Question

the following scatterplot shows the arm - span (in centimeters) and height (in centimeters) for a random sample of 19 twelfth - graders. the correlation is r = 0.902. (a) explain why it is incorrect to say that the correlation is 0.902 centimeter. (b) what would happen to the correlation if height was measured in inches instead of centimeters? explain your answer. (c) what would happen to the correlation if height was used as the explanatory variable and arm - span was used as the response variable?

Explanation:

Step1: Recall correlation property

Correlation coefficient \(r\) is unit - less. It measures the strength and direction of a linear relationship between two variables and is not affected by the units of measurement.

Step2: Answer part (a)

It is incorrect to say that the correlation depends on units because \(r\) is unit - less. Changing units of measurement of the variables does not change the linear relationship between them in terms of direction and strength.

Step3: Answer part (b)

Since \(r\) is unit - less, if height was measured in inches instead of centimeters, the correlation coefficient \(r = 0.902\) would remain the same.

Step4: Answer part (c)

If the roles of the explanatory and response variables are reversed (height as response and arm - span as explanatory), the correlation coefficient \(r = 0.902\) would remain the same. Correlation measures the strength and direction of the linear association between two variables and is symmetric, i.e., \(r_{xy}=r_{yx}\).

Answer:

(a) It is incorrect because the correlation coefficient \(r\) is unit - less and does not depend on the units of the variables.
(b) The correlation would remain \(r = 0.902\) as \(r\) is unit - less.
(c) The correlation would remain \(r = 0.902\) since correlation is symmetric for two variables.