Chapter 3: Q.20 (page 161)

Data on dating A student wonders if tall women tend to date taller men than do short women. She measures herself, her dormitory roommate, and the women in the adjoining rooms. Then she measures the next man each woman dates. Here are the data (heights in inches):
(a) Make a scatterplot of these data. Based on the scatterplot, do you expect the correlation to be positive or negative? Near $\pm 1$ or not?
(b) Find the correlation r step-by-step. First, find the mean and standard deviation of each variable. Then find the six standardized values for each variable. Finally, use the formula for $r$ . Do the data show that taller women tend to date taller men?

Short Answer

Expert verified

a). The correlation should be positive.

b). The value of $r = 0.5653$ .

Step by step solution

Part(a) Step 1: Given Information

Data:

Part(a) Step 2: Explanation

Women $(x)$ is on the horizontal axis, and Men $(y)$ is on the vertical axis.

Part(a) Step 3: Explanation

The scatterplot slopes upwards, indicating that the correlation should be positive.

Because the points do not appear to be on the same axis, the correlation should be less than $\pm 1$ .

Part(b) Step 1: Given Information

Data:

Part(b) Step 2: Explanation

To find $X \cdot Y, X^{2}$ and $Y^{2}$ as it was done in the table below.

Part (b) Step 3: Explanation

Find the sum of every column to get:

$\sum X = 396, \sum Y = 414, \sum X \cdot Y = 27339, \sum X^{2} = 26158, \sum Y^{2} = 28598$

To calculate the correlation coefficient, use the formula below.

localid="1649931991616" $r = \frac{n \cdot \sum X Y - \sum X \cdot \sum Y}{\sqrt{[n \sum X^{2} - {(\sum X)}^{2}] \cdot [n \sum Y^{2} - {(\sum Y)}^{2}]}} = \frac{6 \cdot 27339 - 396 \cdot 414}{\sqrt{[6 \cdot 26158 - 396^{2}] \cdot [6 \cdot 28598 - 414^{2}]}} \approx 0.5653$