Chapter 3, Problem R3.4RE

(a)

To determine

To make a scatterplot that is suitable for predicting when the cherry trees will blossom from the temperature and which variable did you choose as the explanatory variable.

Explanation of Solution

Since we expect the temperature to influence the days in April to first blossom temperature is the explanatory variable and the days in April to first blossom is the response variable. Thus, the scatterplot is as:

  

(b)

To determine

To use the technology to calculate the correlation and the equation of the regression line and interpret the slope and y intercept of the line in the setting.

Answer to Problem R3.4RE

  $\widehat{y}=33.1203-4.6855x$ .

Explanation of Solution

Using calculator, press on STAT and then select $1$ : Edit . and then enter the data of sugar in the list $L_1$ and enter the data of calories in the list $L_2$ .

Next, press on STAT select CALC and then select $\text{Linreg}(a+bx)$ . Next we need to finish the command by entering $L_1{L}_2$ .

  $\text{Linreg}(a+bx)L_1{L}_2$

Finally, pressing on ENTER then gives us the following result:

  $\begin{array}{l}y=a+bx\\ a=33.1203\\ b=-4.6855\\ r=-0.8511\end{array}$

This then implies the regression line as:

  $\begin{array}{l}\widehat{y}=a+bx\\ \Rightarrow \widehat{y}=33.1203-4.6855x\end{array}$

This then implies that on average, the days in April to first blossom decreases by $4.6855$ days per degree Celsius. And the days in April to first blossom is $33.1203$ days when the temperature is zero degree Celsius.

(c)

To determine

To explain would you be willing to use the equation in part (b) to predict the date of the first blossom.

Answer to Problem R3.4RE

No.

Explanation of Solution

The regression line in part (b) is:

  $\widehat{y}=33.1203-4.6855x$

To predict the date to first blossom at ${4.2}^0\text{C}$ is then,

  $\begin{array}{l}\widehat{y}=33.1203-4.6855x\\ =33.1203-4.6855(4.2)\\ =-5.3052\end{array}$

We then note that the predicted day to first blossom in April is $-5.3052$ . However this does not make dense as a day is always a positive integer and thus we are not willing to use the equation in part (b).

(d)

To determine

To calculate and interpret the residual for the year when the average March temperature was ${4.5}^0\text{C}$ .

Answer to Problem R3.4RE

Residual is $-2.033$ .

Explanation of Solution

The regression line in part (b) is:

  $\widehat{y}=33.1203-4.6855x$

Now, the days to first blossom when average March temperature was ${4.5}^0\text{C}$ is:

  $\begin{array}{l}\widehat{y}=33.1203-4.6855x\\ =33.1203-4.6855(4.5)\\ =12.033\end{array}$

And the actual value is $10$ from the table given.

Thus, the residual is as:

  $\begin{array}{l}\text{Residual}=y-\widehat{y}\\ =10-12.033\\ =-2.033\end{array}$

This then implies that we overestimated the number of days in April to first blossom by $2.033$ days when using the regression line with the temperature as the explanatory variable.

(e)

To determine

To use the technology to help construct the residual plot and describe what you see.

Explanation of Solution

The residual plot is as:

  

Thus, there is no obvious pattern in the residual plot and thus the linear regression line seems to be a good fit.