Chapter 3.2, Problem 64E

(a)

To determine

To find: the equation of the least-squares regression line for predicting brain activity

Answer to Problem 64E

Predicted Brain activity for social distress score 2.0 is -0.004516.

Explanation of Solution

Given:

  

Calculation:

From the above output we can clearly see that

Slope $\left(b\right)=0.060782$

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

The study records the brain activity. This is the outcome of the study, the brain activity is the response variable. Now, social distress is a second variable. The entire purpose of the study is to determine how, if at all, social distress affect the brain activity. Social distress is an explanatory variable. We will use social distress (explanatory variable or predictor) to predict brain activity (response or predicted variable). Brain activity is the dependent variable and social distress is the independent variable. The equation of the least-squares regression line for predicting the brain activity from the social distress is,$\text{Brain activity}=-0.12608+0.060782\left(\text{Social distress}\right)$

The brain activity for social distress score 2.0 is, $\begin{array}{l}\left(\text{Predicted}\right)\text{ Brain activity}=-0.12608+0.060782\times 2.0\\ =-0.004516\end{array}$

Predicted Brain activity for social distress score 2.0 is -0.004516.

Conclusion:

Thus, predicted Brain activity for social distress score 2.0 is -0.004516.

(b)

To determine

To find: the percent of the variation in brain activity

Answer to Problem 64E

77.1% of the variation in brain activity among these subjects is explained by the straight line relationship with social distress score.

Explanation of Solution

Calculation:

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

It is interpreted this by saying that 77.1% of the variation in brain activity among these subjects is explained by the straight line relationship with social distress score.

Conclusion:

Therefore, 77.1% of the variation in brain activity among these subjects is explained by the straight line relationship with social distress score.

(c)

To determine

To find: the correlation between social distress score and brain activity and find whether the sign of r is + or -

Answer to Problem 64E

Brain activity and the social distress are positively correlated and the coefficient of correlation between them is 0.878.

Explanation of Solution

Calculation:

From the above figure we can clearly see that

Slope $\left(b\right)=0.060782$

This shows positive correlation between variables.

  $\begin{array}{l}R-sq\left(R^2\right)=77.1\%\hfill \\ r^2=0.771\hfill \end{array}$

Correlation $\begin{array}{l}\left(r\right)=\sqrt{r^2}\\ =\sqrt{0.771}\\ =0.878\end{array}$

Brain activity and the social distress are positively correlated and the coefficient of correlation between them is 0.878.

Conclusion:

Therefore, brain activity and the social distress are positively correlated and the coefficient of correlation between them is 0.878.

(d)

To determine

Tointerpret: the value of s

Answer to Problem 64E

The value of sis$s=0.0250896$

Explanation of Solution

Calculation:

For these data the standard deviation of the residuals$s=0.0250896$

The standard deviation of the residuals s measures the average size of the prediction errors (residuals) when using the regression line. The value of s $s=0.0250896$ shows very less prediction error.

Conclusion:

Therefore, the value of s is $s=0.0250896$