$4.$ Confidence intervals for predictions $-$ an application

Suppose a data set of $4,000$ observations $($nn$=4,000)$ was analyzed using OLS to examine the factors influencing students' college grade point averages. The regression results are as follows, with standard errors in parentheses:

colgpa$=1\mathrm{.}550(0\mathrm{.}09)+0\mathrm{.}0010(0\mathrm{.}00006)$sat$0\mathrm{.}03(0\mathrm{.}0007)$hsperc$0\mathrm{.}05(0\mathrm{.}01)$hsize$+0\mathrm{.}007(0\mathrm{.}003)$hsize$2$colgpa$^=1\mathrm{.}550(0\mathrm{.}09)+0\mathrm{.}0010(0\mathrm{.}00006)$sat$0\mathrm{.}03(0\mathrm{.}0007)$hsperc$0\mathrm{.}05(0\mathrm{.}01)$hsize$+0\mathrm{.}007(0\mathrm{.}003)$hsize$2$

where

colgpacolgpa$=$ GPA in collegesatsat$=$ score on the SAT examhsperchsperc$=$ percentile in high school graduating classhsizehsize$=$ size of graduating class, in hundreds

nn$=4,000$R$2$R$2=0\mathrm{.}254$R$2$R$2=0\mathrm{.}253$$$$^=0\mathrm{.}45$

The predicted college GPA for someone who scored $1,120$ on the SAT and graduated in the $27$th percentile of their $700-$person high school graduating class, is$.$

Suppose you would like to construct a $95\%$ confidence interval around the expected$$college GPA for someone with sat$=1,120$sat$=1,120,$ hsperc$=27$hsperc$=27,$ and hsize$=7$hsize$=7.$

colgpa$=1\mathrm{.}853(0\mathrm{.}01)+0\mathrm{.}0010(0\mathrm{.}00006)$sat$00\mathrm{.}03(0\mathrm{.}0007)$hsperc$00\mathrm{.}05(0\mathrm{.}01)$hsize$0+0\mathrm{.}007(0\mathrm{.}003)$hsize$02$colgpa$^=1\mathrm{.}853(0\mathrm{.}01)+0\mathrm{.}0010(0\mathrm{.}00006)$sat$00\mathrm{.}03(0\mathrm{.}0007)$hsperc$00\mathrm{.}05(0\mathrm{.}01)$hsize$0+0\mathrm{.}007(0\mathrm{.}003)$hsize$02$

where

colgpacolgpa$=$ GPA in collegesat$0$sat$0=$ sat$1,120$sat$1,120$hsperc$0$hsperc$0=$ hsperc$27$hsperc$27$hsize$0$hsize$0=$ hsize$7$hsize$7$

nn$=4,000$R$2$R$2=0\mathrm{.}254$R$2$R$2=0\mathrm{.}253$$$$^=0\mathrm{.}45$

Hint: The $95\%$ critical value of the t$$distribution is $1\mathrm{.}96.$

The confidence interval around this expected value of college GPA would be$1\mathrm{.}961\mathrm{.}96.$

Now suppose you would like to construct a $95\%$ confidence interval around the college GPA for someone with sat$=1,120$sat$=1,120,$ hsperc$=27$hsperc$=27,$ and hsize$=7$hsize$=7,$ and not $$just the expected value of college GPA for these values of the explanatory variables.

This confidence interval around this value of college GPA would be$1\mathrm{.}961\mathrm{.}96.$