Question

A manager is assessing the correlation between the number of employees in a plant and the number of products produced yearly. The table shows the data:Number of employees(x)050100150200250300350400Number of products(y)1001,1002,1003,1004,1005,1006,1007,1008,100Part A: Is there any correlation between the number of employees in the plant and the number of products produced yearly? Justify your answer. (4 points)Part B: Write a function that best fits the data. (3 points)Part C: What does the slope and y-intercept of the plot indicate? (3 points)

Answer 1

Given the table that shows the data of the number of employees and the number of products, you can identify these points:

$(0,100),(50,1100),(100,2100),(150,3100),(200,4100),(250,5100),(300,6100),(350,7100),(400,8100)$

Part A

Plot the points on a Coordinate Plane:

Notice that the relationship is approximately linear. Therefore, you can conclude that there is a perfectly positive correlation between the variables.

Part B

Notice that, in this case, the function that best fits the data must be a Linear Function. It can be written in Slope-Intercept Form:

$y=mx+b$

Where "m" is the slope and "b" is the y-intercept.

In this case, you can identify that the y-intercept is:

$b=100$

Because that is the value of "y" when:

$x=0$

In order to find the slope, substitute the coordinates of one of the points into the equation and solve for "m":

$\begin{gathered} 1100=m(50)+100 \\ \\ \frac{1100-100}{50}=m \\ \\ m=20 \end{gathered}$

Therefore, the function that best fits the data is:

$y=20x+100$

Part C

Notice that the x-values represent the number of employees and the y-values reprsent the number of products.

Therefore, the slope indicates each employee produces 20 products yearly.

And the y-Intercept indicates that 100 products are produced yearly even when there are no employees.

Hence, the answers are:

Part A: Yes, there is a perfectly positive correlation between the variables because their relation is approximately linear.

Part B:

$y=20x+100$

Part C:

- The slope indicates each employee produces 20 products yearly.

- The y-Intercept indicates that 100 products are produced yearly even when there are no employees.