Chapter 7

3. Are there ever any circumstances when a correlation such as Pearson’s r can be interpreted as evidence for a causal connection between two variables? If yes, what circumstances?

Solution: Yes, that could be the case when there are absolutely no confounders.

5. A researcher says that 50% of the variance in blood pressure can be predicted from HR and that blood pressure is positively associated with HR. What is the correlation between blood pressure and HR?

Solution: Notice that we are provided with the following information: \({{r}^{2}}=0.50\). Observe that

But it is also stated that blood pressure is positively associated with HR, so then

Chapter 9

6. How are r, b, and β related to each other?

Solution: We know that all r, b, and β must have the same sign.

8. What is a multiple R? How is multiple R2 interpreted?

Solution: Multiple R corresponds to the correlation between the observed values \({{Y}_{i}}\) and the predicted values \({{\hat{Y}}_{i}}\) . The value of the multiple R2 coefficient is interpreted as an approximation of the proportion of variance in the dependent variable \(Y\) explained by the multiple regression model. When the number of predictors becomes large, the multiple R ² coefficient tends to overestimate the proportion of explained variation, and the Adjusted R ² coefficient should be used instead.

Lab Assignment

2. Table 9.2 displays a small set of (real) data that show GNP per capita (X) and mean life satisfaction for a set of 19 nations (Y).

1. Enter the data into SPSS.
2. Run the < Analyze > → < Descriptive Statistics > → < Descriptives > procedure; obtain the mean and standard deviation for each variable.
3. Do appropriate preliminary data screening, then run the bivariate regression to predict life satisfaction from GNP. Write up the results in paragraph form, including statistical significance, effect size, and nature of relationship. Also, use the < Save > command to save the unstandardized residuals as a new variable in your SPSS worksheet. Include effect size and nature of relationship even if the regression is not statistically significant.
4. Using the < Transform > → < Compute > command, create a new variable that tells you the squared unstandardized residual for each case. Using the < Descriptives > command, obtain the sum of the squared residuals across all 19 cases. Note that this sum should correspond to SSresidual in the ANOVA table in your SPSS regression printout.
5. Show that you can use the SS values in the ANOVA table in the SPSS printout to reproduce the value of R2 on the SPSS printout.

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