Question

Recycle and Review Exercises 29-31 refer to the following setting. 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 the following two tasks in random order while a partner timed his or her performance: (1) Read $32$words aloud as quickly as possible, and (2) say the color in which each of $32$words is printed as quickly as possible. Try both tasks for yourself using the word list given.

Color words (3.1, 3.2, 12.1) Can we use a student's ward task time to predict his or her color task lime?

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

b. Use technology to find the equation of the least-squares regression line. Define any variables you use,

c. Find and interpret the residual for the student who completed the word tack in 4 seconds.

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 $H_0:{\beta }_1=0$versus $H_{\alpha }:{\beta }_1\&gt:0.H_{\alpha }{\beta }_1>0is0.0215$interpret this value.

Note: John Ridley stroop is often credited with the discovery in 1535 of the fae that the caler in which "color words" are printed interferes with people's ability to identify the color. The paper outlining the so-called Stoop effect, though, was originally published by Geman researchers in 1929 .

Short answer

(a)

(b) The equation of the least-squares regression line is $\hat{y}=4.887+1.132x$

(c) Residual$=-2.075$

(d) The P-value represents the probability of the test statistical value being correct.

Step-by-step solution

Part (a) Step 1: Given Information 

Given table is

we have to make an appropriate scatterplot to help answer this question

Part (a) Step 2: Explanation  

Part (b) Step 1: Given Information  

Given table is

we have to find the equation of the least-squares regression line

Part (b) Step 2: Explanation  

The formulas are

$a=\frac{\sum Y·\sum X^2-\sum x·\sum XY}{n·\sum X^2-{\left(\sum x\right)}^2}$

$b=\frac{n.\sum XY-\sum Y·\sum X}{n·\sum X^2-{\left(\sum x\right)}^2}$

a & b values can be computed as

$$\begin{gathered} a=\frac{\sum Y·\sum X^2-\sum x·\sum XY}{n·\sum X^2-{\left(\sum x\right)}^2} \\ =\frac{308.2665-203.4009}{16.2665-{203}^2} \\ =4.887 \end{gathered}$$

$$\begin{gathered} a=\frac{n.\sum XY-\sum Y·\sum X}{n·\sum X^2-{\left(\sum x\right)}^2} \\ =\frac{16.4009-203.308}{16.2665-{203}^2} \\ =1.132 \end{gathered}$$

Therefore the least square regression equation is

$\hat{y}=4.887+1.132x$

Part (b) Step 1: Given Information  

Given that

$x=9$

$y=13$

we have to find the residual value.

Part (c) Step 2: Explanation  

From part (b)

$\hat{y}=4.887+1.132x$

putitng $9ofthex$

$$\begin{gathered} \hat{y}=4.887+1.132\left(9\right) \\ =15.075 \end{gathered}$$

Residual value$y-\hat{y}$

$13-15.075=-2.075$

Part (d) Step 1: Given Information  

Given in the question that

$H_0:\beta =0$

$H_{\alpha }:\beta >0$

$P=2.15\%$

We have to explain the P value.

Part (d) Step 2: Explanation  

The probability that the test statistical value is correct is represented by the P-value. A value that is more extreme if the null hypothesis is true. If the population regression is zero, the chances of obtaining an extreme (slope-wise) sample that is similar to the sample provided are$2.15$