Question

Carrie and Ryan have both computed the slope of the least-squares line using data for which the standard deviation of the x-values and the standard deviation of the y-values are equal. Carrie gets a value of 0.5 for the slope, and Ryan gets a value of 2. One of them is right. Which one?

Answer 1

Answer:

Carrie's slope value is correct.

Step-by-step explanation:

The least square regression line is: $(y -\bar y)=b_{yx} (x-\bar x)$

Here $b_{yx}$ is the slope of the line.

The formula to compute the slope is:

$b_{yx}=r\frac{\sigma_{x}}{\sigma_{y}}$

Here $\sigma_{x}$ = standard deviation of x and $\sigma_{y}$ = standard deviation of y.

It is provided that the standard deviation of x and y are equal.

So the slope of the regression line is:

$b_{yx}=r\frac{\sigma_{x}}{\sigma_{y}}=r\frac{\sigma_{x}}{\sigma_{x}}=r$

Thus, if the standard deviation of x and y are equal the slope of the line is same as the correlation coefficient.

The correlation coefficient is a measure used to determine the strength of the linear relationship between the variables.

It ranges from -1 to 1.

Carrie's slope value was 0.50 and Ryan's slope value was 2.

Since -1 ≤ r ≤ 1 the value of slope cannot be 2.

Thus, Ryan's slope value is incorrect.