Question 5 Page 58

Question 5 Rubric Team Daren ‘Richard Petty’ Starnes – Lawrenceville School Beth ‘Jeff Gordon’ Johnson – George Mason University David ‘Dale Earnhardt Jr’ Neal – Sonoma State University Paul ‘Tony Stewart’ Rodriguez – Troy High School Nate ‘Matt Kenseth’ Wetzel – University of Wisconsin Stevens Point

Plan of Attack Review of question Model solution Detailed scoring instructions Scoring Guidelines Scoring Notes Sample papers Practice papers “What if” questions

Intent of Question The primary goal of this question is to assess students’ ability to identify, set up, perform, and interpret the results of an appropriate hypothesis test to address a particular question. More specific goals were to assess students’ ability to (1) state appropriate hypotheses; (2) identify the appropriate statistical test procedure and check appropriate conditions for inference; (3) calculate the appropriate test statistic and p-value; and (4) draw an appropriate conclusion, with justification, in the context of the study.

The goal of grading this question will be to assign the score for a Complete Response for a Substantial Response for a Developing Response for a Minimal Response 0 for a Response which is not a 4, 3, 2, or 1 This will be accomplished by assigning a score of: Essentially correct (E), Partially correct (P) or Incorrect (I) to each of four steps of the question Each essentially correct (E) step counts as 1 point. Each partially correct (P) step counts as ½ point

Question 5 A researcher conducted a study to investigate whether local car dealers tend to charge women more than men for the same car model. Using information from the county tax collector’s records, the researcher randomly selected one man and one woman from among everyone who had purchased the same model of an identically equipped car from the same dealer. The process was repeated for a total of 8 randomly selected car models. The purchase prices and the differences (woman – man) are shown in the table on the next page. Summary statistics are also shown. Do the data provide convincing evidence that, on average, women pay more than men in the county for the same car model?

Step 1: States a correct pair of hypotheses Solution Notice that the data are paired because each car model in the study includes one purchased by a man and one purchased by a woman. Step 1: States a correct pair of hypotheses Let represent the population mean difference in purchase price (woman – man) for identically equipped cars of the same model, sold to both men and women by the same dealer, in this county. The hypotheses to be tested are versus

Step 2: Identifies a correct test procedure (by name or by formula) and checks appropriate conditions. The appropriate procedure is a paired t-test. Conditions: 1. The sample is randomly selected from the population. The first condition is met because the sample of car models for each sex was randomly selected from the county tax collector’s records. The sample size (n=8 ) is not large, so we need to investigate whether it is reasonable to assume that the population of price differences is normally distributed.

Step 2: Identifies a correct test procedure (by name or by formula) and checks appropriate conditions. Conditions: 2. The population of price differences (woman – man) is normally distributed, or the sample size is large. The dotplot of sample price differences reveals a fairly symmetric distribution, so we will consider the second condition to be met.

Step 3: Correct mechanics, including the value of the test statistic and p-value (or rejection region) The test statistic is The p-value, based on a t-distribution with 8 – 1 = 7 degrees of freedom, is 0.008.

Step 4: States a correct conclusion in the context of the study, using the result of the statistical test. Because the p-value is very small (for instance, smaller than α = 0.05), we reject the null hypothesis. The data provide convincing evidence that, on average, women pay more than men in the county for the same car model.

Step 1: States a correct pair of hypotheses Scoring Step 1: States a correct pair of hypotheses Let represent the population mean difference in purchase price (woman – man) for identically equipped cars of the same model, sold to both men and women by the same dealer, in this county. The hypotheses to be tested are versus

Step 1 is scored as follows: Essentially correct (E) if the response identifies the correct parameter AND states correct hypotheses.   Partially correct (P) if the response identifies the correct parameter OR states correct hypotheses, but not both; Incorrect (I) if the response does not meet the criteria for an E or a P. Note: Defining the parameter symbol in context or simply using common parameter notation is sufficient.

Scoring Notes

Which of the following would receive full credit for Step 1? Ho: m1 - m2 = 0 vs Ha: m1 - m2 > 0 where m1 = true mean price that a women pay for car and m2 = true mean price that a man pay for cars for the same models 2. Ho: md = 0 vs Ha: md > 0 where md = mw - mm 3. Ho: md = 0 vs Ha: md > 0 where md is the true mean population difference in price between what women and men pay for the same car model 4. Ho: m = 0 vs Ha: m > 0 where m = mean of women’s purchase prices - mean of men’s purchase prices. 0 of 0

Which of the following would receive full credit for Step 1? Ho: m1 - m2 = 0 vs Ha: m1 - m2 > 0 where m1 = true mean price that a women pay for car and m2 = true mean price that a man pay for cars for the same models 2. Ho: md = 0 vs Ha: md > 0 where md = mw - mm 3. Ho: md = 0 vs Ha: md > 0 where md is the true mean population difference in price between what women and men pay for the same car model 4. Ho: m = 0 vs Ha: m > 0 where m = mean of women’s purchase prices - mean of men’s purchase prices. Uses a Two-Sample Hypothesis approach Starts off with full credit, but then the student tries to define the parameter and they do it incorrectly. Defines the parameter correctly No subscript, so they must define the parameter. The student, however, defines it as a Two-Sample hypothesis 0 of 0

Which of the following would receive full credit for Step 1? Ho: md = 0 vs Ha: mm < mw Ho: md = 0 vs Ha: md > 0 where md = mw - mm with mw = true mean price that a women pay for cars and mm = true mean price that a man pay for cars Ho: This is no difference between the true average amount that women and men pay for the same model vs Ha: The true mean amount that women pay is greater than that of men for the same model. 4. Ho: mwomen-men = 0 vs Ha: mwomen-men > 0 0 of 313

Which of the following would receive full credit for Step 1? Ho: md = 0 vs Ha: mm < mw Ho: md = 0 vs Ha: md > 0 where md = mw - mm with mw = true mean price that a women pay for cars and mm = true mean price that a man pay for cars Ho: This is no difference between the true average amount that women and men pay for the same model vs Ha: The true mean amount that women pay is greater than that of men for the same model. 4. Ho: mwomen-men = 0 vs Ha: mwomen-men > 0 The null is fine, but there is a two-sample alternative. Mu sub d is GREAT, but the student tries to define it and does it wrong! The two sample procedure written in words. The subscripts are equivalent to just d. 0 of 313

Step 2: Identifies a correct test procedure (by name or by formula) and checks appropriate conditions. Matched Pairs t test. 1. Random Selection – Stated in problem

Step 2: Identifies a correct test procedure (by name or by formula) and checks appropriate conditions. Conditions: 2. The population of price differences (woman – man) is normally distributed, or the sample size is large. The dotplot of sample price differences reveals a fairly symmetric distribution, so we will consider the second condition to be met.

Step 2 is scored as follows: Essentially correct (E) if the response identifies the correct test procedure (by name or by formula) AND checks both conditions correctly. Partially correct (P) if the response correctly completes two of the three components (identification of procedure, check of randomness condition, check of normality condition). Incorrect (I) if the response does not meet the criteria for an E or a P. Note: The randomness condition can be verified by referring to the random selection of car models or to the random selection of male and female car buyers.

Scoring Notes

Which of the following would not receive credit in Step2? I will use a one sample t-test The dotplot given has no outliers or skewness SRS√ Based on the normal probability plot of the differences, the data looks approximately normal 0 of 312

Which of the following would not receive credit in Step2? I will use a one sample t-test The dotplot given has no outliers or skewness SRS√ Based on the normal probability plot of the differences, the data looks approximately normal Stating one-sample t-test counts as naming the test procedure. Which dotplot? Three dotplots were provided to the students. This is poor communication! Random Check is OK for this procedure this year. It may not always get full credit!!! The student clearly identifies they are looking at the differences. Note NPP are not required in AP Statistics. 0 of 312

Step 3: Correct mechanics, including the value of the test statistic and p-value (or rejection region) The test statistic is The p-value, based on a t-distribution with 8 – 1 = 7 degrees of freedom, is 0.008.

Step 3 is scored as follows Essentially correct (E) if the response correctly calculates both the test statistic and the p-value. Partially correct (P) if the response correctly calculates the test statistic but not the p-value; OR if the response calculates the test statistic incorrectly but then calculates the correct p-value for the computed test statistic.   Incorrect (I) if the response does not meet the criteria for an E or a P. Note: If the response identifies a z test for a mean as the correct procedure in step 2, then the response can earn a P in step 3 if both the test statistic and the p-value are calculated correctly.

Scoring Notes Step 3: Mechanics Once a student has established the procedure being used in Step 2, the student is expected to use that procedure when performing calculations in Step 3. If not, then the student loses credit for the test statistic. Two-sample t test results (pooled or unpooled): t = 0.1195 p-value = 0.4533 using df = 13.99 (or 7, 13, 14) Critical region approach: Test procedure Critical value for one-sided test Paired t 1.895 Two-sample t with df = 7 Two-sample t with df = 13 1.771 Two-sample t with df = 14 1.761

Scoring Notes Step 3: Mechanics A response that uses the critical value for a two-sided test should score at best a P. If a student provides the correct test statistic and/or p-value but shows additional incorrect work, such as a wrong formula, reduce the score in this step by one level (i.e., E to P, or P to I). If a student attempts to interpret the p-value, but does so incorrectly, then do not give credit for the linkage component.

Step 4: States a correct conclusion in the context of the study, using the result of the statistical test. Because the p-value is very small (for instance, smaller than α = 0.05), we reject the null hypothesis. The data provide convincing evidence that, on average, women pay more than men in the county for the same car model.

Step 4 is scored as follows: Essentially correct (E) if the response provides a correct conclusion in context, also providing justification based on linkage between the p-value and the conclusion.   Partially correct (P) if the response provides a correct conclusion with linkage to the p-value, but not in context; OR if the response provides a correct conclusion in context, but without justification based on linkage to the p-value. Incorrect (I) if the response does not meet the criteria for E or P.

Step 4 Notes:

Scoring Notes

Scoring Each essentially correct (E) step counts as 1 point. Each partially correct (P) step 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 decide whether to score up or down, depending on the overall strength of the response and communication.

Paper A Page 1

Paper A Page 2

Paper A Step 1: E Step 2: E Step 3: E Step 4: E 4

Paper B Page 1

Paper B Page 2

Paper B 3 Step 1: P Step 2: E Step 3: E Step 4: P 2 sample approach No linkage of p and alpha (and accept Ha is not good!)

Paper C Page 1

Paper C Page 2

Paper C 2 Step 1: P Step 2: I Step 3: E Step 4: P 2 sample approach It is clear this student believes this is a 2 sample t-test Correct 2 sample mechanics The highest score a student could receive if a 2 sample t-test was done is a 2! No context

Paper D Page 1

Paper D Page 2

Paper D 1 Step 1: P Step 2: I Step 3: I Step 4: P 2 sample 2 sample gets an I in step 2 Could be an E if 2 sample t, but student used normal Accept the null

Time for you to try some!

Paper E Page 1

Paper E Page 2

Paper E 3 ½ to 4 Step 1: E Step 2: E Step 3: E Step 4: P No “average” Good communication with the definition of “average” in step 1 No “average”

Paper F Page 1

Paper F Page 2

Paper F 3 ½ to 3 Step 1: P Step 2: E Step 3: E Step 4: E Compared to the previous example, there is not good communication with the definition of “average” in step 1 (it looks like 2 sample hypotheses)

Paper G Page 1

Paper G Page 1

Paper G 3 ½ to 3 Step 1: E Step 2: E Step 3: E Step 4: P No use of average in the conclusion and POOR communication throughout (no definition in step 1, mu sub d is approximately normal?, t* distribution?)

Paper H Page 1

Paper H Page 2

Paper H 2 Step 1: P Step 2: I Step 3: E Step 4: P Name is wrong, random OK, what is this a graph of? (poor communication) 2 sample is wrong, but matches what we penalized in step 2 No average

Paper I Page 1

Paper I Page 2

Paper I 2 ½ to 3 Step 1: P Step 2: P Step 3: P Step 4: E Tried to define the parameter, but was wrong Step 1: P Step 2: P Step 3: P Step 4: E No normal 2 ½ to 3 Minor error in formula