The table below gives data from a study that shows that social exclusion causes
ID: 3200777 • Letter: T
Question
The table below gives data from a study that shows that social exclusion causes "real pain." That is, activity in the area of the brain that responds to physical pain goes up as distress from social exclusion goes up. The scatterplot for this data shows a moderately strong linear relationship. The data are given below. Social Brain Social Brain Subject distress activity Subject distress activity 1 1.24 0.055 8 2.20 0.026 2 1.89 0.040 9 2.58 0.028 3 1.04 0.027 10 2.77 0.033 4 2.52 0.019 11 2.77 0.064 5 2.13 0.019 12 3.39 0.077 6 2.61 0.019 13 3.61 0.124 7 2.01 0.021 (a) What is the equation of the least-squares regression line for predicting brain activity from social distress score? (Round your answers to four decimal places.) brain activity = social distress + (b) On your plot, show the "up and over" lines that predict brain activity for social distress score 1.7. Use the equation of the regression line to get the predicted brain activity level. Verify that it agrees with your plot. (Round your answer to four decimal places.) (c) What percent of the variation in brain activity among these subjects is explained by the straight-line relationship with social distress score? (Round your answer to a whole number.)
Explanation / Answer
Solution:
Here, we have to use the regression analysis for answering the questions. From the given data for the two variables, the regression output is given as below:
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.96534238
R Square
0.931885911
Adjusted R Square
0.925693721
Standard Error
0.01378492
Observations
13
ANOVA
df
SS
MS
F
Significance F
Regression
1
0.028597428
0.02859743
150.4938
9.27033E-08
Residual
11
0.002090264
0.00019002
Total
12
0.030687692
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
-0.071396513
0.008218154
-8.6876581
2.96E-06
-0.089484548
-0.05330848
Social Distress
0.00123321
0.000100526
12.2675896
9.27E-08
0.001011954
0.00145447
(a) What is the equation of the least-squares regression line for predicting brain activity from social distress score?
The least square regression equation or line for predicting brain activity from social distress score is given as below:
Brain activity = 0.0012*Social Distress – 0.0714
(b) On your plot, show the "up and over" lines that predict brain activity for social distress score 1.7. Use the equation of the regression line to get the predicted brain activity level. Verify that it agrees with your plot.
We have regression equation as below:
Brain activity = 0.0012*Social Distress – 0.0714
For social distress = 1.7, brain activity = 0.0012*1.7 – 0.0714 = -0.06936
This prediction is agrees with the plot which is shown below:
(c) What percent of the variation in brain activity among these subjects is explained by the straight-line relationship with social distress score?
We are given a coefficient of determination or the value of R square as 0.9319 or approximately 93%. About 93% of the variation in brain activity among these subjects is explained by the straight line relationship with social distress score.
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.96534238
R Square
0.931885911
Adjusted R Square
0.925693721
Standard Error
0.01378492
Observations
13
ANOVA
df
SS
MS
F
Significance F
Regression
1
0.028597428
0.02859743
150.4938
9.27033E-08
Residual
11
0.002090264
0.00019002
Total
12
0.030687692
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
-0.071396513
0.008218154
-8.6876581
2.96E-06
-0.089484548
-0.05330848
Social Distress
0.00123321
0.000100526
12.2675896
9.27E-08
0.001011954
0.00145447
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