Chapter 4 Practice Free Response - Answers

 

 

1.   The productivity of American agriculture has grown rapidly due to improved technology (crop varieties, fertilizers, mechanization).  Here are data on the output per hour of labor on American farms.  The variable is an “index number” that gives productivity as a percent of the 1967 level.

 

   Year    Productivity           Year      Productivity          

  1940         21                1965           91

  1945         27                1970          113

  1950         35                1975          137

  1955         47                1980          166

  1960         67                1985          217

 

You will now examine whether an exponential model is appropriate for the growth of productivity over time. Be sure to let 1940 = 0, and all other years are set as years since 1940.

 

a.   Perform an appropriate logarithmic transformation on this data. Then do a linear regression on this transformed data and give the values of “a” and “b” for the regression.

 

            Original Data                       Transformed Data                          Regression        

         

 

b.   Use the results of part (a) to determine the coefficients of the exponential model (A and B). Also write the exponential function that results from these coefficients. 

 

A = 10a = 101.335 = 21.63 and B = 10b = 100.023 = 1.054, so y = 21.63(1.054)x

 

c.   Using a residual plot, determine whether an exponential model is appropriate for this data. Explain your reasoning.

 

This residual plot seems to show a pattern; therefore the data does not accurately follow an exponential growth pattern.

 

 

 

2.   Over the past 30 years in the United States there has been a strong positive correlation between cigarette sales and the number of high school graduates.

                                                                                              

a.   Draw a diagram of the relationship and identify all variables.

 

                                                                                                                     

b.   The statement prior to #10 represents (circle the correct answer):

 

            causation                      common response                  confounding

 

 

 

3.   A 1969 study among the Pima Indians of Arizona investigated the relationship between a mother’s diabetic status and the appearance of birth defects in her children.  The results appear in the two-way table below.

 

                                  Diabetic Status                                                  

  Birth Defects    Nondiabetic   Prediabetic   Diabetic

     None               754           362           38

    One or more        31           13            9__

                                                          

 

 

a.   Fill in the row and column totals in the margins of the table.

 

                                  Diabetic Status                                                  

  Birth Defects    Nondiabetic   Prediabetic   Diabetic    

     None               754           362           38    1154

    One or more        31           13            9__    53

                      785           375           47

b.   Compute (in percents) the conditional distributions of birth defects for each diabetic status.

 

Nondiabetic

Prediabetic

Diabetic

31 / 785 = 4%

13 / 375 = 3%

9 / 47 = 19%

 

c.   Comment on any clear associations you see.

 

Mothers who are diabetic are much more likely to have children with birth defects.