Question
1. Which of the following statements is (are) true?
I. In the computer output for regression, \(s\) is the estimator of \(\sigma\), the standard deviation of the residuals.
II. The t-test statistic for the \(\mathrm{H}_0: \beta=0\) has the same value as the t-test statistic for \(\mathrm{H}_0: \rho=0\).
III. The t-test for the slope of a regression line is always two sided \(\left(\mathrm{H}_{\mathrm{A}}: \beta \neq 0\right)\).
a. I only
b. II only
c. III only
d. I and II only
e. I and III only
Use the following output in answering questions 2-4:
A study attempted to establish a linear relationship between IQ score and musical aptitude. The following table is a partial printout of the regression analysis and is based on a sample of 20 individuals.
The regression equation is
$
\begin{array}{lllll}
\widehat{\text { MusApt }}=-22.3+0.493 \text { IQ } & & & \\
\text { Predictor } & \text { Coef } & \text { St Dev } & t \text { ratio } & P \\
\text { Constant } & -22.26 & 12.94 & -1.72 & .102 \\
\text { IQ } & 0.4925 & 0.1215 & & \\
s=6.143 & \text { R-sq }=47.7 \% & \text { R-sq (adj })=44.8 \% & &
\end{array}
$
▶️Answer/Explanation
Ans:D
II is true since it can be shown \(t=\frac{b}{s_b}=r \sqrt{\frac{n-2}{1-r^2}}\). III is not true since, although we often use the alternative \(\mathrm{H}_{\mathrm{A}}: \beta \neq 0\), we can certainly test a null with an alternative that states that there is a positive or a negative association between the variables.
Question
The value of the t-test statistic for \(\mathrm{H}_0: \beta=0\) is
a. 4.05
b. -1.72
c. 0.4925
d. 6.143
e. 0.0802
▶️Answer/Explanation
Ans:
The correct answer is (a).
$
t=\frac{b}{s_b}=\frac{0.4925}{0.1215}=4.05 \text {. }
$
Question
A \(99 \%\) confidence interval for the slope of the regression line is
a. \(0.4925 \pm 2.878(6.143)\)
b. \(0.4925 \pm 2.861(0.1215)\)
C. \(0.4925 \pm 2.861(6.143)\)
d. \(0.4925 \pm 2.845(0.1215)\)
e. \(0.4925 \pm 2.878(0.1215)\)
▶️Answer/Explanation
Ans:The correct answer is (e).
For \(n=20, d f=20-2=18 \Rightarrow t^*=2.878\) for \(C=\) 0.99 .
Question
Which of the following best interprets the slope of the regression line?
a. A student with an IQ one point above another student has a Musical Aptitude score 0.4925 points higher.
b. As IQ score increases, so does the Musical Aptitude score.
c. A student with an IQ one point above another student is predicted to have a Musical Aptitude score 0.4925 points higher.
d. For each additional point of Musical Aptitude, IQ is predicted to increase by 0.4925 points.
e. There is a strong predictive linear relationship between IQ score and Musical Aptitude.
▶️Answer/Explanation
Ans:The correct answer is (c).
Note that (a) is not correct since it doesn’t have “predicted” or “on average” to qualify the increase. (b) is a true statement but is not the best interpretation of the slope. (d) has mixed up the response and explanatory variables. \((\mathrm{e})\) is also true \((\mathrm{t}=4.05 \Rightarrow \mathrm{P}\)-value \(=0.0008)\) but is not an interpretation of the slope.
Question
A group of 12 students take both the SAT Math and the SAT Verbal. The least-squares regression line for predicting Verbal Score from Math Score is determined to be \(\sqrt{\text { Verbal Score }}=106.56+0.74\) (Math Score). Further, \(\mathrm{s}_{\mathrm{b}}=\) 0.11 . Which of the following is a \(95 \%\) confidence interval for the slope of the regression line?
a. \(0.74 \pm 0.245\)
b. \(0.74 \pm 0.242\)
c. \(0.74 \pm 0.240\)
d. \(0.74 \pm 0.071\)
e. \(0.74 \pm 0.199\)
▶️Answer/Explanation
Ans: The correct answer is (a).
A \(95 \%\) confidence interval at \(12-2=10\) degrees of freedom has a critical value of \(t^{\star}=2.228\) (from Table B; if you have a TI-83/84 with the invT function, \(\operatorname{invT}(0.975,10)=2.228)\). The required interval is \(0.74 \pm\) \((2.228)(0.11)=0.74 \pm 0.245\).