Chapter 4, Problem 4.63RE

(a)

To determine

To construct the scatterplot that is suitable for predicting when the cherry trees will bloom from the temperature and describe the direction, form and strength of the relationship.

Explanation of Solution

The scatterplot for predicting when the cherry trees will bloom from the temperature is as follows:

  

There is a moderately strong, negative linear relationship between temperature and the days in April until the first blossom.

(b)

To determine

To find the equation of the least squares regression line and interpret the slope and y -intercept of the line.

Answer to Problem 4.63RE

Days in April in first blossom $=-4.4272(\text{temperature)}+31.588$

The slope is $4.4272$ days.

The y -intercept is $31.588$ days.

Explanation of Solution

The equation of the least squares regression line is:

Days in April in first blossom $=-4.4272(\text{temperature)}+31.588$

The slope says that for every degree increase in temperature, the number of days in April to first bloom decreases on average, by $4.4272$ days. The y -intercept says that at zero degrees, it would take $31.588$ days in April until the first bloom.

(c)

To determine

To predict when the first cherry blossom will appear.

Answer to Problem 4.63RE

The first cherry blossom will appear in $16\text{ days}$ .

Explanation of Solution

The average march temperature this year was $\text{3}{\text{.5}}^0\text{C}$ .

To calculate it,

Days in April to first bloom $\begin{array}{l}=-4.4272(\text{temperature)}+31.588\\ =-4.4272(3.5)+31.588\\ \approx 16\text{ days}\end{array}$

The first cherry blossom will appearin $16\text{ days}$ .

(d)

To determine

To find the residual for the year when the average march temperature was $\text{4}{\text{.5}}^0\text{C}$ .

Answer to Problem 4.63RE

The residual is $-2$ .

Explanation of Solution

The average march temperature was $\text{4}{\text{.5}}^0\text{C}$ .

Days in April to first bloom $\begin{array}{l}=-4.4272(\text{temperature)}+31.588\\ =-4.4272(4.5)+31.588\\ \approx 12\text{ days}\end{array}$

Residual $\begin{array}{l}=10-12\\ =-2\end{array}$

(e)

To determine

To construct a residual plot and describe what we see.

Answer to Problem 4.63RE

Explanation of Solution

The residual plot is as:

  

There is quite a bit of deviation from the x-axis, indicating that the regression model does not do a very good job of predicting the days in April until first bloom.

(f)

To determine

To find and interpret the value of $r^2$ in this setting.

Answer to Problem 4.63RE

  $r^2=0.635$ .

Explanation of Solution

The regression model accurately predicts the days in April until first bloom $63.5\%$ of the time.

  $r^2=0.635$ .