Question

question 6
1 pts
points earned on a biology exam and hours spent studying for the exam were recorded for a random sample of college students. the regression equation, which explains 56.3% of the variability in points earned, is given below.
predicted points earned = 38.60 + 2.70 (hours studied)
based on what you have learned in chapters 14 and 15 about correlation and regression, which one of the following interpretations of this information is most accurate?
the relationship between points earned and hours spent studying is weak.
the correlation between points earned and hours spent studying is approximately 0.56.
it would be extrapolation to predict the points earned for a student who studies for 40 hours.
since ( r^2 = 0.563 ), this means the correlation must be positive.
none of the above statements are accurate.

Explanation:

Step1: Analyze \( r^2 \) and correlation

We know \( r^2 = 0.563 \) (since it explains 56.3% of variability). The correlation \( r \) has the same sign as the slope of the regression line. The slope here is \( 2.70 \) (positive), so \( r \) should be positive. Also, \( r=\pm\sqrt{r^2} \), so \( r = \sqrt{0.563}\approx0.75 \) (not 0.56), and a relationship with \( r^2 = 0.563 \) is moderate, not weak. Extrapolation depends on the range of the original data (not given here), so we can't assume 40 hours is extrapolation. But since the slope is positive, \( r \) must be positive (because \( r^2 \) is non - negative and \( r \) has the same sign as the slope).

Step2: Evaluate each option

• Option 1: A relationship with \( r^2 = 0.563 \) is not weak (moderate), so this is incorrect.
• Option 2: \( r=\sqrt{0.563}\approx0.75

eq0.56 \), so this is incorrect.

• Option 3: We don't know the original range of hours studied, so we can't say 40 hours is extrapolation, so this is incorrect.
• Option 4: The slope of the regression equation is positive (\( 2.70 \)), and \( r^2 = 0.563 \). Since \( r=\pm\sqrt{r^2} \) and the slope and correlation have the same sign (because the slope is positive, the relationship is positive), so \( r \) must be positive. This is correct.
• Option 5: Since option 4 is correct, this is incorrect.

Answer:

D. Since \( r^{2}=0.563 \), this means the correlation must be positive.