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High school seniors with strong academic records apply to the nation's most selective colleges in greater numbers each year. Because the number of slots remains relatively sta- ble, some colleges reject more early applicants. Suppose that for a recent admissions class, an Ivy League college received 2851 applications for early admission. Of this group, it admitted 1033 students early, rejected 854 outright, and deferred 964 to the regular admission pool for further consideration. In the past, this school has admitted 18%% of the deferred early admission applicants during the regular admission process. Counting the students admitted early and the students admitted during the regular admission process, the total class size was 2375 . Let E,R,E,R, and DD represent the events that a student who applies, for early admission is admitted early, rejected outright, or deferred to the regular admis- sions pool. a. Use the data to estimateP(E),P(R),andP(D).b. Are eventsEandDmutually exclusive? FindP(ED).a. Use the data to estimateP(E),P(R),andP(D).b. Are eventsEandDmutually exclusive? FindP(ED). c. For the2375students that were admitted, what is the probability that a randomlyselected student was accepted during early admission?c. For the2375students that were admitted, what is the probability that a randomlyselected student was accepted during early admission? d. Suppose a student applies for early admission. What is the probability that the student will be admitted for early admission or be deferred and later admitted during the regular admission process?

Visa Card USA studied how frequently young consumers, ages 18 to 24,24, use plastic (debit and credit) cards in making purchases (Associated Press, January 16,2006).16,2006). The results of the study provided the following probabilities.

The probability that a consumer uses a plastic card when making a purchase is.37.7.The probability that a consumer uses a plastic card when making a purchase is.37.7. Given that the consumer uses a plastic card, there is a19probability that the con-sumer is18to24vears old.Given that the consumer uses a plastic card, there is a19probability that the con-sumer is18to24vears old. Given that the consumer uses a plastic card, there is a.81 probability that the consumeris more than24years old.Given that the consumer uses a plastic card, there is a.81 probability that the consumeris more than24years old. U.S. Census Bureau data show that 14%% of the consumer population is 18 to 24 years old. a. Given that the consumer is18to24years old, what is the probability that the consumeruses a plistic card?a. Given that the consumer is18to24years old, what is the probability that the consumeruses a plistic card? b. Given that the consumer is over24years old, what is the probability that the consumeruses a plastic card?b. Given that the consumer is over24years old, what is the probability that the consumeruses a plastic card? c. What is the interpretation of the probabilities shown in parts (a) and (b)? d. Should companies such as Visa, MasterCard, and Discover make plastic cards availableto the18- to24-year-old age group before these consumers have had time to establish acredit history? If no, why? If yes, what restrictions might the companies place on thisage groun?d. Should companies such as Visa, MasterCard, and Discover make plastic cards availableto the18- to24-year-old age group before these consumers have had time to establish acredit history? If no, why? If yes, what restrictions might the companies place on thisage group?

The American Council of Education reported that 47%% of college freshmen eam a degree and graduate within five years (Associated Press, May 6,2002).6,2002). Assume that graduation records show that women make up 50%% of the students who graduated within five years, but only 45%% of the students who did not graduate within five years. The students who had not graduated within five years either dropped out or were still working on their degrees.

a.A2WLetA1=the student graduated within five years=the student did not graduate within five years=the student is a female studentUsing the given information, what are the values forP(A1),P(A2),P(W|A1),andP(W|A2)?a.LetA1=the student graduated within five yearsA2=the student did not graduate within five yearsW=the student is a female studentUsing the given information, what are the values forP(A1),P(A2),P(W|A1),andP(W|A2)? b. What is the probability that a female student will graduate within five years?c. What is the probability that a male student will graduate within five years?d. Given the preceding results, what are the percentage of women and the percentage ofmen in the entering freshman class?b. What is the probability that a female student will graduate within five years?c. What is the probability that a male student will graduate within five years?d. Given the preceding results, what are the percentage of women and the percentage ofmen in the entering freshman class?

Two Wharton professors analyzed 1,613,2341,613,234 putts by golfers on the Professional Golfers Association (PGA) Tour and found that 983,764983,764 of the putts were made and 629,470629,470 of the putts were missed. Further analysis showed that for putts that were made, 64.0%% of the time the player was attempting to make a par putt and 18.8%% of the time the player was attempting to make a birdie putt. And, for putts that were missed, 20.3%% of the time the player was attempting to make a par putt and 73.4%% of the time the player was attempting to make a birdie putt (Is Tiger Woods Loss Averse? Persistent Bias in the Face of Experi- ence, Competition, and High Stakes, Pope, D. G., and M. E. Schweitzer, June 2009,2009, The Wharton School, University of Pennsylvania.). a. What is the probability that a PGA Tour player makes a putt?b. Suppose that a PGA Tour player has a putt for par. What is the probability that theplaver will make the putt?a. What is the probability that a PGA Tour player makes a putt?b. Suppose that a PGA Tour player has a putt for par. What is the probability that theplaver will make the putt? c. Suppose that a PGA Tour player has a putt for birdie. What is the probability that theplayer will make the putt?d. Comment on the differences in the probabilities computed in parts (b) and (c).c. Suppose that a PGA Tour player has a putt for birdie. What is the probability that theplayer will make the putt?d. Comment on the differences in the probabilities computed in parts (b) and (c).

A large consumer goods company ran a television advertisement for one of its soap prod- ucts. On the basis of a survey that was conducted, probabilities were assigned to the fol- lowing events. BSBS=individual purchased the product=individual recalls seeing the advertisement=individual purchased the product and recalls seeing the advertisementB=individual purchased the productS=individual recalls seeing the advertisementBS=individual purchased the product and recalls seeing the advertisement The probabilities assigned were P(B)=.20,P(S)=.40,P(B)=.20,P(S)=.40, and P(BS)=.12P(BS)=.12 a. What is the probability of an individual's purchasing the product given that theindividual recalls seeing the advertisement? Does seeing the advertisement increasethe probability that the individual will purchase the product? As a decision maker,would you recommend continuing the advertisement (assuming that the cost isreasonable)?a. What is the probability of an individual's purchasing the product given that theindividual recalls seeing the advertisement? Does seeing the advertisement increasethe probability that the individual will purchase the product? As a decision maker,would you recommend continuing the advertisement (assuming that the cost isreasonable)? b. Assume that individuals who do not purchase the company's soap product buy fromits competitors. What would be your estimate of the company's market share? Wouldyou expect that continuing the advertisement will increase the company's market?share? Why or why not?b. Assume that individuals who do not purchase the company's soap product buy fromits competitors. What would be your estimate of the company's market share? Wouldyou expect that continuing the advertisement will increase the company's market?share? Why or why not? $\begin{array}{l}{\text { c. The company also tested another advertisement and assigned it values of } P(S)=.30} \\ {\text { and } P(B \cap S)=.10 . \text { What is } P(B | S) \text { for this other advertisement? Which advertise- }} \\ {\text { ment seems to have had the bigger effect on customer purchases? }}\end{array}

Cooper Realty is a small real estate company located in Albany, New York, specializing primarily in residential listings. It recently became interested in determining the likeli- hood of one of its listings being sold within a certain number of days. An analysis of com- pany sales of 800 homes in previous vears produced the following data. a. IfAis defined as the event that a home is listed for more than90days before beingsold, estimate the probability ofA.b. IfBis defined as the event that the initial asking price is under$150,000,estimate theprobability ofB.a. IfAis defined as the event that a home is listed for more than90days before beingsold, estimate the probability ofA.b. IfBis defined as the event that the initial asking price is under$150,000,estimate theprobability ofB. c. What is the probability ofAB?d. Assuming that a contract was just signed to list a home with an initial asking price ooless than$150,000,what is the probability that the home will take Cooper Realty morethan90days to sell?c. What is the probability ofAB?d. Assuming that a contract was just signed to list a home with an initial asking price ooless than$150,000,what is the probability that the home will take Cooper Realty morethan90days to sell? e. Are events AA and BB independent?

Statistics from the 2009 Major League Baseball season show that there were 157 players who had at least 500 plate appearances. For this group, 42 players had a batting average of .300 or higher, 53 players hit 25 or more home runs, and 14 players had a batting average of 300 or higher and hit 25 or more home runs. Only four players had 200 or more hits (ESPN website, January 10,2010)10,2010) . Use the 157 players who had at least 500 plate appear- ances to answer the following questions. a. What is the probability that a randomly selected player had a batting average of.300or higher?b. What is the probability that a randomly selected player hit25or more home runs?c. Are the events "having a batting average of.300or higher" and "hitting25or morehome runs" mutually exclusive?a. What is the probability that a randomly selected player had a batting average of.300or higher?b. What is the probability that a randomly selected player hit25or more home runs?c. Are the events "having a batting average of.300or higher" and "hitting25or morehome runs" mutually exclusive? d. What is the probability that a randomly selected player had a batting average of300or higher or hit25or more home runs?e. What is the probability that a randomly selected player had200or more hits? Doesobtaining200or more hits appear to be more difficult than hitting25or more homeruns? Explain.d. What is the probability that a randomly selected player had a batting average of300or higher or hit25or more home runs?e. What is the probability that a randomly selected player had200or more hits? Doesobtaining200or more hits appear to be more difficult than hitting25or more homeruns? Explain.

Statistics from the 2009 Major League Baseball season show that there were 157 players who had at least 500 plate appearances. For this group, 42 players had a batting average of .300 or higher, 53 players hit 25 or more home runs, and 14 players had a batting average of 300 or higher and hit 25 or more home runs. Only four players had 200 or more hits (ESPN website, January 10,2010)10,2010) . Use the 157 players who had at least 500 plate appear- ances to answer the following questions. a. What is the probability that a randomly selected player had a batting average of.300or higher?b. What is the probability that a randomly selected player hit25or more home runs?c. Are the events "having a batting average of.300or higher" and "hitting25or morehome runs" mutually exclusive?a. What is the probability that a randomly selected player had a batting average of.300or higher?b. What is the probability that a randomly selected player hit25or more home runs?c. Are the events "having a batting average of.300or higher" and "hitting25or morehome runs" mutually exclusive? d. What is the probability that a randomly selected player had a batting average of300or higher or hit25or more home runs?e. What is the probability that a randomly selected player had200or more hits? Doesobtaining200or more hits appear to be more difficult than hitting25or more homeruns? Explain.d. What is the probability that a randomly selected player had a batting average of300or higher or hit25or more home runs?e. What is the probability that a randomly selected player had200or more hits? Doesobtaining200or more hits appear to be more difficult than hitting25or more homeruns? Explain.

The Professional Golf Association (PGA) measured the putting accuracy of professional golfers playing on the PGA Tour and the best amateur golfers playing in the World Ama- teur Championship (Golf Magazine, January 2007 ). A sample of 10756 -foot putts by professional golfers found 688 made putts. A sample of 1200 6-foot putts by amateur golfers found 696 made putts. a. Estimate the proportion of made 6 -foot putts by professional golfers. Estimate the proportion of made 6 -foot putts by amateur golfers. Which group had a better putting accuracy? b. What is the point estimate of the difference between the proportions of the two populations? What does this estimate tell you about the percentage of putts made by the two groups of golfers? c. What is the 95$\%$ confidence interval for the difference between the two population proportions? Interpret this confidence interval in terms of the percentage of putts made by the two groups of golfers.

An American Automobile Association (AAA) study imestigated the question of whether a man or a woman was more likely to stop and ask for directions (AAA, Janury 2006 ). The situation referred to in the study stated the following: "If you and your spouse are driving together and become lost, would you stop and ask for directions?" A sample representative of the data used by AAA showed 300 of 811 women said that they would stop and ask for directions, while 255 of 750 men said they would stop and ask for directions. a. The AAA research hypothesis was that women would be more likely to say that they would stop and ask for directions. Formulate the null and alternative hypotheses for this study. b. What is the percentage of women who indicated that they would stop and ask for directions? c. What is the percentage of men who indicated that they would stop and ask for directions? d. $A t \alpha=.05$ , test the hypothesis. What is the $p$ -value, and what conclusion would you expect AAA to draw from this study?