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) 0 50 100 150 200 250 300 350 400 Number of products (y) 100 1,100 2,100 3,100 4,100 5,100 6,100 7,100 8,100 Part 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).

The correlation coefficient helps us to know how strong is the relation between two variables. Its value is always between +1 to -1, where, the numerical value shows how strong is the relation between them and, the '+' or '-' sign shows whether the relationship is positive or negative.

1 indicates a strong positive relationship.
-1 indicates a strong negative relationship.

A result of zero indicates no relationship at all, therefore, independent variable.

A.) In order to find the correlation between the two quantities we need to know the mean of the two values, therefore, the mean of the number of employees can be written as,

The mean of the number of employees

[tex]\text{Mean of the number of employees} = \dfrac{0+50+100+150+200+250+300+350+400}{9}[/tex]

[tex]\text{Mean of the number of employees} = 200[/tex]

The mean of the number of products

Mean of the number of Products [tex]= \dfrac{100+1100+2100 +3100+ 4100 +5100+ 6100+ 7100+ 8100}{9}\\\\\\=4100[/tex]

Now, the table can be written in the following manner as shown below, using the formula of the correlation,

[tex]r =\dfrac{\sum\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)}{\sqrt{\sum\left(x_{i}-\bar{x}\right)^{2} \sum\left(y_{i}-\bar{y}\right)^{2}}}[/tex]

[tex]r =\dfrac{(30 \times 10^4)}{\sqrt{(15 \times 10^4)\times (60 \times 10^6)}}\\\\r = 0.1[/tex]

B.) The given data follows the linear relationship, therefore, the function of the given data will follow the general equation of a line.

y = mx + C

If we look at the value of y when the value of x was 0,

[tex]y = mx +C \\\\100 = m(0) + C\\\\C = 100[/tex]

Substitute the value of C in any other data point we will get,

[tex]y = mx+C\\\\1100 = m(50) +100\\\\m = 20[/tex]

Hence, the function that best fits the data is y=20x+100.

C.) The slope and y-intercept of the plot indicate that as the number of employees in the organization increases the number of products increases as well. The y-intercept shows us that if there are no employees in the company at that point the company will be able to produce 100 products on its own.

Learn more about Correlation Coefficients:

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