Pi Many people know the mathematical constant p is approximately 3.14. But thats not

Which test? For each of the following situations, state whether youd use a chi-square goodness-of-fit test, a chi-square test of homogeneity, a chi-square test of independence, or some other statistical test: a) A brokerage firm wants to see whether the type of account a customer has (Silver, Gold, or Platinum) affects the type of trades that customer makes (in person, by phone, or on the Internet). It collects a random sample of trades made for its customers over the past year and performs a test. b) That brokerage firm also wants to know if the type of account affects the size of the account (in dollars). It performs a test to see if the mean size of the account is the same for the three account types. c) The academic research office at a large community college wants to see whether the distribution of courses chosen (Humanities, Social Science, or Science) is different for its residential and nonresidential students. It assembles last semesters data and performs a test.

Which test again? For each of the following situations, state whether youd use a chi-square goodness-of-fit test, a chi-square test of homogeneity, a chi-square test of independence, or some other statistical test: a) Is the quality of a car affected by what day it was built? A car manufacturer examines a random sample of the warranty claims filed over the past two years to test whether defects are randomly distributed across days of the work week. b) A medical researcher wants to know if blood cholesterol level is related to heart disease. She examines a database of 10,000 patients, testing whether the cholesterol level (in milligrams) is related to whether or not a person has heart disease. c) A student wants to find out whether political leaning (liberal, moderate, or conservative) is related to choice of major. He surveys 500 randomly chosen students and performs a test.

Dice After getting trounced by your little brother in a childrens game, you suspect the die he gave you to roll may be unfair. To check, you roll it 60 times, recording the number of times each face appears. Do these results cast doubt on the dies fairness? a) If the die is fair, how many times would you expect each face to show? b) To see if these results are unusual, will you test goodness-of-fit, homogeneity, or independence? c) State your hypotheses. d) Check the conditions. e) How many degrees of freedom are there? f) Find x2 and the P-value. g) State your conclusion.

M&Ms As noted in an earlier chapter, the Master-foods Company says that until very recently yellow candies made up 20% of its milk chocolate M&Ms, red another 20%, and orange, blue, and green 10% each. The rest are brown. On his way home from work the day he was writing these exercises, one of the authors bought a bag of plain M&Ms. He got 29 yellow ones, 23 red, 12 orange, 14 blue, 8 green, and 20 brown. Is this sample consistent with the companys stated proportions? Test an appropriate hypothesis and state your conclusion. a) If the M&Ms are packaged in the stated proportions, how many of each color should the author have expected to get in his bag? b) To see if his bag was unusual, should he test goodnessof- fit, homogeneity, or independence? c) State the hypotheses. d) Check the conditions. e) How many degrees of freedom are there? f) Find x2 and the P-value. g) State a conclusion.

Human births If there is no seasonal effect on human births, we would expect equal numbers of children to be born in each season (winter, spring, summer, and fall). A student takes a census of her statistics class and finds that of the 120 students in the class, 25 were born in winter, 35 in spring, 32 in summer, and 28 in fall. She wonders if the excess in the spring is an indication that births are not uniform throughout the year. a) What is the expected number of births in each season if there is no seasonal effect on births? b) Compute the x2 statistic. c) How many degrees of freedom does the x2 statistic have? d) Find the a = 0.05 critical value for the x2 distribution with the appropriate number of df. e) Using the critical value, what do you conclude about the null hypothesis at a = 0.05?

Bank cards At a major credit card bank, the percentages of people who historically apply for the Silver, Gold, and Platinum cards are 60%, 30%, and 10%, respectively. In a recent sample of customers responding to a promotion, of 200 customers, 110 applied for Silver, 55 for Gold, and 35 for Platinum. Is there evidence to suggest that the percentages for this promotion may be different from the historical proportions? a) What is the expected number of customers applying for each type of card in this sample if the historical proportions are still true? b) Compute the x2 statistic. c) How many degrees of freedom does the x2 statistic have? d) Find the a = 0.05 critical value for the x2 distribution with the appropriate number of df. e) Using the critical value, what do you conclude about the null hypothesis at a = 0.05?

Nuts A company says its premium mixture of nuts contains 10% Brazil nuts, 20% cashews, 20% almonds, and 10% hazelnuts, and the rest are peanuts. You buy a large can and separate the various kinds of nuts. Upon weighing them, you find there are 112 grams of Brazil nuts, 183 grams of cashews, 207 grams of almonds, 71 grams of hazelnuts, and 446 grams of peanuts. You wonder whether your mix is significantly different from what the company advertises. a) Explain why the chi-square goodness-of-fit test is not an appropriate way to find out. b) What might you do instead of weighing the nuts in order to use a x2 test?

Mileage A salesman who is on the road visiting clients thinks that, on average, he drives the same distance each day of the week. He keeps track of his mileage for several weeks and discovers that he averages 122 miles on Mondays, 203 miles on Tuesdays, 176 miles on Wednesdays, 181 miles on Thursdays, and 108 miles on Fridays. He wonders if this evidence contradicts his belief in a uniform distribution of miles across the days of the week. Explain why it is not appropriate to test his hypothesis using the chi-square goodness-of-fit test.

NYPD and race Census data for New York City indicate that 29.2% of the under-18 population is white, 28.2% black, 31.5% Latino, 9.1% Asian, and 2% other ethnicities. The New York Civil Liberties Union points out that, of 26,181 police officers, 64.8% are white, 14.5% black, 19.1% Latino and 1.4% Asian. Do the police officers reflect the ethnic composition of the citys youth? Test an appropriate hypothesis and state your conclusion.

Violence against women In its study When Men Murder Women: An Analysis of 2009 Homicide Data, 2011, the Violence Policy Center (www.vpc.org) reported that 1818 women were murdered by men in 2009. Of these victims, a weapon could be identified for 1654 of them. Of those for whom a weapon could be identified, 861 were killed by guns, 364 by knives or other cutting instruments, 214 by other weapons, and 215 by personal attack (battery, strangulation, etc.). The FBIs Uniform Crime Report says that, among all murders nationwide, the weapon use rates were as follows: guns 63.4%, knives 13.1%, other weapons 16.8%, personal attack 6.7%. Is there evidence that violence against women involves different weapons than other violent attacks in the United States?

Fruit flies Offspring of certain fruit flies may have yellow or ebony bodies and normal wings or short wings. Genetic theory predicts that these traits will appear in the ratio 9:3:3:1 (9 yellow, normal: 3 yellow, short: 3 ebony, normal: 1 ebony, short). A researcher checks 100 such flies and finds the distribution of the traits to be 59, 20, 11, and 10, respectively. a) Are the results this researcher observed consistent with the theoretical distribution predicted by the genetic model? b) If the researcher had examined 200 flies and counted exactly twice as many in each category118, 40, 22, 20what conclusion would he have reached? c) Why is there a discrepancy between the two conclusions?

Pi Many people know the mathematical constant p is approximately 3.14. But thats not exact. To be more precise, here are 20 decimal places: 3.14159265358979323846. Still not exact, though. In fact, the actual value is irrational, a decimal that goes on forever without any repeating pattern. But notice that there are no 0s and only one 7 in the 20 decimal places above. Does that pattern persist, or do all the digits show up with equal frequency? The table shows the number of times each digit appears in the first million digits. Test the hypothesis that the digits 0 through 9 are uniformly distributed in the decimal representation of p.

Hurricane frequencies The National Hurricane Center provides data that list the numbers of large (category 3, 4, or 5) hurricanes that have struck the United States, by decade since 1851 (www.nhc.noaa.gov/dcmi.shtml). The data are summarized below. Recently, theres been some concern that perhaps the number of large hurricanes has been increasing. The natural null hypothesis would be that the frequency of such hurricanes has remained constant. a) With 96 large hurricanes observed over the 16 periods, what are the expected value(s) for each cell? b) What kind of chi-square test would be appropriate? c) State the null and alternative hypotheses. d) How many degrees of freedom are there? e) The value of x2 is 12.67. Whats the P-value? f) State your conclusion.

Lottery numbers The fairness of the South African lottery was recently challenged by one of the countrys political parties. The lottery publishes historical statistics at its Website (http://www.nationallottery.co.za/lotto/ statistics.aspx). Here is a table of the number of times each of the 49 numbers has been drawn in the main lottery and as the bonus ball number as of June 2007: We wonder if all the numbers are equally likely to be the bonus ball. a) What kind of test should we perform? b) There are 655 bonus ball observations. What are the appropriate expected value(s) for the test? c) State the null and alternative hypotheses. d) How many degrees of freedom are there? e) The value of x2 is 34.5. Whats the P-value? f) State your conclusion.

Childbirth, part There is some concern that if a woman has an epidural to reduce pain during childbirth, the drug can get into the babys bloodstream, making the baby sleepier and less willing to breastfeed. In December 2006, the International Breastfeeding Journal published results of a study conducted at Sydney University. Researchers followed up on 1178 births, noting whether the mother had an epidural and whether the baby was still nursing after 6 months. Here are their results: a) What kind of test would be appropriate? b) State the null and alternative hypotheses.

Does your doctor know? A survey8 of articles from the New England Journal of Medicine (NEJM) classified them according to the principal statistics methods used. The articles recorded were all noneditorial articles appearing during the indicated years. Lets just look at whether these articles used statistics at all. Has there been a change in the use of Statistics? a) What kind of test would be appropriate? b) State the null and alternative hypotheses.

Childbirth, part In Exercise 15, the table shows results of a study investigating whether aftereffects of epidurals administered during childbirth might interfere with successful breastfeeding. Were planning to do a chi-square test. a) How many degrees of freedom are there? b) The smallest expected count will be in the epidural/ no breastfeeding cell. What is it? c) Check the assumptions and conditions for inference.

Does your doctor know? (part ) The table in Exercise 16 shows whether NEJM medical articles during various time periods included statistics or not. Were planning to do a chi-square test. a) How many degrees of freedom are there? b) The smallest expected count will be in the 1989/No cell. What is it? c) Check the assumptions and conditions for inference.

Childbirth, part In Exercises 15 and 17, weve begun to examine the possible impact of epidurals on successful breastfeeding. a) Calculate the component of chi-square for the epidural/ no breastfeeding cell. b) For this test, x2 = 14.87. Whats the P-value? c) State your conclusion.

Does your doctor know? (part ) In Exercises 16 and 18, weve begun to examine whether the use of statistics in NEJM medical articles has changed over time. a) Calculate the component of chi-square for the 1989/ No cell. b) For this test, x2 = 25.28. Whats the P-value? c) State your conclusion.

Childbirth, part In Exercises 15, 17, and 19, weve tested a hypothesis about the impact of epidurals on successful breastfeeding. The table shows the tests residuals. a) Show how the residual for the epidural/no breastfeeding cell was calculated. b) What can you conclude from the standardized residuals?

Does your doctor know? (part ) In Exercises 16, 18, and 20, weve tested a hypothesis about whether the use of statistics in NEJM medical articles has changed over time. The table shows the tests residuals. a) Show how the residual for the 1989/No cell was calculated. b) What can you conclude from the patterns in the standardized residuals?

Childbirth, part In Exercises 15, 17, 19, and 21, weve looked at a study examining epidurals as one factor that might inhibit successful breastfeeding of newborn babies. Suppose a broader study included several additional issues, including whether the mother drank alcohol, whether this was a first child, and whether the parents occasionally supplemented breastfeeding with bottled formula. Why would it not be appropriate to use chi- square methods on the 2 * 8 table with yes/no columns for each potential factor?

Does your doctor know? (part ) In Exercises 16, 18, 20, and 22, we considered data on articles in the NEJM. The original study listed 23 different Statistics methods. (The list read: t-tests, contingency tables, linear regression, . . . .) Why would it not be appropriate to use a chi-square test on the 23 * 3 table with a row for each method?

Internet use poll A Pew Research poll in April 2009 from a random sample of U.S. adults asked the questions Did you use the Internet yesterday? and Are you White, Black, or Hispanic/Other? Is the response to the question about the Internet independent of race? a) Under the null hypothesis, what are the expected values? b) Compute the x2 statistic. c) How many degrees of freedom does it have? d) Find the P-value. e) What do you conclude?

Internet use poll, II The same poll as in Exercise 25 also asked the questions Did you use the Internet yesterday? and What is your educational level? Is the response to the question about the internet independent of educational level? a) Under the null hypothesis, what are the expected values? b) Compute the x2 statistic. c) How many degrees of freedom does it have? d) Find the P-value. e) What do you conclude?

Titanic Here is a table we first saw in Chapter 2 showing who survived the sinking of the Titanic based on whether they were crew members, or passengers booked in first-, second-, or third-class staterooms: a) If we draw an individual at random, whats the probability that we will draw a member of the crew? b) Whats the probability of randomly selecting a thirdclass passenger who survived? c) Whats the probability of a randomly selected passenger surviving, given that the passenger was a first-class passenger? d) If someones chances of surviving were the same regardless of their status on the ship, how many members of the crew would you expect to have lived? e) State the null and alternative hypotheses. f) Give the degrees of freedom for the test. g) The chi-square value for the table is 187.8, and the corresponding P-value is barely greater than 0. State your conclusions about the hypotheses.

NYPD and sex discrimination The table below shows the rank attained by male and female officers in the New York City Police Department (NYPD). Do these data summaries indicate that men and women are equitably represented at all levels of the department? a) Whats the probability that a person selected at random from the NYPD is a female? b) Whats the probability that a person selected at random from the NYPD is a detective? c) Assuming no bias in promotions, how many female detectives would you expect the NYPD to have? d) To see if there is evidence of differences in ranks attained by males and females, will you test goodnessof- fit, homogeneity, or independence? e) State the hypotheses. f) Test the conditions. g) How many degrees of freedom are there? h) The chi-square value for the table is 290.1 and the P-value is less than 0.0001. State your conclusion about the hypotheses.

Titanic again Examine and comment on this table of the standardized residuals for the chi-square test you looked at in Exercise 27.

NYPD again Examine and comment on this table of the standardized residuals for the chi-square test you looked at in Exercise 28.

Cranberry juice Its common folk wisdom that drinking cranberry juice can help prevent urinary tract infections in women. In 2001 the British Medical Journal reported the results of a Finnish study in which three groups of 50 women were monitored for these infections over 6 months. One group drank cranberry juice daily, another group drank a lactobacillus drink, and the third drank neither of those beverages, serving as a control group. In the control group, 18 women developed at least one infection, compared to 20 of those who consumed the lactobacillus drink and only 8 of those who drank cranberry juice. Does this study provide supporting evidence for the value of cranberry juice in warding off urinary tract infections? a) Is this a survey, a retrospective study, a prospective study, or an experiment? Explain. b) Will you test goodness-of-fit, homogeneity, or independence? c) State the hypotheses. d) Test the conditions. e) How many degrees of freedom are there? f) Find x2 and the P-value. g) State your conclusion. h) If you concluded that the groups are not the same, analyze the differences using the standardized residuals of your calculations.

Cars A random survey of autos parked in the student lot and the staff lot at a large university classified the brands by country of origin, as seen in the table. Are there differences in the national origins of cars driven by students and staff? a) Is this a test of independence or homogeneity? b) Write appropriate hypotheses. c) Check the necessary assumptions and conditions. d) Find the P-value of your test. e) State your conclusion and analysis.

Montana A poll conducted by the University of Montana classified respondents by whether they were male or female and political party, as shown in the table. We wonder if there is evidence of an association between being male or female and party affiliation. Democrat Republican Independent Male 36 45 24 Female 48 33 16 a) Is this a test of homogeneity or independence? b) Write an appropriate hypothesis. c) Are the conditions for inference satisfied? d) Find the P-value for your test. e) State a complete conclusion.

Fish diet Medical researchers followed 6272 Swedish men for 30 years to see if there was any association between the amount of fish in their diet and prostate cancer. (Fatty Fish Consumption and Risk of Prostate Cancer, Lancet, June 2001) Fish Consumption Total Subjects Prostate Cancers Never/Seldom 124 14 Small Part of Diet 2621 201 Moderate Part 2978 209 Large Part 549 42 a) Is this a survey, a retrospective study, a prospective study, or an experiment? Explain. b) Is this a test of homogeneity or independence? c) Do you see evidence of an association between the amount of fish in a mans diet and his risk of developing prostate cancer? d) Does this study prove that eating fish does not prevent prostate cancer? Explain.

Montana revisited The poll described in Exercise 33 also investigated the respondents party affiliations based on what area of the state they lived in. Test an appropriate hypothesis about this table and state your conclusions. Democrat Republican Independent West 39 17 12 Northeast 15 30 12 Southeast 30 31 16

Working parents In April 2009, Gallup published results from data collected from a large sample of adults in the 27 European Union member states. One of the questions asked was, Which is the most practicable and realistic option for child care, taking into account the need to earn a living? The counts below are representative of the entire collection of responses. Male Female Both Parents Work Full Time 161 140 One Works Full Time, Other Part Time 259 308 One Works Full Time, Other Stays Home for Kids 189 161 Both Parents Work Part Time 49 63 No Opinion 42 28 Source: www.gallup.com/poll/117358/Work-Life-Balance-Tilts-Against-Women- Single-Parents.aspx a) Is this a survey, a retrospective study, a prospective study, or an experiment? b) Will you test goodness-of-fit, homogeneity, or independence? c) Based on these results, do you think men and women have differing opinions when it comes to raising children?

Maryland lottery In the Maryland Pick-3 Lottery, three random digits are drawn each day. A fair game depends on every value (0 to 9) being equally likely to show up in all three positions. If not, someone who detects a pattern could take advantage of that. The table shows how many times each of the digits was drawn during a recent 32-week period, and some of them4 and 7, for instanceseem to come up a lot. Could this just be a result of randomness, or is there evidence the digits arent equally likely to occur?

Stock market Some investors believe that stock prices show weekly patterns, claiming for example that Fridays are more likely to be up days. From the trading sessions since October 1, 1928 we selected a random sample of 1000 days on which the Dow Jones Industrial Average (DJIA) showed a gain in stock prices. The table shows how many of these fell on each day of the week. Sure enough, more of them are Fridaysand Tuesday looks like a bad day to own stocks. Can this be explained as just randomness, or is there evidence here to help an investor?

Grades Two different professors teach an introductory Statistics course. The table shows the distribution of final grades they reported. We wonder whether one of these professors is an easier grader. a) Will you test goodness-of-fit, homogeneity, or independence? b) Write appropriate null hypotheses. c) Find the expected counts for each cell, and explain why the chi-square procedures are not appropriate.

Full moon Some people believe that a full moon elicits unusual behavior in people. The table shows the number of arrests made in a small town during weeks of six full moons and six other randomly selected weeks in the same year. We wonder if there is evidence of a difference in the types of illegal activity that take place. Full Moon Not Full Violent (murder, assault, rape, etc.) 2 3 Property (burglary, vandalism, etc.) 17 21 Drugs/Alcohol 27 19 Domestic Abuse 11 14 Other Offenses 9 6 a) Will you test goodness-of-fit, homogeneity, or independence? b) Write appropriate null hypotheses. c) Find the expected counts for each cell, and explain why the chi-square procedures are not appropriate.

Grades again In some situations where the expected cell counts are too small, as in the case of the grades given by Professors Alpha and Beta in Exercise 39, we can complete an analysis anyway. We can often proceed after combining cells in some way that makes sense and also produces a table in which the conditions are satisfied. Here we create a new table displaying the same data, but calling Ds and Fs Below C: Prof. Alpha Prof. Beta A 3 9 B 11 12 C 14 8 Below C 12 3 a) Find the expected counts for each cell in this new table, and explain why a chi-square procedure is now appropriate. b) With this change in the table, what has happened to the number of degrees of freedom? c) Test your hypothesis about the two professors, and state an appropriate conclusion.

Full moon, next phase In Exercise 40 you found that the expected cell counts failed to satisfy the conditions for inference. a) Find a sensible way to combine some cells that will make the expected counts acceptable. b) Test a hypothesis about the full moon and state your conclusion.

Racial steering A subtle form of racial discrimination in housing is racial steering. Racial steering occurs when real estate agents show prospective buyers only homes in neighborhoods already dominated by that familys race. This violates the Fair Housing Act of 1968. According to an article in Chance magazine (Vol. 14, no. 2 [2001]), tenants at a large apartment complex recently filed a lawsuit alleging racial steering. The complex is divided into two parts: Section A and Section B. The plaintiffs claimed that white potential renters were steered to Section A, while African-Americans were steered to Section B. The table describes the data that were presented in court to show the locations of recently rented apartments. Do you think there is evidence of racial steering? New Renters White Black Total Section A 87 8 95 Section B 83 34 117 Total 170 42 212

Titanic, redux Newspaper headlines at the time, and traditional wisdom in the succeeding decades, have held that women and children escaped the Titanic in greater proportions than men. Heres a summary of the relevant data. Do you think that survival was independent of whether the person was male or female? Explain. Sex Female Male Total Survival Alive 343 367 710 Dead 127 1364 1491 Total 470 1731 2201

Steering revisited You could have checked the data in Exercise 43 for evidence of racial steering using twoproportion z procedures. a) Find the z-value for this approach, and show that when you square your z-value, you get the value of x2 you calculated in Exercise 37. b) Show that the resulting P-values are the same.

Survival on the Titanic, one more time In Exercise 44 you could have checked for a difference in the chances of survival for men and women using two-proportion z procedures. a) Find the z-value for this approach. b) Show that the square of your calculated value of z is the value of x2 you calculated in Exercise 44. c) Show that the resulting P-values are the same.

Pregnancies Most pregnancies result in live births, but some end in miscarriages or stillbirths. A June 2001 National Vital Statistics Report examined those outcomes in the United States during 1997, broken down by the age of the mother. The table shows counts consistent with that report. Is there evidence that the distribution of outcomes is not the same for these age groups?

Education by age Use the survey results in the table to investigate differences in education level attained among different age groups in the United States.