AP® Statistics 2013 Scoring GuidelinesThe College BoardThe College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 1900, the College Board was created to expand access to higher education. Today, the membership association is made up of over 6,000 of the world’s leading educational institutions and is dedicated to promoting excellence and equity in education. Each year, the College Board helps more than seven million students prepare for a successful transition to college through programs and services in college readiness and college success — including the SAT® and the Advanced Placement Program®. The organization also serves the education community through research and advocacy on behalf of students, educators, and schools. 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.© 2013 The College Board. College Board, Advanced Placement Program, AP, SAT and the acorn logo are registered trademarks of the College Board. All other products and services may be trademarks of their respective owners. Visit the College Board on the Web: www.collegeboard.org.AP Central is the official online home for the AP Program: apcentral.collegeboard.org.www.mymathscloud.com
AP® STATISTICS2013 SCORING GUIDELINES© 2013 The College Board. Visit the College Board on the Web: www.collegeboard.org.Question 1Intent of QuestionThe primary goals of this question were to assess a student’s ability to (1) use a stem-and-leaf plot to answer a question about a distribution of data; (2) identify and compute an appropriate confidence interval after checking the necessary conditions; and (3) interpret the interval in the context of the data. SolutionPart (a):Four of the 23 crows in the sample had a lead level greater than 6.0 ppm. Therefore, the proportion of crows in the sample that were classified as unhealthy is ≈40.174.23Part (b):Step 1: Identifies the appropriate confidence interval (by name or by formula) and checks appropriate conditions.The appropriate procedure is a one-sample t-interval for a population mean. Conditions: 1. The sample is randomly selected from the population. 2. The population has a normal distribution, or the sample size is large. The first condition is met because we were told that the crows were randomly selected. The sample size of 23 is not considered large, so we need to examine the sample data to assess whether it is reasonable to assume that the population distribution of lead levels for all crows in this region is normal. The stem-and-leaf plot shows no strong skewness or outliers, so we will consider the second condition to be met.Step 2: Correct mechanics A 95% confidence interval for the population mean μ is given by: *.sxtn± The critical value for 95% confidence, based on =23 – 122 degrees of freedom, is =*2.074.t The 95% confidence interval for μ is therefore 1.124.902.0744.900.484,23± ×≈±which is the interval (4.416, 5.384) ppm.Using the raw data rather than the given summary statistics, the 95% confidence interval for μ is (4.411, 5.3803).Step 3: InterpretationWe can be 95% confident that the population mean lead level among all crows in this region is between 4.416 and 5.384 parts per million.www.mymathscloud.com
AP® STATISTICS2013 SCORING GUIDELINES© 2013 The College Board. Visit the College Board on the Web: www.collegeboard.org.Question 1 (continued)ScoringThis question is scored in four sections. Section 1 consists of part (a); section 2 consists of part (b), step 1; section 3 consists of part (b), step 2; and section 4 consists of part (b), step 3. Each section is scored as essentially correct (E), partially correct (P), or incorrect (I). Section 1is scored as follows:Essentially correct (E) if the response provides the correct numerical answer, as a decimal or as a fraction, with work shown. Partially correct (P) if the response provides the correct numerical answer as a decimal but does not show the fraction that produced the answer, OR shows a fraction with the correct numerator but an incorrect denominator, OR shows the correct fraction but computes an incorrect answer.Incorrect (I) if the response does not meet the criteria for E or P. Section 2is scored as follows:Essentially correct (E) if the response identifies a one-sample t-interval for a population mean (either by name or formula) AND also checks both the random sampling and the normality/large sample condition correctly. Note: Any reasonable comment about the normality displayed in the stem-and-leaf plot (or another appropriately sketched plot) is acceptable. Partially correct (P) if the response identifies the correct procedure AND checks only one of the two conditions correctly OR does not identify the correct procedure but does check both conditions correctly. Incorrect (I) if the response identifies the correct procedure but does not check conditions correctly OR does not identify the correct procedure and checks at most one condition correctly. Section 3is scored as follows:Essentially correct (E) if the response gives the correct confidence interval. Supporting work is not required, but if included, it must be correct. Partially correct (P) if the response gives an incorrect but reasonable confidence interval with appropriate supporting work shown OR gives a correct confidence interval with incorrect (but appropriate) supporting work shown.Note: If the response identifies a one-sample z-interval as the correct procedure in Section 2, then the response earns a P in Section 3 if this interval is calculated correctly. Incorrect (I) if the response makes use of an inappropriate procedure for a confidence interval about a population mean.www.mymathscloud.com
AP® STATISTICS2013 SCORING GUIDELINES© 2013 The College Board. Visit the College Board on the Web: www.collegeboard.org.Question 1 (continued)Section 4is scored as follows:Essentially correct (E) if the response gives a reasonable interpretation of the interval that includes four elements: 1.Estimating a mean 2.Inference about a population 3.95% confidence 4.Context (lead level/ppm and crows). Partially correct (P) if the response gives a reasonable interpretation of the interval that includes both of the first two elements and one of the last two elements; ORif the response gives a correct interpretation of the confidence level in context (lead level/ppm andcrows) but does not attempt to interpret the confidence interval.Incorrect (I) if the response does not meet the criteria for an E or a P. Each essentially correct (E) response counts as 1 point, and a partially correct (P) response counts as ½ point.4 Complete Response3 Substantial Response2 Developing Response1 Minimal ResponseIf 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 strength of the response and communication.www.mymathscloud.com
AP® STATISTICS2013 SCORING GUIDELINES© 2013 The College Board. Visit the College Board on the Web: www.collegeboard.org.Question 2 Intent of QuestionThe primary goals of this question were to assess a student’s ability to (1) recognize and explain why a particular sampling method is likely to be biased; (2) describe a method for selecting a simple random sample from a population using a computer random number generator; and (3) demonstrate an understanding of the principle of stratification by describing circumstances in which one stratification variable would be better than another. SolutionPart (a):The first 500 students who enter the football stadium were not likely to be representative of the population of all students at the university. In other words, these 500 students were likely to differ systematically from the population with regard to many variables. For example, these 500 students might have more school pride than the population of students as a whole, which might be related to their opinions about the appearance of university buildings and grounds. Perhaps their school pride is related to having more positive opinions about the appearance of university buildings and grounds, in which case the sample proportion of students who were satisfied would be biased toward overestimating the population proportion of students who were satisfied.Part (b):Obtain a list of all 70,000 students at the university. Assign an identification number from 1 to 70,000 to each student. Then use a computer to generate 500 random integers between 1 and 70,000 without replacement. The students whose ID numbers correspond to those numbers were then selected for the sample. Part (c):Stratifying by campus would be more advantageous than stratifying by gender provided that opinions about appearance of university buildings and grounds between the two campuses differ more than the opinions about appearance of university buildings and grounds between the two genders. ScoringParts (a), (b), and (c) were scored as essentially correct (E), partially correct (P), or incorrect (I). Part (a) is scored as follows: Essentially correct (E) if the response correctly includes the following three components: 1.Provides a reasonable explanation for why the sample might not be representative of the population; 2.Mentions a link between the nonrepresentative nature of the convenience sample and the variable of interest (opinion about appearance of university buildings and grounds); www.mymathscloud.com
AP® STATISTICS2013 SCORING GUIDELINES© 2013 The College Board. Visit the College Board on the Web: www.collegeboard.org.Question 2 (continued) 3.Indicates a plausible direction for the bias of the estimator by: oExplicitly identifying the direction of the bias in the estimate of the population proportion of students satisfied with the appearance of the buildings and grounds, ORoStating or implying that the students in the sample were more (or less) likely to be satisfied with the appearance of the buildings and grounds than those not in the sample. Partially correct (P) if the response correctly provides exactly two of the three components listed above.Incorrect (I) if the response correctly provides one or none of the three components listed above. Part (b) is scored as follows: Essentially correct (E) if the response correctly includes the following three components: 1.Assigns numbers to the student names; 2.Uses a computer random number generator to randomly generate 500 distinct/unique numbers between 1 and 70,000; 3.Selects students whose names correspond to the 500 random numbers for the sample. Partially correct (P) if the response correctly includes two of the three components listed above (with the exception of the second reason given for an (I) below). Incorrect (I) if the response correctly includes no more than one of these three components; ORif the response proposes implementing a sampling method other than simple random sampling (for example, systematic sampling). Part (c) is scored as follows: Essentially correct (E) if the response correctly notes that the circumstance described requires more variability in opinions about appearance of university buildings and grounds between the twocampuses than between the two genders.Partially correct (P) if the response says that the circumstance described requires considerable variability in opinions about appearance of university buildings and grounds between the two campuses without explicitly comparing to variability between the two genders, OR if the response only says that the circumstance described requires more variability between the two campuses than between the two genders without referring to opinions about appearance of university buildings and grounds, OR if the response notes that the circumstance described requires homogeneity of opinions about appearance of university buildings and grounds within the two campuses. Incorrect (I) if the response does not meet the criteria for E or P. www.mymathscloud.com
