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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 \(\chi^{2}\) statistic. c. How many degrees of freedom does the \(\chi^{2}\) statistic have?

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. \(\ln\) 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 \(\chi^{2}\) statistic. c. How many degrees of freedom does the \(\chi^{2}\) statistic have?

Which test? For each of the following situations, state whether you'd use a chi-square goodness-of-fit test, a chisquare 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 semester's data and performs a test.

Which test, again? For each of the following situations, state whether you'd 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 workweek. 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.

M\&M's As noted in an earlier chapter, Mars Inc. says that until very recently yellow candies made up \(20 \%\) of its milk chocolate M\&M's, 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\&M's. He got 29 yellow ones, 23 red, 12 orange, 14 blue, 8 green, and 20 brown. Is this sample consistent with the company's stated proportions? Test an appropriate hypothesis and state your conclusion. a. If the M\&M's 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 goodness-of-fit, homogeneity, or independence? c. State the hypotheses. d. Check the conditions. e. How many degrees of freedom are there? f. Find \(\chi^{2}\) and the P-value. g. State a conclusion.

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. On 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 \(\chi^{2}\) test?

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 FBl's 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?

Childbirth, part 1 There is some concern that if a woman has an epidural to reduce pain during childbirth, the drug can get into the baby's bloodstream, making the baby sleepier and less willing to breastfeed. 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. Below are their results. a. What kind of test would be appropriate? b. State the null and alternative hypotheses.

Cranberry juice It's 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. Check the conditions. e. How many degrees of freedom are there? f. Find \(\chi^{2}\) 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.