Exercises for Question Type 11: Determining the type of validity

Question 1

Skill question

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

Student	SAT score ( $x$ )	GPA ( $y$ )
A	1100	3.0
B	1250	3.2
C	1300	3.5
D	950	2.8
E	1400	3.7

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

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Question 2

Skill question

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, $r_{s}$ .

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Question 3

Skill question

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

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Question 4

Skill question

The following summary statistics were calculated for a group of students' SAT scores, $x$ , and their first-year university GPAs, $y$ :

$\overset{x}{ˉ} = 1200$ $\overset{y}{ˉ} = 3.24$ $\sum (x_{i} - \overset{x}{ˉ})^{2} = 125, 000$ $\sum (x_{i} - \overset{x}{ˉ}) (y_{i} - \overset{y}{ˉ}) = 250$

Determine the least-squares regression line of $y$ on $x$ in the form $y = a + b x$ , and use your equation to predict the GPA for a student who scored 1200 on the SAT.

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Question 5

Skill question

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.

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Question 6

Skill question

A regression slope for predicting GPA from SAT is estimated as $b = 0.003$ with standard error $S E_{b} = 0.001$ . Construct a $95%$ confidence interval for $b$ (use $t_{0.025, 48} \approx 2.01$ ).

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Question 7

Skill question

A logistic regression model predicts the probability of a student achieving a $GPA \geq 3.0$ based on their $SAT$ score. The model has a coefficient $β = 0.004$ and an intercept $γ = 2$ .

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

Calculate the predicted probability for a student with an $SAT$ score of $1300$ .

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Question 8

Skill question

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 $α = 0.05$ level whether there is a significant linear relationship.

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Question 9

Skill question

Bivariate analysis and linear regression.

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

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Question 10

Skill question

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.

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Question 11

Skill question

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.

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IB Mathematics Applications & Interpretation Question Type 11: Determining the Type of Validity Exercises