A researcher found a study relating the mortaly rates for women aped 65 to 74 , y. to the proportion of calories fiom swoeteners in their diet \( x \). When researchers botoed at the association of \( \mathrm{x} \) and \( \mathrm{x} \) they found that the coefficient of determination was \( r^{2}=0,486 \) Select a conclusion that the researcher can make from this dath. a.) The comrebtion coefficient, t, is 0697 . b.) The cometation coetticient, is is 0.236 . c) About \( 70 \% \) of the variation in propertion of calories in smotbeners is explained by a linear relationahip with mortality reses. d) About St2 of the variation in mortalty rales is explained by a inear rebtationthip with proportion of
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If the coefficient of determination is 0.486, this means that 48.6% of the total variation in y can be explained by the linear relationship between x and y. So, the correct conclusion that the researcher can make from this data is: Show more…
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Use the data in BWGHT2 for this exercise. (i) Estimate the equation $\log (b w g h t)=\beta_{0}+\beta_{1} n p v i s+\beta_{2} n p v i s^{2}+u$ by OLS, and report the results in the usual way. Is the quadratic term significant? (ii)Show that, based on the equation from part (i), the number of prenatal visits that maximizes log(bwght) is estimated to be about $22 .$ How many women had at least 22 prenatal visits in the sample? (iii) Does it make sense that birth weight is actually predicted to decline after 22 prenatal visits? Explain. (iv) Add mother's age to the equation, using a quadratic functional form. Holding npvis fixed, at what mother's age is the birth weight of the child maximized? What fraction of women in the sample are older than the "optimal"" age? (v) Would you sat mother's age and number of prenatal visits explain a lot of the variation in log(bwght)? (vi) Using quadratics for both npvis and age, decide whether using the natural log or the level of bwght is better for predicting bwght.
Use the data in CPS78 $\mathrm{CPS} 78_{-} 85$ for this exercise. (i) How do you interpret the coefficient on y85 in equation $(13.2) ?$ Does it have an interesting interpretation? (Be careful here; you must account for the interaction terms y 85$\cdot$educ and $y 85$ female.) (ii) Holding other factors fixed, what is the estimated percent increase in nominal wage for a male with 12 years of education? Propose a regression to obtain a confidence interval for this estimate. [Hint: To get the confidence interval, replace y 85$\cdot$educ with $y 85 \cdot(e d u c-12) ;$ refer to Example $6.3 . ]$ (iii) Reestimate equation $(13.2)$ but let all wages be measured in 1978 dollars. In particular, define the real wage as rwage $=$ wage for 1978 and as rwage $=$ wagel 1.65 for $1985 .$ Now, use log(rwage) in place of log(wage) in estimating $(13.2) .$ Which coefficients differ from those in equation $(13.2) ?$ (iv) Explain why the $R$ -squared from your regression in part (iii) is not the same as in equation $(13.2) .$ Hint: The residuals, and therefore the sum of squared residuals, from the two regressions are identical.) (v) Describe how union participation changed from 1978 to $1985 .$ (vi) Starting with equation $(13.2),$ test whether the union wage differential changed over time. This should be a simple $t$ test.) (vii) Do your findings in parts (v) and (vi) conflict? Explain.
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