AP Statistics 2007 FreeResponse Questions Form B


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1 AP Statistics 2007 FreeResponse Questions Form B The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded in 1900, the association is composed of more than 5,000 schools, colleges, universities, and other educational organizations. Each year, the College Board serves seven million students and their parents, 23,000 high schools, and 3,500 colleges through major programs and services in college admissions, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its bestknown programs are the SAT, the PSAT/NMSQT, and the Advanced Placement Program (AP ). The College Board is committed to the principles of excellence and equity, and that commitment is embodied in all of its programs, services, activities, and concerns The College Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Central, SAT, and the acorn logo are registered trademarks of the College Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation. Permission to use copyrighted College Board materials may be requested online at: Visit the College Board on the Web: AP Central is the official online home for the AP Program: apcentral.collegeboard.com.
2 Formulas begin on page 3. Questions begin on page 6. Tables begin on page
3 Formulas (I) Descriptive Statistics x = Â x i n ( ) 2 1 sx = Â x x n  1 i  sp = (  1) + ( ) ( n  1) + ( n 1) n s2 2 1 n s2 1 2 ŷ = b + b x 0 1 ( x  x i )( y  y i ) Â( x  x) Â b = 1 2 i b = y  b x 0 1 r = 1 Ê x  x y y i ˆÊ  ˆ i Â n  1 Á s Á s Ë Ë x y b 1 s y = r s x s b 1 = ( y  yˆ ) 2 Â i i n Â( x  x i ) 3
4 (II) Probability PA (» B) = PA ( ) + PB ( )  PA ( «B) PAB ( ) = PA ( «B) PB ( ) EX ( ) = μ x = Â xp i i ( ) 2 Var( X) = s 2 x = Â x  μ p i x i If X has a binomial distribution with parameters n and p, then: Ênˆ PX ( = k) = p k (1 p) nk Á Ëk  μ x = np s x = np(1  p) μ p ˆ = p s pˆ = p(1  p) n If x is the mean of a random sample of size n from an infinite population with mean μ and standard deviation, s then: μ x = μ s x = s n 4
5 (III) Inferential Statistics Standardized test statistic: statistic  parameter standard deviation of statistic Confidence interval: statistic ± ( critical value) ( standard deviation of statistic) SingleSample Statistic Sample Mean Standard Deviation of Statistic s n Sample Proportion p(1  p) n TwoSample Statistic Difference of sample means Standard Deviation of Statistic s n s2 + 1 n2 Special case when s1 = s 2 s n n 1 2 Difference of sample proportions p1(1  p1) p2(1  p2) + n n 1 2 Special case when p1 = p2 p ( 1  p) n n 1 2 Chisquare test statistic = Â ( observed  expected) 2 expected 5
6 STATISTICS SECTION II Part A Questions 15 Spend about 65 minutes on this part of the exam. Percent of Section II grade 75 Directions: Show all your work. Indicate clearly the methods you use, because you will be graded on the correctness of your methods as well as on the accuracy and completeness of your results and explanations. 1. The Better Business Council of a large city has concluded that students in the city s schools are not learning enough about economics to function in the modern world. These findings were based on test results from a random sample of 20 twelfthgrade students who completed a 46question multiplechoice test on basic economic concepts. The data set below shows the number of questions that each of the 20 students in the sample answered correctly (a) Display these data in a stemplot. (b) Use your stemplot from part (a) to describe the main features of this score distribution. (c) Why would it be misleading to report only a measure of center for this score distribution? 2007 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for students and parents). 6 GO ON TO THE NEXT PAGE.
7 2. The graph below displays the relative frequency distribution for X, the total number of dogs and cats owned per household, for the households in a large suburban area. For instance, 14 percent of the households own 2 of these pets. (a) According to a local law, each household in this area is prohibited from owning more than 3 of these pets. If a household in this area is selected at random, what is the probability that the selected household will be in violation of this law? Show your work. (b) If 10 households in this area are selected at random, what is the probability that exactly 2 of them will be in violation of this law? Show your work. (c) The mean and standard deviation of X are 1.65 and 1.851, respectively. Suppose 150 households in this area are to be selected at random and X, the mean number of dogs and cats per household, is to be computed. Describe the sampling distribution of X, including its shape, center, and spread The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for students and parents). 7 GO ON TO THE NEXT PAGE.
8 3. The United States Department of Energy is conducting an experiment to compare the heat gain in houses using two different types of windows, A and B. Six windows of each type are available for the experiment. The Department has constructed a house with twelve windows as shown on the floor plan below. In the interior of the house, each window is surrounded by a window box to capture and measure the amount of heat coming in through that window and to isolate the heat gain for each window. (a) A randomized block experiment will be used to compare the heat gain for the two types (A and B) of windows. How would you group the window boxes into blocks? (Clearly indicate your blocks using the window box numbers.) Justify your choice of blocks. (b) For the design in part (a), describe how you would assign window types (A and B) to the numbered window boxes The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for students and parents). 8 GO ON TO THE NEXT PAGE.
9 4. Each of 25 adult women was asked to provide her own height (y), in inches, and the height (x), in inches, of her father. The scatterplot below displays the results. Only 22 of the 25 pairs are distinguishable because some of the xy, pairs were the same. The equation of the least squares regression line is yˆ = x. ( ) (a) Draw the least squares regression line on the scatterplot above. (b) One father s height was x = 67 inches and his daughter s height was y = 61 inches. Circle the point on the scatterplot above that represents this pair and draw the segment on the scatterplot that corresponds to the residual for it. Give a numerical value for the residual. (c) Suppose the point x = 84, y = 71 is added to the data set. Would the slope of the least squares regression line increase, decrease, or remain about the same? Explain. (Note: No calculations are necessary to answer this question.) Would the correlation increase, decrease, or remain about the same? Explain. (Note: No calculations are necessary to answer this question.) 2007 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for students and parents). 9 GO ON TO THE NEXT PAGE.
10 5. A serum cholesterol level above 250 milligrams per deciliter (mg/dl) of blood is a risk factor for cardiovascular disease in humans. At a medical center in St. Louis, a study to test the effectiveness of a new cholesterollowering drug was conducted. One hundred people with cholesterol levels between 250 mg/dl and 300 mg/dl were available for this study. Fifty people were assigned at random to each of two treatment groups. One group received the standard cholesterollowering medication and the other group received the new drug. After taking the drug for three weeks, the 50 subjects who received the standard treatment had a mean decrease in cholesterol level of 10 mg/dl with a standard deviation of 8 mg/dl, and the 50 subjects who received the new drug had a mean decrease of 18 mg/dl with a standard deviation of 12 mg/dl. Does the new drug appear to be more effective than the standard treatment in lowering mean cholesterol level? Give appropriate statistical evidence to support your conclusion The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for students and parents) GO ON TO THE NEXT PAGE.
11 STATISTICS SECTION II Part B Question 6 Spend about 25 minutes on this part of the exam. Percent of Section II grade 25 Directions: Show all your work. Indicate clearly the methods you use, because you will be graded on the correctness of your methods as well as on the accuracy and completeness of your results and explanations. 6. Scientists interested in preserving natural habitats and minimizing the possible extinction of certain bird species conducted a study to determine if it is better for conservation groups to purchase a few large nature preserves or many small preserves in order to meet these goals. The scientists studied 13 randomly selected islands of different sizes to determine the risk of extinction for bird species. Islands are thought to be a good imitation of what would happen in a nature preserve because of their isolation. If a species lived on only one island, it was considered to be at risk. Scientists have determined that whether or not one species becomes extinct is independent of whether or not another species becomes extinct. In 1990 scientists counted the number of atrisk species on each of the selected islands. They returned to each of these islands in the year 2000 to see whether the species still existed on the islands. Species that were present in 1990 but absent in 2000 were considered extinct. Data collected by the scientists are given in the table below. Island Area (in sq km) Species at Risk in 1990 Species Extinct by 2000 Proportion Extinct The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for students and parents) GO ON TO THE NEXT PAGE.
12 (a) One scientist involved in the study believes that large islands (those with areas greater than 25 square kilometers) are more effective than small islands (those with areas of no more than 25 square kilometers) for protecting atrisk species. The scientist noted that for this study, a total of 19 of the 208 species on the large islands became extinct, whereas a total of 66 of the 299 species on the small islands became extinct. Assume that the probability of extinction is the same for all atrisk species on large islands and the same for all atrisk species on small islands. Do these data support the scientist's belief? Give appropriate statistical justification for your answer. (b) Another scientist who worked on this study thinks that the proportion of species that become extinct is more directly related to the size of the islands than simply to whether the islands are grouped as large or small. This scientist investigated the relationship between the proportion of extinct birds and the area, in square kilometers, of islands. A least squares analysis was conducted on the proportion extinct and ln(area). The regression analysis output, the scatterplot, and the residual plot are shown below. Predictor Constant ln(area) Coef StDev T P S = RSq = 87.1% Estimate the slope of the least squares regression line using a 95 percent confidence interval. Interpret your answer in the context of this situation. (c) In part (a), the scientist assumed that the probability of a species becoming extinct is the same for each of the large islands. Similarly, the scientist assumed that the probability is the same for each of the small islands. Based on your answer in part (b), do you think this is a reasonable assumption? Explain. (d) A conservation group with a longterm goal of preserving species believes that all atrisk species will disappear whenever land inhabited by those species is developed. It has an opportunity to purchase land in an area about to be developed. The group has a choice of creating one large nature preserve with an area of 45 square kilometers and containing 70 atrisk species, or 5 small nature preserves, each with an area of 3 square kilometers and each containing 16 atrisk species unique to that preserve. Which choice would you recommend and why? STOP END OF EXAM 2007 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for students and parents). 12
13 Probability Table entry for z is the probability lying below z. z Table A Standard normal probabilities z
14 Probability Table entry for z is the probability lying below z. Table A (Continued) z z
15 Table entry for p and C is the point t* with probability p lying above it and probability C lying between t * and t*. Probability p t* Table B t distribution critical values Tail probability p df % 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8% 99.9% Confidence level C 15
16 Table entry for p is the point ( χ 2 ) with probability p lying above it. (χ 2 ) Probability p Table C 2 c critical values Tail probability p df
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