression box type in the following: (c7-100)/(c8/sqrt(5)) Click on OK You should now see the t values for the 200 samples in c10. Name c10 t.

Graphs, at last! Now construct histograms of the 200 z values and the 200 t values. These two graphical displays will provide insight about how each of these two statistics behaves in repeated sampling. Use the same scale for the two histograms so that it will be easier to compare the two distributions. Graph S Histogram In the Graph variables box, enter c10 for graph 1 and c11 for graph 2 Click the Frame dropdown menu and select multiple graphs. Then under the scale choices, select Same X and same Y.

Now use the histograms from Step 5 to answer the following questions: a. Write a brief description of the shape, center and spread for the histogram of the z values. Is what you see in the histogram consistent with what you would have expected to see? Explain. (Hint: In theory, what is the distribution of the z statistic?) b. How does the histogram of the t values compare to the z histogram? Be sure to comment on center, shape, and spread. c. Is your answer to Part (b) consistent with what would be expected for a statistic that has a t distribution? Explain. d. The z and t histograms are based on only 200 samples, and they only approximate the corresponding sampling distributions. The 5th percentile for the standard normal distribution is 1.645 and the 95th percentile is 1.645. For a t distribution with df
5  1
4, the 5th and 95th percentiles are 2.13 and 2.13, respectively. How do these percentiles compare to those of the distributions displayed in the histograms? (Hint: Sort the 200 z values in MINITAB, choose Sort from the Manip menu. Once the values are sorted, percentiles from the histogram can be found by counting in 10 [which is 5% of 200] values from either end of the sorted list. Then repeat this with the t values.) e. Are the results of your simulation and analysis consistent with the statement that the statistic has a standard normal (z) distribution and the statistic t
has a t distribution? Explain.

The authors of the article Perceived Risks of Heart Disease and Cancer Among Cigarette Smokers (Journal of the American Medical Association [1999]: 10191021) expressed the concern that a majority of smokers do not view themselves as being at increased risk of heart disease or cancer. A study of 737 current smokers selected at random from U.S. households with telephones found that of 737 smokers surveyed, 295 indicated that they believed they have a higher than average risk of cancer. Do these data suggest that p, the true proportion of smokers who view themselves as being at increased risk of cancer is in fact less than .5, as claimed by the authors of the paper? Test the relevant hypotheses using a
.05.

A number of initiatives on the topic of legalized gambling have appeared on state ballots. Suppose that a political candidate has decided to support legalization of casino gambling if he is convinced that more than twothirds of U.S. adults approve of casino gambling. USA Today (June 17, 1999) reported the results of a Gallup poll in which 1523 adults (selected at random from households with telephones) were asked whether they approved of casino gambling. The number in the sample who approved was 1035. Does the sample provide convincing evidence that more than two-thirds approve?

The article Credit Cards and College Students: Who Pays, Who Benefits? (Journal of College Student Development [1998]: 5056) described a study of credit card payment practices of college students. According to the authors of the article, the credit card industry asserts that at most 50% of college students carry a credit card balance from month to month. However, the authors of the article report that, in a random sample of 310 college students, 217 carried a balance each month. Does this sample provide sufficient evidence to reject the industry claim?

Although arsenic is known to be a poison, it also has some beneficial medicinal uses. In one study of the use of arsenic to treat acute promyelocytic leukemia (APL), a rare type of blood cell cancer, APL patients were given an arsenic compound as part of their treatment. Of those receiving arsenic, 42% were in remission and showed no signs of leukemia in a subsequent examination (Washington Post, November 5, 1998). It is known that 15% of APL patients go into remission after the conventional treatment. Suppose that the study had included 100 randomly selected patients (the actual number in the study was much smaller). Is there sufficient evidence to conclude that the proportion in remission for the arsenic treatment is greater than .15, the remission proportion for the conventional treatment? Test the relevant hypotheses using a .01 significance level.

According to the article Which Adults Do Underage Youth Ask for Cigarettes? (American Journal of Public Health [1999]: 15611564), 43.6% of the 149 18- to 19-year-olds in a random sample have been asked to buy cigarettes for an underage smoker. a. Is there convincing evidence that fewer than half of 18- to 19-year-olds have been approached to buy cigarettes by an underage smoker? b. The article went on to state that of the 110 nonsmoking 18- to 19-year-olds, only 38.2% had been approached to buy cigarettes for an underage smoker. Is there evidence that less than half of nonsmoking 18- to 19-year-olds have been approached to buy cigarettes?

Many people have misconceptions about how profitable small, consistent investments can be. In a survey of 1010 randomly selected U.S. adults (Associated Press, October 29, 1999), only 374 responded that they thought that an investment of $25 per week over 40 years with a 7% annual return would result in a sum of over $100,000 (the correct amount is $286,640). Is there sufficient evidence to conclude that less than 40% of U.S. adults are aware that such an investment would result in a sum of over $100,000? Test the relevant hypotheses using a
.05.

The same survey described in Exercise 10.77 also asked the individuals in the sample what they thought was their best chance to obtain more than $500,000 in their lifetime. Twenty-eight percent responded win a lottery or sweepstakes. Does this provide convincing evidence that more than one-fourth of U.S. adults see a lottery or sweepstakes win as their best chance of accumulating $500,000? Carry out a test using a significance level of .01.

The state of Georgias HOPE scholarship program guarantees fully paid tuition to Georgia public universities for Georgia high school seniors who have a B average in academic requirements as long as they maintain a B average in college. Of 137 randomly selected students enrolling in the Ivan Allen College at the Georgia Institute of Technology (social science and humanities majors) in 1996 who had a B average going into college, 53.2% had a GPA below 3.0 at the end of their first year (Who Loses HOPE? Attrition from Georgias College Scholarship Program, Southern Economic Journal [1999]: 379390). Do these data provide convincing evidence that a majority of students at Ivan Allen College who enroll with a HOPE scholarship lose their scholarship?

Speed, size, and strength are thought to be important factors in football performance. The article Physical and Performance Characteristics of NCAA Division I Football Players (Research Quarterly for Exercise and Sport [1990]: 395401) reported on physical characteristics of Division I starting football players in the 1988 football season. Information for teams ranked in the top 20 was easily obtained, and it was reported that the mean weight of starters on top-20 teams was 105 kg. A random sample of 33 starting players (various positions were represented) from Division I teams that were not ranked in the top 20 resulted in a sample mean weight of 103.3 kg and a sample standard deviation of 16.3 kg. Is there suffi- cient evidence to conclude that the mean weight for nontop-20 starters is less than 105, the known value for top- 20 teams?

Are young women delaying marriage and marrying at a later age? This question was addressed in a report issued by the Census Bureau (Associated Press, June 8, 1991). The report stated that in 1970 (based on census results) the mean age of brides marrying for the first time was 20.8 years. In 1990 (based on a sample, because census results were not yet available), the mean was 23.9. Suppose that the 1990 sample mean had been based on a random sample of size 100 and that the sample standard deviation was 6.4. Is there sufficient evidence to support the claim that in 1990 women were marrying later in life than in 1970? Test the relevant hypotheses using a
.01. (Note: It is probably not reasonable to think that the distribution of age at first marriage is normal in shape.)

According to the article Workaholism in Organizations: Gender Differences (Sex Roles [1999]: 333346), the following data were reported on 1996 income for random samples of male and female MBA graduates from a certain Canadian business school: N s Males 258 $133,442 $131,090 Females 233 $105,156 $98,525 Note: These salary figures are in Canadian dollars. a. Test the hypothesis that the mean salary of male MBA graduates from this school was in excess of $100,000 in 1996. b. Is there convincing evidence that the mean salary for all female MBA graduates is above $100,000? Test using a
.10. c. If a significance level of .05 or .01 were used instead of .10 in the test of Part (b), would you still reach the same conclusion? Explain.

Duck hunting in populated areas faces opposition on the basis of safety and environmental issues. The San Luis Obispo Telegram-Tribune (June 18, 1991) reported the results of a survey to assess public opinion regarding duck hunting on Morro Bay (located along the central coast of x California). A random sample of 750 local residents included 560 who strongly opposed hunting on the bay. Does this sample provide sufficient evidence to conclude that the majority of local residents oppose hunting on Morro Bay? Test the relevant hypotheses using a
.01

Seat belts help prevent injuries in automobile accidents, but they certainly dont offer complete protection in extreme situations. A random sample of 319 front-seat occupants involved in head-on collisions in a certain region resulted in 95 people who sustained no injuries (Influencing Factors on the Injury Severity of Restrained Front Seat Occupants in Car-to-Car Head-on Collisions, Accident Analysis and Prevention [1995]: 143150). Does this suggest that the true (population) proportion of uninjured occupants exceeds .25? State and test the relevant hypotheses using a significance level of .05.

White remains the most popular car color in the United States, but its popularity appears to be slipping. According to an annual survey by DuPont (Los Angeles Times, February 22, 1994), white was the color of 20% of the vehicles purchased during 1993, a decline of 4% from the previous year. (According to a DuPont spokesperson, white represents innocence, purity, honesty, and cleanliness.) A random sample of 400 cars purchased during this period in a certain metropolitan area resulted in 100 cars that were white. Does the proportion of all cars purchased in this area that are white appear to differ from the national percentage? Test the relevant hypotheses using a
.05. Does your conclusion change if a
.01 is used?

When a published article reports the results of many hypothesis tests, the P-values are not usually given. Instead, the following type of coding scheme is frequently used: *p
.05, **p
.01, ***p
.001, ****p
.0001. Which of the symbols would be used to code for each of the following P-values? a. .037 c. .072 b. .0026 d. .0003

A random sample of n
44 individuals with a B.S. degree in accounting who started with a Big Eight accounting firm and subsequently changed jobs resulted in a sample mean time to change of 35.02 months and a sample standard deviation of 18.94 months (The Debate over Post-Baccalaureate Education: One Universitys Experience, Issues in Accounting Education [1992]: 1836). Can it be concluded that the true average time to change exceeds 2 years? Test the appropriate hypotheses using a significance level of .01.

What motivates companies to offer stock ownership plans to their employees? In a random sample of 87 companies having such plans, 54 said that the primary rationale was tax related (The Advantages and Disadvantages of ESOPs: A Long-Range Analysis, Journal of Small Business Management [1991]: 1521). Does this information provide strong support for concluding that more than half of all such firms feel this way?

The article Caffeine Knowledge, Attitudes, and Consumption in Adult Women (Journal of Nutrition Education [1992]: 179184) reported the following summary statistics on daily caffeine consumption for a random sample of adult women: n
47, mg, s
235 mg, and the data values ranged from 5 to 1176. a. Does it appear plausible that the population distribution of daily caffeine consumption is normal? Is it necessary to assume a normal population distribution to test hypotheses about the value of the population mean consumption? Explain your reasoning. b. Suppose that it had previously been believed that mean consumption was at most 200 mg. Does the given information contradict this prior belief? Test the appropriate hypotheses at significance level .10.

Past experience has indicated that the true response rate is 40% when individuals are approached with a request to fill out and return a particular questionnaire in a stamped and addressed envelope. An investigator believes that if the person distributing the questionnaire is stigmatized in some obvious way, potential respondents would feel sorry for the distributor and thus tend to respond at a rate higher than 40%. To investigate this theory, a distributor is fitted with an eye patch. Of the 200 questionnaires distributed by this individual, 109 were returned. Does this strongly suggest that the response rate in this situation exceeds the rate in the past? State and test the appropriate hypotheses at significance level .05.

An automobile manufacturer who wishes to advertise that one of its models achieves 30 mpg (miles per gallon) decides to carry out a fuel efficiency test. Six nonprofessional drivers are selected, and each one drives a car from Phoenix to Los Angeles. The resulting fuel efficiencies (in miles per gallon) are: 27.2 29.3 31.2 28.4 30.3 29.6 x
215 Assuming that fuel efficiency is normally distributed under these circumstances, do the data contradict the claim that true average fuel efficiency is (at least) 30 mpg?

A student organization uses the proceeds from a particular soft-drink dispensing machine to finance its activities. The price per can had been $0.75 for a long time, and the average daily revenue during that period had been $75.00. The price was recently increased to $1.00 per can. A random sample of n
20 days after the price increase yielded a sample average daily revenue and sample standard deviation of $70.00 and $4.20, respectively. Does this information suggest that the true average daily revenue has decreased from its value before the price increase? Test the appropriate hypotheses using a
.05.

A hot tub manufacturer advertises that with its heating equipment, a temperature of 100F can be achieved in at most 15 min. A random sample of 25 tubs is selected, and the time necessary to achieve a 100F temperature is determined for each tub. The sample average time and sample standard deviation are 17.5 min and 2.2 min, respectively. Does this information cast doubt on the companys claim? Carry out a test of hypotheses using signifi- cance level .05.

Let p denote the proportion of voters in a certain state who favor a particular proposed constitutional amendment. Consider testing H0: p
.5 versus Ha: p .5 at significance level .05 based on a sample of size n
50. a. Suppose that H0 is in fact true. Use Appendix Table 1 (our table of random numbers) to simulate selecting a sample, and use the resulting data to carry out the test. b. If you repeated Part (a) a total of 100 times (a simulation consisting of 100 replications), how many times would you expect H0 to be rejected? c. Now suppose that p
.6, which implies that H0 is false. Again, use Appendix Table 1 to simulate selecting a sample, and carry out the test. If you repeated this a total of 100 times, would you expect H0 to be rejected more frequently than when H0 is true?

A type of lie detector that measures brain waves was developed by a professor of neurobiology at Northwestern University (Associated Press, July 7, 1988). He said, It would probably not falsely accuse any innocent people and it would probably pick up 70% to 90% of guilty people. Suppose that the result of this lie detector test is allowed as evidence in a criminal trial as the sole basis of a decision between two rival hypotheses: accused is innocent versus accused is guilty. Although these are not statistical hypotheses (statements about a population characteristic), the possible decision errors are analogous to Type I and Type II errors. In this situation, a Type I error is finding an innocent person guiltyrejecting the null hypothesis of innocence when it is in fact true. A Type II error is finding a guilty person innocentnot rejecting the null hypothesis of innocence when it is in fact false. If the developer of the lie detector is correct in his statements, what is the probability of a Type I error, a? What can you say about the probability of a Type II error, b?