The following table shows the running speed , in centimeters per second, of ants as a func- tion of the temperature $T$ in degrees Celsius. Use linear regression to model running speed as a function of
Temperature. $$\begin{array} { | l | l | l | l | l | }
\hline T & 25.6 & 30.3 & 32.3 & 34.8 \
\hline R & 2.22 & 3.85 & 4.06 & 5.29 \
\hline
\end{array}$$
A) $R = 0.35 T + 2.22$
B) $T = 0.35 R + 2.22$
C) $R = 0.32 T - 5.97$
D) $T = 0.32 R - 5.97$

Question

The following table shows the running speed  , in centimeters per second, of ants as a func- tion of the temperature $T$ in degrees Celsius. Use linear regression to model running speed as a function of
Temperature. $$\begin{array} { | l | l | l | l | l | } 
\hline T & 25.6 & 30.3 & 32.3 & 34.8 \
\hline R & 2.22 & 3.85 & 4.06 & 5.29 \
\hline
\end{array}$$
A) $R = 0.35 T + 2.22$ 
B) $T = 0.35 R + 2.22$ 
C) $R = 0.32 T - 5.97$ 
D) $T = 0.32 R - 5.97$

Quizplus · Accepted Answer

We want to model the running speed as a linear function of temperature using linear regression. We can use technology to find the line of best fit, or we can use the formula for the slope of the line:

$$b = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^n (x_i - \bar{x})^2}$$

where $x$ represents temperature, $y$ is running speed, and $n$ is the number of data points. $\bar{x}$ and $\bar{y}$ are the mean values of the $x$ and $y$ data points.

Substituting the given values into this formula, we get:

$$b = \frac{(25.6 - 30.25)(2.22 - 3.855) + (30.3 - 30.25)(3.85 - 3.855) + (32.3 - 30.25)(4.06 - 3.855) + (34.8 - 30.25)(5.29 - 3.855)}{(25.6 - 30.25)^2 + (30.3 - 30.25)^2 + (32.3 - 30.25)^2 + (34.8 - 30.25)^2} \approx 0.32$$

So the slope of the line of best fit is approximately 0.32. To find the y-intercept, we can use the point-slope form of a line:

$$y - y_1 = b(x - x_1)$$

where $x_1$ and $y_1$ are the coordinates of any data point. Using $(25.6, 2.22)$, we get:

$$y - 2.22 = 0.32(x - 25.6)$$

Simplifying and solving for $y$, we get:

$$y = 0.32x - 5.97$$

So the best choice is C.

The Following Table Shows the Running Speed , in Centimeters