Question Type 11: Determining the type of validity Exercises

Given the following data for five students: SAT scores [1100, 1250, 1300, 950, 1400] and first-year GPAs [3.0, 3.2, 3.5, 2.8, 3.7], calculate the Pearson correlation coefficient between SAT and GPA.

From the regression line $\hat y=0.84+0.002x$, calculate the residual for a student with SAT $x=1300$ and actual GPA $3.10$.

Given a sample correlation $r=0.75$ between SAT and GPA, compute the coefficient of determination and interpret its meaning in context.

Using the data from Question 1, determine the least‐squares regression line predicting GPA $y$ from SAT score $x$, and predict the GPA for a student who scored 1200 on the SAT.

Model A yields $r=0.70$ and Model B yields $r=0.50$ for predicting first-year GPA from two different entrance tests. Which model shows stronger predictive validity? Compute and compare $R^2$ values.

Assume the Pearson correlation coefficient between SAT score and first-year GPA is $r=0.60$ for a sample of $n=25$ students. Test at the $\alpha=0.05$ level whether there is a significant linear relationship.

Five students are ranked by SAT and by GPA as follows: SAT ranks [1,2,3,4,5], GPA ranks [2,1,4,5,3]. Compute the Spearman rank correlation coefficient.

A regression slope for predicting GPA from SAT is estimated as $b=0.003$ with standard error $SE_b=0.001$. Construct a 95% confidence interval for $b$ (use $t_{0.025,48}\approx2.01$).

In a multiple regression predicting GPA, SAT score has a partial correlation $r_{y,x_1\cdot x_2}=0.40$ controlling for high school GPA. Compute the incremental $R^2$ attributable to SAT.

A logistic regression predicting probability of achieving GPA≥3.0 from SAT yields coefficient $\beta=0.004$. For a student with SAT=1300, compute the predicted probability. (Use logistic function $p=\frac{1}{1+e^{-z}}$, $z=\beta x-\gamma$, assume intercept $\gamma=2$.)

Given an observed correlation $r_{xy}=0.60$ between SAT and GPA and a reliability coefficient of SAT test $r_{xx'}=0.81$, compute the disattenuated (true‐score) correlation.