Coefficient of Determination (R²)

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Show your work for each problem.

1.
The correlation coefficient of a data set is $r = 0.6$ . What is $R^2$ ?
1. a)1.2
2. b)0.36
3. c)0.6
4. d)0.06
2.
The correlation between two variables is $r = -0.5$ . What is $R^2$ ? Give your answer as a decimal (like 0.3).
3.
A regression of plant height on hours of sunlight has $R^2 = 0.72$ . Which statement is correct?
1. a)Sunlight causes 72% of a plant's height
2. b)72% of the data points lie exactly on the regression line
3. c)72% of the variation in plant height is explained by the linear relationship with sunlight
4. d)The slope of the regression line is 0.72
4.
For a regression line, $SS_{\text{tot}} = 200$ and $SS_{\text{res}} = 50$ . What is $R^2$ ? Give your answer as a decimal.
5.
The correlation between hours of practice and points scored is $r = 0.9$ . Find $R^2$ and the percent of variation explained.
6.
A bakery's regression of daily bread sales on temperature has $SS_{\text{tot}} = 60$ and $SS_{\text{res}} = 21$ . Find $R^2$ .
7.
Which of these values is NOT possible for $R^2$ of a least squares regression line?
1. a)1
2. b)0.5
3. c)0
4. d)1.2
8.
A regression line has a negative slope and $R^2 = 0.49$ . What is the correlation coefficient $r$ ? Give your answer as a decimal.
9.
A model has $R^2 = 0.64$ . What percent of the variation in $y$ is NOT explained by the linear relationship with $x$ ? Give the percent as a number only (like 25).
10.
A regression of phone battery charge on hours of use has slope $-2.5$ and $R^2 = 0.36$ . Find the correlation coefficient $r$ .
11.
The $y$ -values of a data set are $5, 9, 6, 12, 8$ . The regression line on $x$ has $SS_{\text{res}} = 12$ . Find $R^2$ .
12.
The residuals of a regression line are $2, -2, 1, 0, -1$ and $SS_{\text{tot}} = 40$ . Find $R^2$ .
13.
In a summer study of 30 towns, a regression of sunburn cases on ice cream sales gives $R^2 = 0.85$ . What is the best conclusion?
1. a)The line explains 85% of the variation, but a third variable such as sunny weather may drive both
2. b)Selling less ice cream would prevent sunburns
3. c)Eating ice cream causes 85% of sunburns
4. d)85% of the towns lie exactly on the regression line
14.
The residuals of a regression line are $3, -1, -2, 1, -1$ and $SS_{\text{tot}} = 80$ . What is $R^2$ ? Give your answer as a decimal.
15.
A straight line fitted to some data has $R^2 = 0.95$ , but the residual plot shows a clear U shape. What should you conclude?
1. a)The line is a perfect model because $R^2$ is so high
2. b)The calculation of $R^2$ must be wrong
3. c) $x$ causes 95% of the changes in $y$
4. d)The data follows a curve, so a non-linear model would fit better
16.
The data $(1, 3)$ , $(2, 2)$ , $(3, 6)$ , $(4, 10)$ , $(5, 9)$ has the regression line $\hat{y} = 2x$ . Calculate $R^2$ .
17.
Model A has $r = -0.8$ and model B has $r = 0.7$ for the same response variable. Which model explains more of the variation in $y$ , and by how many percentage points?
18.
A regression has $SS_{\text{tot}} = 250$ and $R^2 = 0.92$ . Find $SS_{\text{res}}$ and the explained variation $SS_{\text{tot}} - SS_{\text{res}}$ .