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MA 2612 Test 2 B '99 Solutions







NAME:





1.
In an effort to study the health implications of lead in gasoline, researchers regressed mean lead concentrations ($\mu$g/dl) in umbilical-cord blood of babies born at a major Boston hospital (variable BABYLEAD) on monthly gasoline lead sales (metric tons) in Massachusetts over a 14 month period in 1980-81 (variable SOLDLEAD). SAS/INSIGHT Regression output is shown in Figures 1 and 2.
(a)
(9 points) What is the response? The predictor? The regressor?

ANS: Response: BABYLEAD; predictor, regressor: SOLDLEAD

(b)
(10 points) Write out the equation of the fitted model. Interpret the slope.

ANS:

\begin{displaymath} \hat{\mbox{BABYLEAD}}=4.1082+0.0149 \mbox{SOLDLEAD}\end{displaymath}

The change in predicted BABYLEAD per unit change in SOLDLEAD is 0.0149

(c)
(10 points) Is it wise to interpret the intercept? Why or why not? If it is, Interpret the intercept.

ANS: It is not wise, since a SOLDLEAD value of 0 is well outside the range of the data.

(d)
(10 points) Construct a 95% confidence interval for the slope. Does the result indicate that there is a significant relationship between BABYLEAD and SOLDLEAD?

\begin{displaymath} 0.0149\pm (0.0047)(2.1788)=(0.00466,0.02514)\end{displaymath}

ANS: Since the interval does not contain 0, it indicates a significant relationship between BABYLEAD and SOLDLEAD.

(e)
(5 points) By what proportion is the uncertainty in predicting the response reduced by using the regression model?

ANS: 0.4533

(f)
(5 points) What is the correlation between BABYLEAD and SOLDLEAD?

ANS: 0.6733

(g)
(10 points) Evaluate the fit.

There is no evidence from the residual plot of lack of fit.

(h)
(5 points) Estimate the standard deviation of the random errors.

ANS: 0.6162

2.
For the 1997-98 academic year, a new statewide ban on racial and gender preferences in public college admissions in California went into effect. The following table displays the number of new students admitted to the University of California law schools for the 1996-97 and 1997-98 academic years, broken down by race.

Year Ethnicity          
           
Frequency          
Percent          
Row Pct.          
Col Pct.          
Expected Native Asian  
Residual Black Latino American American White Total
1996-97 *43 89 10 116 464 722
  2.8 *5.8 0.7 7.5 30.1 46.8
  6.0 12.3 *1.4 16.1 64.3  
  72.9 60.1 71.4 *43.1 44.1  
  27.6 69.3 6.6 125.9 *492.7  
  2.9 2.4 1.4 -0.9 -1.3  
1997-98 16 59 4 153 589 821
  1.0 3.8 0.3 9.9 38.2 53.2
  2.0 7.2 0.5 18.6 71.7  
  27.1 39.9 28.6 56.9 55.9  
  31.4 78.7 7.4 143.1 560.3  
  *-2.7 -2.2 -1.3 0.8 1.2  
Total 59 148 14 269 1053 1543
  3.8 9.6 0.9 17.4 68.2 100

(a)
(6 points) Fill in the six values missing from the table. ANS: See table

(b)
(10 points) What is the conditional distribution of ethnicity given the year is 1996-97? 1997-98? Compare these conditional distributions: where do you find differences?

ANS: For 1996-97 the percentage distributions of Blacks, Latinos, Asian Americans and Whites are 6.0, 12.3 1.4, 16.1, and 64.2; for 1997-98, these percentages are 2.0, 7.2, 0.5, 18.6, and 71.7. These percentages suggest

  • Substantial declines in the percentage of Black (4%) and Latino (5.1%) students admitted.
  • A modest decline (0.9%) in the percentage of Native American students admitted.
  • A modest gain (2.5%) in the percentage of Asian American students admitted.
  • A substantial gain (7.5%) in the percentage of White students admitted.

(c)
(10 points) At the 0.05 level of significance, do you conclude that the admission mix is different for the two years? Use the fact that the $\chi^2$ test statistic takes the value 34.53. Justify your answer.

ANS: Since 34.52 exceeds 9.488, the 0.95 quantile of the $\chi^2_4$ distribution, we conclude the admission mix is significantly different for the two years.

(d)
(10 points) Use the Pearson residual values to assess the major sources of the differences in the admission mix.

ANS: The major sources of differences in the Pearson residuals are Blacks and Latinos. Blacks have Pearson residuals of 2.9 for 1996-97 and -2.7 for 1997-98, and Latinos have Pearson residuals of 2.2 for 1996-97 and -2.4 for 1997-98, corresponding to the substantial decline in admissions to those ethnic groups.


 
Figure 1: Regression output for lead data, problem 1.
\begin{figure} \centerline{ \includegraphics *[height=8in,width=6in]{t2q1a.eps}} \vspace{2ex}\end{figure}


 
Figure 2: Normal quantile plot and summary measures for Studentized residuals, lead data, problem 1.
\begin{figure} \centerline{ \includegraphics *[height=5in,width=6in]{t2q1b.eps}} \vspace{2ex}\end{figure}

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Joseph D Petruccelli
12/5/1999