Lighthouse Initiative for Texas Classrooms

Introduction to AP* Statistics Problem 6, 1997

On the AP* Statistics exam, the free-response section consists of five short-answer questions and one investigative task, a question that students usually spend about 30 minutes answering. The investigative task question from the 1997 AP Statistics exam follows. The 1997 question focuses on the concepts of analyzing data, finding models that correctly describe the data, writing equations for these models, and then interpreting the results.

When scoring the exam papers for this question, it became obvious that students had difficulty in working with data in contextual situations and seemed to have had little experience in exploring data that did not graph into a line. Many students did not seem to be familiar with the use of square roots and logarithms in data analysis. Although square roots and logarithms are introduced in Algebra II and further explored in Precalculus, it is usually not done in this context. Including this use of square roots and logarithms in the Pre-AP* classroom is a simple and worthwhile addition to the curriculum.

Many students upon seeing the data in this problem immediately entered it into their calculators and, only then, read the question. In this particular problem, all necessary computations had been given in the stem of the question. What students needed to do was to utilize the given models to determine an appropriate asking price for an automobile. Students in Pre-AP mathematics should have practice in producing various types of graphs, in determining appropriate mathematical models for data, and in utilizing a given model to make predictions. They should also know when a question is asking them to do one of the above. Finally, students had difficulty in coding data. They did not know that 1980, 1984, 1988, etc. could be written as 80, 84, and 88 and that the coded data was used in this problem.

Another major area of difficulty for students was being able to express their thoughts in coherent, concise sentences. This should be one of the goals of all Pre-AP mathematics courses. Many students could not explain their reasoning in part (d). Students also had difficulty in correctly interpreting the questions and therefore did not respond appropriately. In part (e), students seemed to think that the question was asking for their opinion about buying a car and did not understand that the prompt "Use some or all of the given data" meant that they had to base their answer strictly on the data and information given in the problem. Students were more likely to base their answer on personal experience.

AP* Statistics Problem 6, 1997

You are planning to sell a used 1988 automobile and want to establish an asking price that is competitive with that of other cars of the same make and model that are on the market. A review of newspaper advertisements for used cars yields the following data for 12 different cars of this make and model. You want to fit a least square regression model to these data for use as a model in establishing the asking price for your car.

Production Year 1990 1991 1992 1987 1993 1991 1993 1985 1984 1982 1986 1979
Asking Price (in thousands of dollars) 6.0 7.7 8.8 3.4 9.8 8.4 8.9 1.5 1.6 1.4 2.0 1.0

The computer printouts for three different linear regression modles are shown below. Model 1 fits the asking price as a function of the production year, Model 2 fits the natural logarithm of the asking price as a function of the production year, and Model 3 fits the square root of the asking price as a function of the production year. Each printout also includes a plot of the residuals from the linear model versus the fitted values, as well as additional descriptive data produced from the least squares procedure.

Model 1

Figure 1a

The regression equation is Price = -58.1 + 0.179 Year.

Predictor

Coef

Stdev

t-ratio

p

Constant

-58.050

7.205

-8.06

0.000

Year

0.71900

0.08200

8.77

0.000

S= 0.1255                  R-sq = 88.5%

Figure 1b

Analysis of Variance

SOURCE

DF

SS

MS

F

p

Regression

1

121.10

121.10

76.88

0.000

Error

10

15.75

1.58

   

Total

11

136.85

     

 

Model 2

Figure 1a

The regression equation is LnPrice = -14.9 + 0.185 Year.

Predictor

Coef

Stdev

t-ratio

p

Constant

-14.924

1.223

-12.21

0.000

Year

0.18502

0.01392

13.30

0.000

S= 0.2130                 R-sq = 94.6%

Figure 1b

Analysis of Variance

SOURCE

DF

SS

MS

F

p

Regression

1

8.0190

8.0190

176.77

0.000

Error

10

0.4536

0.0454

   

Total

11

8.4726

     

Model 3

Figure 1a

The regression equation is Sqr = -13.3 + 0.176 Year.

Predictor

Coef

Stdev

t-ratio

p

Constant

-13.313

1.447

-9.20

0.000

Year

0.17559

0.01647

10.66

0.000

S= 0.2520                 R-sq = 91.9%

Figure 1b

Analysis of Variance

SOURCE

DF

SS

MS

F

p

Regression

1

7.2221

7.2221

113.72

0.000

Error

10

0.6351

0.0635

   

Total

11

7.8572

     
  1. Use Model 1 to establish an asking price for your 1988 automobile.
  2. Use Model 2 to establish an asking price for your 1988 automobile.
  3. Use Model 3 to establish an asking price for your 1988 automobile.
  4. Describe any shortcomings you see in these three models.
  5. Use some or all of the given data to find a better method for establishing an asking price for your 1988 automobile. Explain why your method is better.

Source: Copyright © 2006. The College Board. Reproduced with permission.

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