1. The file Enquirer_703Sp2011.xls contains 608 observations on variables relating to passage rates on exams taken in early 1996 by students in 608 Ohio school districts.

The variables are:

%_passed percentage of students in fourth-, sixth-, ninth-, and twelfth- grades who passed exams relating to math, reading, science, writing, and citizenship proficiency exams in 608 Ohio school districts. The passage rates were combined to obtain an overall percentage of students that past the tests for each district. The exams were taken in early 1996.

%_on_ADC Percentage of students on Aid for Dependent Children (ADC) in each school district.

%_Free_Lunch Percentage of students who qualify for free or reduced-price lunches in each school district.

Median_Income The school district’s median income.

Rank School district’s ranking based on percent passed

School District Name of the school district

County County where the school district is located.

An explanation of this data set can also be found on page 574 of the text.

Estimate a simple regression model showing the relationship between percent passed and percent on ADC:

%_Passed = \(\beta \) ₀ + \(\beta \) ₁ *%_on_ADC + e

1. What percent of the variation in passage rates is explained by the variation in percent of students on ADC .
2. What is your estimated value for \(\beta \) ₁ ? Interpret what this says about the relationship between the percent on ADC and the percent of students who passed the exams.
3. Test the hypothesis that the percent of students on ADC has no influence on the percent of students passing the exam against the alternative that it does have a negative effect on the percent of students passing the exam. Assume  = .01.
4. Use a 95 percent confidence interval estimate of \(\beta \) ₁ to discuss the precision of this parameter’s estimate.
5. Develop a 95 percent prediction interval estimate of the percent of students passing the exam in a school district where %_on_ADC = 13.00 percent.
   Now estimate the multiple regression model:
   %_Passed = \(\beta \) ₀ + \(\beta \) ₁ *%_on_ADC + \(\beta \) ₂ *%_Free_Lunch + \(\beta \) ₃ *Median_Income + e
6. Using the estimate \(\beta \) ₁ obtained from the above multiple regression model, compute a 95 percent confidence interval estimate of \(\beta \) _1. Does this interval contain the value of \(\beta \) ₁ that was estimated in the simple regression model (the answer to part b )? If not, then what can explain there being so much difference between the value of \(\beta \) ₁ estimated in the simple model and the value of  ₁ estimated in the multiple regression model?
7. Explain and interpret what the coefficient estimates of \(\beta \) ₂ and \(\beta \) ₃ mean.
8. Are \(\beta \) ₂ and \(\beta \) ₃ each significant? Explain.
9. How has adding %_Free_Lunch and Median_Income into the model improved the explanatory power of the model?
10. What percent of students are expected to pass the exam in a school district where %_on_ADC = 13.00 percent, %_Free_Lunch = 25.00 percent, and the median income = $22,500?

Solution: (a) The following table is obtained:

The regression coefficients are computed

and

This means that the regression equation is

The Pearson correlation r is computed as

The coefficient of determination is

which means that 57% of the variation in % Passed is explained by % on ADC.

(b) Based on the regression equation obtained, we conclude that for an increase of one percent in the percentage on ADC there is a decrease of 0.6908 units in % Passed, on the average.

(c) Using SPSS:

The t-statistic associated to \({{\beta }_{1}}\) is t = -28.341, and the corresponding left-tailed p-value is p = 0.000/2 = 0.000 < 0.01, which means that we reject the null hypothesis, and hence, we have enough evidence to claim that the percent on ADC has a negative influence on the percent of students passing the exam.

(d) The 95% CI for \({{\beta }_{1}}\) is \(\left( -0.739,\,\,-0.643 \right)\).

(e) The predicted value of % Passed for % on ADC = 13 is

Finally, we have that the following table is required: