Question

For a sample of 8 employees, a personnel director has collected the following data on ownership of company stock, y, versus years with the firm, x.

X 6 12 14 6 9 13 15 9
Y 300 408 560 252 288 650 630 522
Determine the least squares regression line and interpret its slope

Answer 1

The least squares regression line is ŷ = 38.75581x + 44.31395, where ŷ is the predicted ownership of company stock, and x is the years with the firm. The slope of 38.75581 suggests a yearly stock ownership increase of about 38.76 units.

To find the least squares regression line, we need to calculate the slope (b) and the y-intercept (a) using the following formulas:

`\[ b = \frac{\sum_(i=1)^(n) (x_i - \bar{x})(y_i - \bar{y})}{\sum_(i=1)^(n) (x_i - \bar{x})^2} \]\[ a = \bar{y} - b\bar{x} \]`

where:

- n is the number of data points (in this case, 8),

-
`\( x_i \) and \( y_i \)` are the individual data points,

-
`\( \bar{x} \) and \( \bar{y} \)` are the mean values of x and y respectively.

Let's calculate these values step by step:

1. Calculate the mean

`\[ \bar{x} = (6 + 12 + 14 + 6 + 9 + 13 + 15 + 9)/(8) = 10.5 \]\[ \bar{y} = (300 + 408 + 560 + 252 + 288 + 650 + 630 + 522)/(8) = 451.25 \]`

2. Calculate the slope

`\[ b = (\sum_(i=1)^(8) (x_i - 10.5)(y_i - 451.25))/(\sum_(i=1)^(8) (x_i - 10.5)^2) \]\[ b = ((6-10.5)(300-451.25) + (12-10.5)(408-451.25) + \ldots + (9-10.5)(522-451.25))/((6-10.5)^2 + (12-10.5)^2 + \ldots + (9-10.5)^2) \]`

After calculating this expression, you should get b to be approximately 38.75581.

3. Calculate the y-intercept a:

`\[ a = 451.25 - 38.75581 * 10.5 \]`

After calculating this expression, you should get a to be approximately 44.31395.

Therefore, the least squares regression line is:

`\[ \hat{y} = 38.75581x + 44.31395 \]`

Now, let's interpret the slope:

The slope of the regression line b represents the change in the dependent variable y for a one-unit change in the independent variable x. In this context, the slope of approximately 38.75581 means that, on average, for each additional year an employee spends with the firm x increases by 1, their ownership of company stock y is expected to increase by approximately 38.75581 units.