Question

Does the color in which words are printed affect your ability to read them? Do the words themselves affect your ability to name the color in which they are printed? Mr. Starnes designed a study to investigate these questions using the $16$students in his AP Statistics class as subjects. Each student performed two tasks in random order while a partner timed: (1) read $32$words aloud as quickly as possible, and (2) say the color in which each $32$word is printed as quickly as possible. Try both tasks for yourself using the word list below.

Can we use a student’s word task time to predict his or her color task time?

(a) Make an appropriate scatterplot to help answer this question. Describe what you see.

(b) Use your calculator to find the equation of the least-squares regression line. Define any symbols you use.

(c) What is the residual for the student who completed the word task in $9$seconds? Show your work.

(d) Assume that the conditions for performing inference about the slope of the true regression line are met. The $p$-value for a test of $H_0:\beta =0$versus $H_a:\beta >0$is $0.0215$. Explain what this value means in context.

Note: John Ridley Stroop is often credited with the discovery in $1935$ of the fact that the color in which "color words" are printed interferes with people's ability to identify the color. The so-called Stroop Effect, though, was originally published by German researchers in $1929$.

Short answer

(a) The scatterplot indicates that the words and colors have a moderately favorable linear connection.

(b) The equation is $\hat{y}=4.887+1.132x$.

(c) The residual who completed the word task in 9 seconds is $-2.075$.

(d) f the slope of the population regression line is zero, the likelihood of getting a sample that is identical to or more extreme than the given sample is $2.15\%$.

Step-by-step solution

Part(a) Step 1: Given Information

Part(a) Step 2: Explanation

Mr. Starnes performed research to see if the colour of written words impacts your ability to read them and if the words themselves affect your ability to name the colour they are printed in. As a result, this data's scatterplot will appear like this:

As a result of the scatterplot, we may determine that,

The scatterplot slopes upward, indicating a positive direction.

Because the points appear to line up roughly along a line, it has a linear form.

Because the points are not too far apart but not too close together, it has a medium strength.

Part(b) Step 1: Given Information

Given in the question that, 32 words are printed in different colors:

Part(b) Step 2: Explanation

Mr. Starnes conducted research to investigate if the colour of written words impacts your ability to read them and if the words themself impact your performance to name the shade in which they are printed. From the calculator's START menu, choose 1: Edit, then enter the sugar data in the list $L_1$and the data of the calorie in the list $L_2$.

Then press STAT, then CALC, then Linerg $(a+bx)$After that, you must finish the command by typing $L_1{L}_2$.

Finally, pressing ENTER results in the output:

$y=a+bx$

$a=4.887$

$b=1.132$

The regression line then becomes:

localid="1652800910422" $$\begin{gathered} \hat{y}=a+bx \\ =4.887+1.132x \end{gathered}$$

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

Part(c) Step 1: Given Information

Given in the question that, 32 words are printed for a study as given below :

Part(c) Step 2: Explanation

Mr. Starnes performed research to see if the colour of written words impacts your ability to read them and if the words themselves affect your ability to name the colour they are printed in. The regression line looks like this:

$$\begin{gathered} \hat{y}=a+bx \\ =4.887+1.132x \end{gathered}$$

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

Given that

$x=9$

$y=13$

We arrive to the following conclusion using the regression line from component (b):

$$\begin{gathered} \hat{y}=a+bx \\ =4.887+1.132x \\ =4.887+1.132\times 9 \\ =15.075 \end{gathered}$$

The residual seems to be the difference between actual and projected figures, as we all know.

$$\begin{gathered} \text{Residual}=y-\hat{y} \\ =13-15.075 \\ =-2.075 \end{gathered}$$

Part(d) Step 1: Given Information

Given in the question that,

Part(d) Step 2:Explanation

Mr. Starnes performed research to see if the colour of written words impacts your ability to read them, and if the words themselves affect your ability to name the colour.

$H_0:\beta =0$

$H_a:\beta >0$

localid="1652801661707" $$\begin{gathered} P=0.0215 \\ =2.15\% \end{gathered}$$

If the null hypothesis is accepted, the P-value is the chances of obtaining the test statistic result or a value that is more extreme.