Chapter 12.1, Problem 30E

(a)

To determine

To make an appropriate scatterplot to help answer this question and describe what you see.

Explanation of Solution

In the question Mr. Starnes designed a study to examine the questions that does the color in which words are printed affect your ability to read them and so the words themselves affect your ability to name the color in which they are printed. Thus, the scatterplot for this data will be as:

  

Thus, by looking at the scatterplot we can say that,

Direction: It is positive because the scatterplot slopes upward.

Form: It is linear because the points seem to roughly line about a line.

Strength: It is moderate because the points do not lie apart but not close together either.

Thus the scatterplot suggest a moderate positive linear relationship between the words and colors.

(b)

To determine

To use your calculator to find the equation of the least squares regression line.

Answer to Problem 30E

  $\begin{array}{l}\widehat{y}=a+bx\\ \Rightarrow \widehat{y}=4.887+1.132x\end{array}$

Explanation of Solution

In the question Mr. Starnes designed a study to examine the questions that does the color in which words are printed affect your ability to read them and so the words themselves affect your ability to name the color in which they are printed. 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:

  $\begin{array}{l}y=a+bx\\ a=4.887\\ b=1.132\end{array}$

This then implies the regression line as:

  $\begin{array}{l}\widehat{y}=a+bx\\ \Rightarrow \widehat{y}=4.887+1.132x\end{array}$

with $\widehat{y}$ the predicted time for the colors and x the time for the words.

(c)

To determine

To find out what is the residual who completed the word task in $9$ seconds.

Answer to Problem 30E

Residual is $-2.075$ .

Explanation of Solution

In the question Mr. Starnes designed a study to examine the questions that does the color in which words are printed affect your ability to read them and so the words themselves affect your ability to name the color in which they are printed. And we have the regression line as:

  $\begin{array}{l}\widehat{y}=a+bx\\ \Rightarrow \widehat{y}=4.887+1.132x\end{array}$

with $\widehat{y}$ the predicted time for the colors and x the time for the words.

Now, it is given that,

  $\begin{array}{l}x=9\\ y=13\end{array}$

Thus, using the regression line in the part (b), we evaluate the result as:

  $\begin{array}{l}\widehat{y}=a+bx\\ \Rightarrow \widehat{y}=4.887+1.132x\\ =4.887+1.132\times 9\\ =15.075\end{array}$

Thus, as we know that the residual is actual minus the predicted values, then,

  $\begin{array}{l}\text{Residual}=y-\widehat{y}\\ =13-15.075\\ =-2.075\end{array}$

Thus, the predicted number of colors for the student who completed the word task in $9$ seconds was overestimated by $2.075$ from the actual number of colors.

(d)

To determine

To explain what does this value means in this context.

Explanation of Solution

In the question Mr. Starnes designed a study to examine the questions that does the color in which words are printed affect your ability to read them and so the words themselves affect your ability to name the color in which they are printed. It is given in the question that,

  $\begin{array}{l}{H}_0:\beta =0\\ H_a:\beta >0\end{array}$

  $P=0.0215=2.15\%$

Thus, the P-value is the probability of obtaining the value of test statistics or a value more extreme if the null hypothesis is true. So, if the slope of the population regression line is zero, then the probability of obtaining a similar or more extreme sample than the given sample is $2.15\%$ .