Question

Crickets chirping The scatterplot shows the relationship between x = temperature in degrees Fahrenheit and y = chirps per minute for the striped ground cricket, along with the regression line y^=−0.31+0.212x

a. Calculate and interpret the residual for the cricket who chirped 20 times per minute when the temperature was 88.6°F.

b. About how many additional chirps per minute do you expect cricket to make if the temperature increases by 10°F?

Short answer

Part (a) Residual is $1.5268$

Part (b) $2.12$ chirps per minute we expect a cricket to make.

Step-by-step solution

Part (a) Step 1: Given information

The regression line is given as follows in the question:

y= −0.31 + 0.212x

Part (a) Step 2: Calculation

Also, when the temperature was $88.6$degrees Fahrenheit, the insect chirped$20$ times per minute.

Thus,

$$\begin{gathered} x=88.6 \\ y=20 \end{gathered}$$

Thus, the predicted number of times cricket chirped is as:

$$\begin{gathered} \stackrel{⏜}{y}=-0.31+0.212x \\ =-0.31+0.212(88.6) \\ =18.4732 \end{gathered}$$

Thus, the residual will be calculated as:

$$\begin{gathered} Residual=y-\stackrel{⏜}{y} \\ =20-18.4732 \\ =1.5268 \end{gathered}$$

This means that while using the regression line to make a prediction for the cricket, we underestimated the amount of chirps per minute by $1.5268$chirps per minute.

Part (b) Step 1: Calculation

The regression line is given as follows in the question:

$\stackrel{⏜}{y}=-0.31+0.212x$

The slope in the least square regression equation is the coefficient of $x$and it indicates the average rise or decrease of $y$per unit of $x$Thus,

$b_1=0.212$

As a result, the average number of chirps per minute increases by $0.212$ per degree Fahrenheit. In this scenario, we raise the temperature by, and we anticipate an increase in the number of chirps per minute of:

$$\begin{gathered} 10\times 0.212 \\ =2.12chirpsperminute \end{gathered}$$