Section 2.2 Solutions

2.23

There is definitely a linear relationship between the variables, so the correlation won’t be as low as 0.1. But there is quite a bit of variation (i.e., the points are in a wide linear band, rather than a tight one), so the correlation won’t be as high as 0.9. Hence, the correlation must be close to 0.6.

2.24

1. Both the S&P 500 and the Dividend Growth fund contain stocks of large U.S. companies. Hence, they should be most similar, and their correlation is probably 0.98. Compared to stocks in developing nations, stocks of small US companies are probably much more similar to stocks of large US companies. Hence the correlation of 0.81 is probably for S&P 500 and the Small Cap fund. That leaves the correlation of 0.35 as the correlation between S&P 500 and the Emerging Markets fund.

2. Positive correlations do not indicate that stocks went up. Rather, they indicate that when the S&P 500 index rose, the other funds often did, too. And when the S&P 500 index fell, the other funds were likely to fall, too.

2.31

1. Any two points form a line (made by simply connecting the two points). The correlation coefficient is exactly 1 (or -1) whenever the points fall exactly on a straight line. Hence, for any two points, the correlation will either be 1 or -1.

2. Parts b-d show how relationships with different shapes can have the same correlation. Hence, you should always graph your data first, and then calculate the correlation (if it’s an appropriate numerical summary).

2.35

One possible graph is shown below. Since the points lie exactly on a straight line with positive slope (y = 2 + x), the correlation coefficient must be 1.

2.38

1. Gender is a categorical variable, and correlation only measures the linear relationship between quantitative variables.

2. The correlation cannot exceed 1.

3. The correlation is unitless (since the values are standardized when calculating the correlation), thus it cannot be expressed in bushels.