Does social rejection hurt? Exercise $14$ (page $160$) gives data from a study that shows that social exclusion causes “real pain.” That is, activity in an area of the brain that responds to physical pain goes up as distress from social exclusion goes up. A scatterplot shows a moderately strong, linear relationship. The figure below shows the Minitab regression output for these data.

(a) What is the equation of the least-squares regression line for predicting brain activity from social distress score? Use the equation to predict brain activity for social distress score of $2.0$

(b) What percent of the variation in brain activity among these subjects is explained by the straight-line relationship with the social distress score?

(c) Use the information in the figure to find the correlation r between social distress score and brain

activity. How do you know whether the sign of r is + or −?

(d) Interpret the value of s in this setting.

A regression line is a straight line that depicts the relationship between an explanatory variable $x$and a response variable $y$ By entering any value of $x$ into the equation of the line, you may use a regression line to anticipate the value of $y$ for any value of $x$

We can readily see from the above output that $slope\left(b\right)=0.060782$

$y-intercept\left(a\right)=-0.12608$

The brain activity is monitored during the examination. This is the study's outcome, with brain activity as the response variable. Now there's a third factor to consider: social discomfort. The study's main purpose is to discover if social stress has any impact on brain function. Social pain is an explanatory variable. We will use social discomfort as an explanatory variable or predictor to predict brain activity (response or predicted variable). The independent variable is social discomfort, while the dependent variable is brain activity. The equation for predicting brain activity from social discomfort using the least-squares regression line is,$Brainactivity=-0.12608+0.060782(socialdistress)$

The brain activity for social distress score of $2.0$is, $$\begin{gathered} \left(predicted\right)brainactivity=-0.12608+0.060782\times 2.0 \\ =-0.004516 \end{gathered}$$

Predicted Brain activity for social distress score of $2.0$is $-0.004516$

From the above output, we can clearly see that$R-sq=77.1\%$

According to this interpretation, the straight-line association with the social distress score accounts for $77.1$ percent of the diversity in brain activity across these patients.

From the above figure, we can clearly see that$Slope\left(b\right)=0.060782$

This shows the positive correlation between variables. $$\begin{gathered} R-sq(R^2)=77.1\% \\ {r}^2=0.771 \\ \left(r\right)=\sqrt{r^2} \end{gathered}$$

Correlation=$\sqrt{0.771}$

$=0.878$

The correlation coefficient between brain activity and social distress is $0.878$, indicating a positive relationship.

The residuals' standard deviation for these data is $s=0.0250896$

When utilizing the regression line, The average size of the prediction mistakes is measured by the standard deviation of the residuals$s$(residuals). The value of $s=0.0250896$indicates that there is relatively little prediction error.