Investigative Task Student Saturday Session

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1 Student Notes: Prep Session Topic Strategies for Investigative Tasks The 90 minute free response section of the AP Statistics exam consists of five open ended problems and one investigative task. Students are expected to spend 12 minutes per problem (on average) on problems 1 through 5; they are advised to save at least 25 minutes for the investigative task, which is always the last problem on the exam. The investigative task involves more extended reasoning than other problems and offers students the opportunity to apply what they have learned to new situations. In addition, it is weighted more heavily in the score computation than other free response questions (weight of for investigative task versus weight of for other free response problems). The score on this question is worth 1/8 of your total score. (Don t leave it blank!) There is little you can do to study for the investigative task, but there are some strategies you can follow to help improve your score with this problem. When you open the free response questions, take a quick look at all the questions and work the one you are most confident with. Then work one or two more free response questions that seem easy to you. Do not spend time (yet) working on questions that seem puzzling. While you still have several free response questions left to do (the ones that seem a little harder), begin work on the investigative task. Almost always, the initial parts of the investigative task will look much like other free response questions and will not involve more extended reasoning. Be sure to take the time to do these parts well. As you work through the investigative task, expect the unexpected. Do not panic when you see something that you have not covered in your class. Often there are hints and guidance provided that will help you with the more creative parts of the investigative task. Read the problem carefully so that you do not overlook the guidance you are provided. If you are stuck on one part of the investigative task, move on to the next part. Sometimes the final parts are not the hardest parts of the problem. Leave the investigative task when you have done all that you can do not stew over it. Return to the other free response questions and work on those that you skipped. After you make a first pass at all the problems, return to the investigative task and think a little more about the parts you did not answer. Try to do something, even if you are not sure your answer is correct. Your statistical reasoning ability and understanding are being tested here often your statistical intuition will guide you to earn some points even when you are unsure of the answer. As you work through the free response section of the exam, make sure you budget your time so that you can complete every part of every problem that you know how to work. Keep in mind that there is no benefit in providing two solutions to a problem the weaker response will always be the one that is graded.

2 2009 #6 STATISTICS SECTION II Part B Question 6 Spend about 25 minutes on this part of the exam. Percent of Section II score 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. A consumer organization was concerned that an automobile manufacturer was misleading customers by overstating the average fuel efficiency (measured in miles per gallon, or mpg) of a particular car model. The model was advertised to get 27 mpg. To investigate, researchers selected a random sample of 10 cars of that model. Each car was then randomly assigned a different driver. Each car was driven for 5,000 miles, and the total fuel consumption was used to compute mpg for that car. A. Define the parameter of interest and state the null and alternative hypotheses the consumer organization is interested in testing. One condition for conducting a one-sample t-test in this situation is that the mpg measurements for the population of cars of this model should be normally distributed. However, the boxplot and histogram shown below indicate that the distribution of the 10 sample values is skewed to the right. Sample Mean B. One possible statistic that measures skewness is the ratio. What values of that statistic Sample Median (small, large, close to one) might indicate that the population distribution of mpg values is skewed to the right? Explain.

3 C. Even though the mpg values in the sample were skewed to the right, it is still possible that the population distribution of mpg values is normally distributed and that the skewness was due to sampling variability. To investigate, 100 samples, each of size 10, were taken from a normal distribution with the same mean and standard deviation as the original sample. For each of those 100 samples, the statistic Sample Mean was calculated. A dotplot of the 100 simulated statistics is shown below. Sample Median Sample Mean In the original sample, the value of the statistic was Based on the value of 1.03 Sample Median and the dotplot above, is it plausible that the original sample of 10 cars came from a normal population, or do the simulated results suggest the original population is really skewed to the right? Explain. D. The table below shows summary statistics for mpg measurements for the original sample of 10 cars. Minimum Q1 Median Q3 Maximum Choosing only from the summary statistics in the table, define a formula for a different statistic that measures skewness. What values of that statistic might indicate that the distribution is skewed to the right? Explain.

4 AP(R) STATISTICS 2009 SCORING GUIDELINES Question 6 Intent of Question The primary goals of this investigative task were to assess a student s ability to (1) define a parameter and state a correct pair of hypotheses; (2) explain how a particular statistic measures skewness; (3) use the observed value of the statistic and a simulated sampling distribution to make a conclusion about the shape of the population; and (4) create a new statistic and explain how it measures skewness. Solution Part (a): The parameter of interest is µ = population mean miles per gallon (mpg) of a particular car model. The null and alternative hypotheses are as follows: H O : µ = 27 H A : µ < 27 Part (b): If the distribution is right-skewed, one would expect the mean to be greater than the median. Sample Mean Therefore the ratio should be large (greater than 1). Sample Median Part (c): Because we are testing for right-skewness, the estimated p-value will be the proportion of the simulated statistics that are greater than or equal to the observed value of The dotplot shows that 14 of the 100 values are more than Because this simulated p-value (0.14) is larger than any reasonable significance level, we do not have convincing evidence that the original population is skewed to the right and conclude that it is plausible that the original sample came from a normal population. Part (d): One possible statistic is Maximum-Median. If the distribution is right-skewed, one would expect the distance from Median-Minimum the median to the maximum to be larger than the distance from the median to the minimum; thus the ratio should be greater than 1. Scoring Parts (a), (b), (c), and (d) are scored as essentially correct (E), partially correct (P), or incorrect (I). Part (a) is scored as follows: Essentially correct (E) if the student correctly states the hypotheses with a lower-tailed alternative hypothesis AND correctly defines the parameter of interest by referring to the mean of the population in context (e.g., population mean mpg, true mean fuel efficiency, mean mpg of all cars of this model, mean mpg of this car model). Partially correct (P) if only one component is correct (hypotheses or definition of parameter). Incorrect (I) otherwise. Notes: If a student attempts to define the parameter more than once (e.g., saying the parameter is... and then later saying m =... ), then these are treated as parallel solutions, and the worst attempt is scored. A symbol other than m in the hypotheses must be explicitly and correctly defined to get credit for the parameter component. If words are used in the hypotheses to describe the parameter, o the words do not count as the definition of the parameter, and o the words must be consistent with the definition of the parameter, or no credit is given for the parameter component.

5 Part (b) is scored as follows: Essentially correct (E) if the student states that large values (or values greater than 1) of the statistic indicate rightskewness AND justifies the answer with a correct statement of how right-skewness affects the relationship between the mean and median. Partially correct (P) if the student: states that large values of the statistic indicate right-skewness BUT only argues that in a normal (or symmetric) distribution the ratio should be close to 1 (i.e., does not discuss the rightskewness) OR makes a correct statement of how right-skewness affects the relationship between the mean and median BUT does not state that large values of the statistic indicate right-skewness OR has reversed the relationship between mean and median in right-skewed distributions or reversed left- and right-skewness AND states that small values of the statistic indicate rightskewness. Incorrect (I) if the student says large values indicate right-skewness but gives no explanation or an incorrect explanation. Part (c) is scored as follows: Essentially correct (E) if the student states that it is plausible that the sample came from a normal population AND justifies the choice with specific numerical evidence from the dotplot describing the relative location of the value 1.03 (e.g., 14 percent of the values are above 1.03). Partially correct (P) if the student: states that it is plausible that the sample came from a normal population AND justifies the choice by describing the relative location of the value 1.03 in the dotplot without specific numerical evidence (e.g., 1.03 is toward the middle of the distribution, 1.03 is within two standard deviations of the mean) OR states that the sample came from a population that is skewed to the right AND justifies the choice by describing the relative location of the value 1.03 in the dotplot (e.g., only 14 percent of the values are above 1.03, 1.03 is in the tail of the distribution). Incorrect (I) if the student does not refer to the relative location of 1.03 in the dotplot. Note: Common incorrect responses include the following: Simply describing the shape of the dotplot to justify normality. Saying that the dots are centered around 1, so the sample came from a normal population. Arguing that 1.03 is close to 1 (without describing its relative position in the dotplot). Stating that the sample size (or number of samples) is large, so the distribution is normal, or that the sample size is too small to make a conclusion. Stating an answer with no explanation. Part (d) is scored as follows: Essentially correct (E) if the student defines a reasonable statistic AND identifies the values that indicate rightskewness with a correct explanation of how the right-skewness affects the components of the statistic. Partially correct (P) if the student: defines a reasonable statistic but fails to adequately justify the values that indicate rightskewness OR does not define a reasonable statistic but uses values from the table (min, Q1, med, Q3, max) to describe a reasonable method for identifying right-skewness (e.g., if the distance from the median to maximum is greater than the distance from the minimum to the median, the distribution is skewed to the right). Incorrect (I) if the method does not include a comparison (e.g., simply checking for outliers on one side only). Notes: The statistic must be a formula that produces a single numerical value. For example, (maximum - median) > (median - minimum) is not a statistic. Any statistic that uses summary values not in the table (such as the mean) is incorrect.

6 Some other acceptable variations of the statistic Maximum-Median include the reciprocal (values < 1 indicate Median-Minimum right-skewness), reversing the order of subtraction in the numerator or denominator (values < -1 indicate rightskewness), and the difference of the numerator and denominator (values > 0 indicate right-skewness). Some other acceptable statistics are listed below. For each statistic, values > 1 indicate rightskewness. Other variations, including reciprocals and statistics based on differences are also acceptable. 1. Maximum-Q3 Q1-Minimum 3.! Q1+Q3$ " # 2 % & Median 2. Q3-Median Median-Q1 4.! Minimum+Maximum$ " # 2 % & Median Each essentially correct (E) response counts as 1 point, and each partially correct (P) response counts as 1/2 point. 4 Complete Response 3 Substantial Response 2 Developing Response 1 Minimal Response If a response is between two scores (for example, 21/2 points), use a holistic approach to determine whether to score up or down, depending on the strength of the response and communication.

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9 Overview The primary goals of this investigative task were to assess a student s ability to (1) define a parameter and state a correct pair of hypotheses; (2) explain how a particular statistic measures skewness; (3) use the observed value of the statistic and a simulated sampling distribution to make a conclusion about the shape of the population; and (4) create a new statistic and explain how it measures skewness. Sample: 6A Score: 4 In part (a) the student correctly defines the parameter of interest, including the concepts of mean and population in context, and correctly states the null and alternative hypotheses using standard notation. Part (a) was scored as essentially correct. In part (b) the student correctly states that large values of the statistic indicate that the distribution is skewed to the right and gives a justification that describes the relationship between the mean and median when the distribution is skewed right. Part (b) was scored as essentially correct. In part (c) the student correctly states that there is not convincing evidence that the original sample came from a skewed population and gives specific numerical evidence from the dotplot ( 14 points out of 100 ). Although not necessary, the student goes on to correctly explain when there would be convincing evidence that the distribution was skewed right. Part (c) was scored as essentially correct. In part (d) the student provides a reasonable statistic to measure skewness. The student then correctly identifies the values of the statistic that indicate right-skewness and justifies the response by discussing how the components of the statistic are affected by right-skewness. Part (d) was scored as essentially correct. The entire answer, based on all four parts, was judged a complete response and earned a score of 4 points.

10 2008 #6 STATISTICS SECTION II Part B Question 6 Spend about 25 minutes on this part of the exam. Percent of Section II score 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. Administrators in a large school district wanted to determine whether students who attended a new magnet school for one year achieved greater improvement in science test performance than students who did not attend the magnet school. Knowing that more parents would want to enroll their children in the magnet school than there was space available for those children, the district administrators decided to conduct a lottery of all families who expressed interest in participating. In their data analysis, the administrators would then compare the change in test scores of those children who were selected to attend the magnet school with the change in test scores of those who applied to attend the magnet school but who were not selected. The tables below show the scores on the same science pretest and the same science posttest for 20 students. Of the 20 students, 8 were randomly selected from the magnet school and 12 were randomly selected from those who applied to attend the magnet school but who were not selected and then attended their original school.

11 A. Perform a test to determine whether students who attend the magnet school demonstrate a significantly higher mean difference in test scores (Posttest Pretest) than students who applied to attend the magnet school but who were not selected and then attended their original school.

12 Administrators were also interested in using pretest scores on this test as a predictor of posttest scores on the test. The following computer output contains the results from separate regression analyses on the magnet school scores and on the original school scores. The accompanying graph displays the data and separate regression lines for the magnet and original schools. B. (i) State the equation of the regression line for the magnet school and interpret its slope in the context of the question. (ii) State the equation of the regression line for the original school and interpret its slope in the context of the question.

13 C. To determine whether there is a significant correlation between pretest score and posttest score, a test of the following hypotheses will be performed. H O : There is no correlation between pretest score and posttest score (true slope = 0) versus H A : There is a correlation between pretest score and posttest score (true slope! 0) (i) Using the regression output, state the p-value and conclusion for this test at the magnet school. Assume the conditions for inference have been met. (ii) Using the regression output, state the p-value and conclusion for this test at the original school. Assume the conditions for inference have been met. D. What additional information do the regression analyses give you about student performance on the science test at the two schools beyond the comparison of mean differences in part (A)?

14 AP STATISTICS 2008 SCORING GUIDELINES Question 6 Intent of Question The primary goals of this investigative task were to assess a student s ability to (1) identify and conduct an appropriate inference based on the differences in the posttest and pretest scores; (2) identify and interpret appropriate information from statistical software; (3) make an inference based on separate regression analyses; and (4) recognize and explain the additional information provided from the different analyses. Solution Part (a): Component 1: States a correct pair of hypotheses. We want to test H O : µ DiffM = µ DiffO versus H A : µ DiffM > µ DiffO, where µ DiffM is the mean difference (posttest pretest) for all students at the magnet school and µ DiffO μ is the mean difference (posttest-pretest) for all students who applied to attend the magnet school but were not selected and then attended the original school. Component 2: Identifies a correct test (by name or formula) and checks the conditions. A two-sample t-test for means, or shows the formula: t = x M! x O + s O n M n O 1. We need to assume randomness of the sampling used. It was stated in the stem that the students from the two different schools were randomly selected. 2. We need to check the assumption that the distributions of differences (posttest pretest) for each of the two schools are normally distributed. Based on histograms and boxplots of these differences, there are no outliers or extreme skewness. Because these graphs reveal no obvious departures from normality, it appears reasonable to proceed with the t-test. s M 2 2 Component 3: Performs correct mechanics, which include the value of the test statistic and p-value (or rejection region): t = x! x M O 11.75! 3 2 s M + s = O " (with df=8.69) n M n O 8 12 and p =

15 Component 4: Draws an appropriate conclusion in context and with linkage to the p-value (or rejection region): Using α = 0.05, we reject H0 because < We conclude that the sample data provide convincing evidence that students who attend the magnet school have a higher mean difference in test scores than students who attend the original school. Part (b): Let y = posttest score and x = pretest score. (i). The predicted regression equation for the magnet school is ŷ = x. For students at the magnet school, a 1-point increase in the pretest score is associated with a predicted increase of points on the posttest (i.e., the slope is positive but close to zero). (ii). The predicted regression equation for the original school is ŷ = x. For students at the original school, a 1-point increase in the pretest score is associated with a predicted increase of points on the posttest (i.e., the slope is positive and close to 1). Part (c): (i). The test statistic is t = 0.40 with a p-value of Because the p-value is greater than any reasonable significance level, say 0.05, we fail to reject H O, We conclude that there is insufficient evidence to state that pretest score is a significant predictor of posttest score at the magnet school. The data do not support a conclusion that a correlation exists between pretest and posttest scores at the magnet school. (ii). The test statistic is t = 6.09 with a p-value of Because the p-value is less than any reasonable significance level, say 0.05, we reject H O and conclude that there is sufficient evidence to state that pretest score is a significant predictor of posttest score at the original school. The data support a conclusion that a correlation exists between pretest and posttest scores at the original school. Part (d): Unlike the two-sample analysis of differences in part (a), the regression analyses allow us to explore the relationship between pretest and posttest scores at each school. From the regression output and graph, we see that students with low pretest scores benefit more from attending magnet schools, as compared with students with low pretest scores at the original school. Also at the magnet school, students with low pretest scores benefit more than students with high pretest scores. In other words, students at the magnet school all score high on the posttest, regardless of how they scored on the pretest. But at the original school, only students who scored high on the pretest scored high on the posttest. Scoring Parts (a), (b), (c), and (d) are scored as essentially correct (E), partially correct (P), or incorrect (I). Part (a) is scored as follows: Essentially correct (E) if all four components are correct. Partially correct (P) if two or three components are correct. Incorrect (I) if at most one component is correct. Part (b) is scored as follows: Essentially correct (E) if all four components both equations and both interpretations in (i) and (ii) are correct. Partially correct (P) if two or three components are correct. Incorrect (I) if at most one component is correct. Part (c) is scored as follows: Essentially correct (E) if all four components both p-values and both conclusions in (i) and (ii) are correct. Partially correct (P) if two or three components are correct. Incorrect (I) if at most one component is correct. Part (d) is scored as follows: Essentially correct (E) if the response clearly explains how the regression analyses provide additional information in this context by addressing the impact of the magnet school on students with low pretest scores. Partially correct (P) if the response clearly describes how the regression analyses provide additional information in context but does not explain the impact of the magnet school on students with low pretest scores. Incorrect (I) if the response does not meet the criteria for an E or P.

16 Each essentially correct (E) response counts as 1 point. Each partially correct (P) response counts as ½ point. 4 Complete Response 3 Substantial Response 2 Developing Response 1 Minimal Response If a response is between two scores (for example, 2½ points) use a holistic approach to determine whether to score up or down, depending on the overall strength of the response and communication.

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19 Sample: 6A Score: 4 This response is correct, complete, and well expressed in all parts. The hypotheses in part (a) are stated correctly, with the parameter symbols defined clearly as the means of the differences. The two-sample t-test is identified by name at the top of the page. The random sampling condition is mentioned, and the normality condition is checked by examining boxplots. The response indicates that the sample sizes are small enough to require normality of the differences in order to apply the two-sample t-test. The mechanics of calculating the test statistic and p-value are correct; notice that the formula for calculating the test statistic is not necessary. The conclusion is very well expressed in context, being carefully worded in terms of the mean difference in test scores. The equations in (b) are reported correctly, using good ŷ notation to denote a predicted value. The variables are not clearly defined, but the response makes clear which variable (pretest score) is represented by x and which (posttest score) is represented by y. The interpretations of slope are both good. The interpretation in (i) uses the phrase on average to convey randomness/variability, while the interpretation in (b) uses the phrase expect to. The p-values in part (c) are reported correctly, and the conclusions are presented clearly. The response is not required to show linkage between the p-value and conclusion, because the question simply asks students to state the p-value and conclusion rather than to conduct a full hypothesis test. The response to part (d) indicates what the regression analyses reveal about student performance on the science test at these two schools. The first sentence captures the essential point by observing that a magnet school student that scores poorly on the pretest is likely to score much higher on the post test. The last three sentences of the response to this part give a specific example, pointing out that a student with a pretest score of 65 is expected to score an 85 on the posttest in the magnet school and only a 69 in the original school. The use of the words likely and expected to in these sentences indicates a recognition and understanding of regression lines as models. This answer was judged complete in all its parts and especially impressive for its clear communication and concise description of the fundamental point in part (d). It merited 4 points.