Question Type 11: Determining the type of validity Exercises

The SAT scores ($x$) and first-year GPAs ($y$) for five students are given in the following table:

Calculate the Pearson correlation coefficient, $r$, for this data.

Five students are ranked by their SAT score and by their GPA as follows:

SAT ranks: $1, 2, 3, 4, 5$ GPA ranks: $2, 1, 4, 5, 3$

Calculate the Spearman rank correlation coefficient, $r_s$.

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

Determine the least-squares regression line of $y$ on $x$ in the form $y = a + bx$, and use your equation to 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.

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 $t_{0.025, 48} \approx 2.01$).

A logistic regression model predicts the probability of a student achieving a $\text{GPA} \ge 3.0$ based on their $\text{SAT}$ score. The model has a coefficient $\beta = 0.004$ and an intercept $\gamma = 2$.

The predicted probability, $p$, is given by $p = \frac{1}{1 + e^{-z}}$, where $z = \beta x - \gamma$ and $x$ is the $\text{SAT}$ score.

Calculate the predicted probability for a student with an $\text{SAT}$ score of $1300$.

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.

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

Given a sample correlation $r = 0.75$ between SAT scores and first-year college GPA, calculate the coefficient of determination and interpret its meaning in this context.

In a multiple regression model predicting first-year university GPA ($y$), the SAT score ($x_1$) has a partial correlation coefficient of $r_{y,x_1 \cdot x_2} = 0.40$, controlling for high school GPA ($x_2$).

Calculate the coefficient of partial determination for the SAT score and interpret its value in the context of the model.