# 2011 # AP Exam Solutions 2011 # # # #1 5/16/2011

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1 2011 AP Exam Solutions 1. A professional sports team evaluates potential players for a certain position based on two main characteristics, speed and strength. (a) Speed is measured by the time required to run a distance of 40 yards, with smaller times indicating more desirable (faster) speeds. From previous speed data for all players in this position, the times to run 40 yards have a mean of 4.60 seconds and a standard deviation of 0.15 seconds, with a minimum time of 4.40 seconds, as shown in the table below. Based on the relationship between the mean, standard deviation, and minimum time, is it reasonable to believe that the distribution of 40- yard running times is approximately normal? Explain. Time to run 40 yards 4.60 seconds Standard Deviation 0.15 seconds Minimum 4.40 seconds Time to run 40 yards Standard Deviation Minimum 4.60 seconds 0.15 seconds 4.40 seconds Z-score for the minimum i is ( )/0.15= )/ We would expect about 10% of the data to be at this point or less if the data were normal. An assumption of normality does not seem appropriate here. (b) Strength is measured by the amount of weight, with more weight indicating more desirable (greater) strength. From previous strength data for all players in this position, the amount of weight has a mean of 310 pounds and a standard deviation of 25 pounds, as shown in the table below. Standard Deviation 310 pounds 25 pounds Calculate and interpret the z-score for a player in this position who can lift a weight of 370 pounds. Calculate and interpret the z-score for a player in this position who can lift a weight of 370 pounds. Standard Deviation 310 pounds 25 pounds Z=( )/25 = 2.4 A player with a 370 pound bench press is 2.4 standard deviations better than the mean score. 1

2 (c) The characteristics of speed and strength are considered to be of equal importance to the team in selecting a player for the position. Based on the information about the means and standard deviations of the speed and strength data for all players and the measurements listed in the table below for Players A and B, which player should the team select if the team can only select one of the two players? Justify your answer. Time to run 40 yards Player A Player B 4.42 seconds 4.57 seconds 370 pounds 375 pounds Time to run 40 yards Player A Player B 4.42 seconds 4.57 seconds 370 pounds 375 pounds Player A 40 yd dash time is 1.2 sd above average. Weight lift is 2.4 sd above average. Player B 40 yd dash time is 0.2 sd above average. Weight lift is 2.6 sd above average. Player A is the best choice, with a small deficit in the weight lift more than made up for by a far superior 40 yd dash time. 2. The table below shows the political party registration by gender of all 500 registered voters in Franklin Township. PARTY REGISTRATION FRANKLIN TOWNSHIP (a) Given that a randomly selected registered voter is a male, what is the probability that he is registered for Party Y? Female Male (a) Given that a randomly selected registered voter is a male, what is the probability that he is registered for Party Y? 48/200 or 0.24 Female Male (b) Among the registered voters of Franklin Township, are the events is a male and is registered for Party Y independent? Justify your answer based on probabilities calculated from the table above. 2

3 Female Male (b) Among the registered voters of Franklin Township, are the events is a male and is registered for Party Y independent? Justify your answer based on probabilities calculated from the table above. Independence means P(male) and P(male registered for party Y) are equal and vice versa. From part A, P(party Y male) = P(is registered for party Y)=168/500=0.336 Therefore the events are not independent. (c) One way to display the data in the table is to use a segmented bar graph. The following segmented bar graph, constructed from the data in the party registration Franklin Township table, shows party-registration distributions for males and females in Franklin Township. In Lawrence Township, the proportions of all registered voters for Parties W, X, and Y are the same as for Franklin Township, and party registration is independent of gender. Complete the graph below to show the distributions of party registration by gender in Lawrence Township. Independence means that for both genders the proportions of people registered in each party are the same. Proportion 88/500= /500= /500= Proportion 88/500= /500= /500= An apartment building has nine floors and each floor has four apartments. The building owner wants to install new carpeting in eight apartments to see how well it wears before she decides whether to replace the carpet in the entire building. The figure below shows the floors of apartments in the building with their apartment numbers. Only the nine apartments indicated with an asterisk (*) have children in the apartment. 3

5 The two groups are separate and randomly selected. A 2-sample t test is appropriate. Conditions: The data come from an SRS. With n 1 =n 2 =10, we ll check each data set for deviations from normality. Data Boxplot of Placebo, Drug Placebo Drug The two groups are separate and randomly selected. A 2-sample t test is appropriate. Conditions: The data come from an SRS and are selected independently. With n=10, we ll check each data set for deviations from normality. The data has some skew but no signs of outliers. Proceed with caution. H 0 : Mu p =Mu d The true mean reduction in cholesterol for the placebo group is the same as for the drug group. H a: Mu p < Mu d The true mean reduction in cholesterol for the placebo group is less than the drug group. Two-sample T for Placebo vs Drug N StDev SE Placebo Drug Difference = mu (Placebo) - mu (Drug) Estimate for difference: % CI for difference: ( , ), although not appropriate for a 1 sided test. T-Test of difference = 0 (vs not =): T-Value = -1.62, 1 sided P-Value = DF = 17 Since my calculated (0.0619) p is greater than my alpha of 0.01, We fail to reject the null hypothesis and cannot conclude the drug is better than the placebo #5 Windmills generate electricity by transferring energy from wind to a turbine. A study was conducted to examine the relationship between wind velocity in miles per hour (mph) and electricity production in amperes for one particular windmill. For the windmill, measurements were taken on twenty-five randomly selected days, and the computer output for the regression analysis for predicting electricity production based on wind velocity is given below. The regression model assumptions were checked and determined to be reasonable over the interval of wind speeds represented in the data, which were from 10 miles per hour to 40 miles per hour. 5

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