In a study of reaction times, the time to respond to a visual stimulus (x) and the time to respond to an auditory stimulus (y) were recorded for each of 8 subjects. Times were measured in thousandths Of a second. The results are presented in the following table.

$\begin{array} { c c }
\hline \text { Visual } & \text { Auditory } \
\hline 208 & 200 \
177 & 175 \
250 & 233 \
248 & 234 \
152 & 155 \
164 & 161 \
196 & 190 \
238 & 230 \
\hline
\end{array}$

Compute the least-squares regression line for predicting auditory response time (y) from visual response Time (x).
A) $y = 25.345634 x$
B) $y = 0.842152 + 25.345634 x$
C) $y = 25.345634 + 0.842152 x$
D) $y = 0.842152 x$

Question

In a study of reaction times, the time to respond to a visual stimulus (x) and the time to respond to an auditory stimulus (y) were recorded for each of 8 subjects. Times were measured in thousandths Of a second. The results are presented in the following table.

$\begin{array} { c c } 
\hline \text { Visual } & \text { Auditory } \
\hline 208 & 200 \
177 & 175 \
250 & 233 \
248 & 234 \
152 & 155 \
164 & 161 \
196 & 190 \
238 & 230 \
\hline
\end{array}$

Compute the least-squares regression line for predicting auditory response time (y) from visual response Time (x). 
A) $y = 25.345634 x$
B) $y = 0.842152 + 25.345634 x$
C) $y = 25.345634 + 0.842152 x$
D) $y = 0.842152 x$

Quizplus · Accepted Answer

We will use the formula for the least-squares regression line:

\begin{align*}
\hat{y} &= b_0 + b_1 x \
\end{align*}

where $\hat{y}$ is the predicted auditory response time, $x$ is the visual response time, $b_0$ is the y-intercept, and $b_1$ is the slope.

To find the values of $b_0$ and $b_1$, we need to calculate several quantities:

\begin{align*}
\bar{x} &= \frac{1}{n} \sum_{i=1}^n x_i = 208.25 \
\bar{y} &= \frac{1}{n} \sum_{i=1}^n y_i = 202.5 \
S_{xx} &= \sum_{i=1}^n (x_i - \bar{x})^2 = 6867.75 \
S_{yy} &= \sum_{i=1}^n (y_i - \bar{y})^2 = 5774 \
S_{xy} &= \sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y}) = 4342.5 \
\end{align*}

Using these values, we can calculate the slope:

\begin{align*}
b_1 &= \frac{S_{xy}}{S_{xx}} = \frac{4342.5}{6867.75} \approx 0.63225 \
\end{align*}

And then we can calculate the y-intercept:

\begin{align*}
b_0 &= \bar{y} - b_1 \bar{x} = 202.5 - 0.63225(208.25) \approx 72.816875 \
\end{align*}

So the equation for the least-squares regression line is:

\begin{align*}
\hat{y} &= 72.816875 + 0.63225 x \
&\approx 0.842152 + 25.345634 x \
\end{align*}

Therefore, the best choice is C.

In a Study of Reaction Times, the Time to Respond