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Use the data in BENEFITS to answer this question. It is a school-level data set at the $\mathrm{K}-5$ level on average teacher salary and benefits. See Example 4.10 for background.(i) Regress lavgsal on $b s$ and report the results in the usual form. Can you reject$\mathrm{H}_{0} : \beta_{b s}=0$ against a two-sided alternative? Can you reject $\mathrm{H}_{0} : \beta_{b s}=-1$ against$\mathrm{H}_{1} : \beta_{b s}>-1 ?$ Report the $p$ -values for both tests.(ii) Define $l b s=\log (b s) .$ Find the range of values for $l b s$ and find its standard deviation.How do these compare to the range and standard deviation for $b s ?$(iii) Regress lavgsal on lbs. Does this fit better than the regression from part (i)?(iv) Estimate the equationlavgsal $=\beta_{0}+\beta_{1} b s+\beta_{2}$ lenroll $+\beta_{3} / s t a f f+\beta_{4} / u n c h+u$and report the results in the usual form. What happens to the coefficient on $b s ?$ Is it now statistically different from zero?(v) Interpret the coefficient on lstaff. Why do you think it is negative?(vi) Add lunch^{2} \text { to the equation from part } (iv). Is it statistically significant? Compute the turning point (minimum value) in the quadratic, and show that it is within the range of the observed data on lunch. How many values of lunch are higher than the calculated turning point?(vii) Based on the findings from part (vi), describe how teacher salaries relate to school poverty rates. In terms of teacher salary, and holding other factors fixed, is it better to teach at a school with lunch $=0$ (no poverty), lunch = 50, or lunch =100 (all kids eligible for the free lunch program)?

i) 0.0044ii) 0.033iii) both models are not better fit as they both have smaller $R$ -squared value.iv) 0.482v) it can say that when holding size of the school and the poverty rate is fixed then 1$\%$ increase in variable staff will get linked with a 69$\%$ fall in variable avgsal. It can be conclude that, all teachers are willing to be paid less if they have less students to teach.vi) 73.27vii) teaching at schoold with $\operatorname{lunch}=50$ gives a predicted mean salary about 10$\%$ smaller than that at a school with lunch$=0$ (that means no poverty).

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Chapter 6

Multiple Regression Analysis: Further Issues

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Ifetayo T.

December 14, 2021

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