AP Statistics 9.2 Confidence Intervals for the Slope of a Regression Model- MCQs - Exam Style Questions
▶️ Answer/Explanation
Let \(b\) be the sample slope and \(\mathrm{SE}_b\) its standard error. A \(95\%\) CI for the population slope is \(b \pm t^{\star}\,\mathrm{SE}_b\) with \(df=n-2\).
Here, \(b=1.054\), \(\mathrm{SE}_b=0.322\), and \(n=31\Rightarrow df=31-2=29\).
For \(df=29\), \(t^{\star}\approx 2.045\) (two-sided \(95\%\)).
Margin of error \(= t^{\star}\,\mathrm{SE}_b = 2.045\times 0.322 \approx 0.659\).
Interval \(= 1.054 \pm 0.659 \Rightarrow (1.054-0.659,\; 1.054+0.659)\).
This gives \((0.396,\; 1.712)\).
✅ Answer: (B)
▶️ Answer/Explanation
The sample slope, \(b\), is the point estimate and therefore the center of the confidence interval.
Sample slope \(b = \frac{-0.181 + 1.529}{2} = \frac{1.348}{2} = 0.674\)
Since the sample slope (\(b\)) is positive (\(0.674\)), the sample correlation coefficient (\(r\)) must also be positive, as they always share the same sign.
✅ Answer: (A)