Does social rejection hurt? Exercise 14 (page 160) gives...

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 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 2.0 . (b) What percent of the variation in brain activity among these subjects is explained by the straight-line relationship with 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.

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 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 2.0 .
(b) What percent of the variation in brain activity among these subjects is explained by the straight-line relationship with 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.

Key Concepts

Least-Squares Regression Line

This is a statistical method used to find the best-fitting straight line through a set of data points by minimizing the sum of the squares of the vertical distances (residuals) from each point to the line. It provides an equation that predicts the response variable from the explanatory variable and helps in understanding the linear relationship between them.

Prediction Using Regression Equation

Once the regression line is established, its equation can be used to predict the value of the response variable for any given value of the explanatory variable. This prediction is based on the assumption that the relationship between the variables is approximately linear and that the model fits the data well.

Coefficient of Determination (R²)

The coefficient of determination, often denoted as R², quantifies the proportion of the variance in the response variable that is predictable from the explanatory variable. It is expressed as a percentage and provides a measure of how well the regression line approximates the real data points, with higher values indicating a better fit.

Correlation Coefficient (r)

The correlation coefficient measures the strength and direction of the linear relationship between two variables. Its value ranges from -1 to +1, where values closer to +1 indicate a strong positive relationship and values closer to -1 indicate a strong negative relationship. The sign of the coefficient reflects the direction of the association, which can be deduced from the slope of the regression line.

Standard Error of the Estimate

This statistic represents the typical distance that the observed values fall from the regression line. It measures the spread of the residuals or errors and provides an estimate of the uncertainty associated with predictions made by the regression model. A lower standard error indicates that the data points are closer to the regression line, suggesting a better fit.

Transcript

00:01 As you referring to an output window that is in your textbook, i do not have access to it, so feel free to follow along in there.

00:09 The first thing we need to do is find the least square regression line.

00:15 Anytime you see this, it is going to be y equals mx plus b.

00:20 So we need to find something on that output window that has the form y equals mx plus b.

00:27 And you should see an equation y is equal to 0 .0607 .02x minus 0 .1 -2 -8.

00:42 And this has the form y equals mx plus b.

00:48 So now we need to use this equation to predict what would happen if social distress score was 2.

00:56 So is two going to be our x or our y? so let's read the question more carefully.

01:05 It says to predict brain activity given social distress score.

01:10 So the given part is our x and the other part is our y.

01:16 So we're given social distress score.

01:19 So that's going to be the x value.

01:22 So we'll plug 2 in 4x.

01:25 0 .0602 times 2 minus 0 .126 .48.

01:34 And you should get negative.

01:37 0 .00576.

01:43 So that's it for part a.

01:46 For part b, what percentage of variance is explained by the equation? any time it asks for percentage of variance explained by a least square regression, line, that means r squared.

02:01 You're going to look for the part in the output window that has r squared...

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