Can you find the problem? Serious Links click here. Old Test 1 Introduction through Section 2. Test 2, Oct. Test 4 Sections 5. Test 5 Sections 6. Muller -- Test 5 merged version. Test 5 Sections 7. Test 6 Sections 7. Test 7 Sections 8. Test 8 Section 9. Binomial : What are the hallmarks and differences? Includes many example problems, with solutions.

Return to Mr. Return to Mathematics Department home page. Return to St. Albans home page. If we think that the sample mean nitrogen removed at a particular buffer width might reasonably be any value in the intervals shown, a sample regression line will result from connecting any point in the interval above 6 to any point in the interval above With this in mind, the dashed lines in the plots on the next page represent extreme cases for possible sample regression lines.

From these plots, we can see that there is a wider range of possible slopes in the second plot on the bottom than in the first plot on the top. Because of this, the variability in the sampling distribution of b , the estimator for the slope of the regression line, will be smaller for the first study plan with four observations at 6 feet and four observations at 13 feet than it would be for the second study plan with four observations at 8 feet and four observations at 10 feet.

Therefore, the first study plan on the top would provide a better estimator of the slope of the regression line than the second study plan on the bottom. Although this assumption was motivated by prior experience, it may not be correct. Describe another way of choosing the widths of the buffer strips at eight locations that would enable the researchers to check the assumption of a straight-line relationship. To assess the linear relationship between width of the buffer strip and the amount of nitrogen removed from runoff water, more widths should be used.

To detect a nonlinear relationship, it would be best to use buffer widths that were spaced out over the entire range of interest. For example, if the range of interest is 6 to 13 feet, eight buffers with widths 6, 7, 8, 9, 10, 11, 12 and 13 feet could be used.

As you look back on this question in its entirety, consider the number of mental transitions that a student would need to make in order to navigate all six parts of the question successfully. Well-prepared AP Statistics students should be able to interpret the slope of a linear regression model in context in part a , and to explain why extrapolation is inappropriate in part b.

Students need to shift their thinking to the underlying assumption that the values of the response variable are normally distributed about the regression line at each value of the explanatory variable prior to parts c and d. The novel reasoning required in part e is typical of the investigative task. To answer this part correctly, students need to recognize how the slopes could vary with each of the proposed study plans.

Part f requires students to pivot one final time in considering a different aspect of the design: how to establish whether the relationship between the variables is linear. I suspect that many statistics teachers would have difficulty making all the moves expected in the last four parts of this question! In the directions that precede the investigative task, students are reminded that the quality of their response includes both statistical accuracy and clear communication.

Directions: Show all your work. Indicate clearly the methods you use, because you will be scored on the correctness of your methods as well as on the accuracy and completeness of your results and explanations. Rather than providing separate sub-scores for each of the six parts of this question, readers are instructed to group parts a and b into a single component, and parts c and d into a single component, with parts e and f counting as one component each. Scoring This question is scored in four sections.

Section 1 consists of parts a and b ; section 2 consists of parts c and d ; section 3 consists of part e ; section 4 consists of part f. Each of the four sections is scored as essentially correct E , partially correct P , or incorrect I.

The E-P-I scoring system for individual components of questions was another clever invention in the early years of the AP Statistics exam. Partially correct P is a broad category that acknowledges a wide range of student answers that include some, but not all, of the required statistical elements, or that suffer from weak communication. Recall that Section 1 consists of part a —interpreting the slope of the regression line in context—and part b —explaining why it is not appropriate to use the model to make predictions far outside the domain of values for the explanatory variable in this study.

Section 1 is scored as follows: Essentially correct E if the response includes the following two components: 1. The response in part a is correct, as evidenced by the correct interpretation of the slope, in context. The response in part b is correct, as evidenced by the identification of extrapolation as the reason that the model should not be used and the response is in context.

It is quite common for question teams to provide additional scoring notes like the ones above to assist readers in applying the rubric consistently for each component of a question.

Sample Responses Q4. Sample Responses Q5. Sample Responses Q6. Score Distributions. Score Distributions. Scoring Guidelines. Grade Distributions. Sample Responses. Solve all your PDF problems. Convert from PDF. Convert to PDF.

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