SOLUTION: Each box of one brand of breakfast cereal contains a coupon entitling purchasers to a free package of vegetable seeds. At the home office, they use the weight of the incoming mail

Question 1170562: Each box of one brand of breakfast cereal contains a coupon entitling purchasers to a free package of vegetable seeds. At the home office, they use the weight of the incoming mail to determine the number of employees to be assigned to collecting coupons and sending out seed packages each day. (The company policy is that all mail received will be answered the same day.) Following is the data for a random sample of 8 days:
Weight of mail (pounds) 11 20 16 6 12 18 23 25
Number of employees 6 10 9 5 8 14 13 16
(a) Draw a scatter diagram for the data-discuss its shape (5 marks)
(b) Using Excel conduct the following regression models for the data
I. LineaR
II. Quadratic
III. Third degree polynomial
IV. Exponential
V. Logarithmic
VI. Power
(c) Determine which model is the best-state the reason for your belief. (5 marks)
(d) What percentage of the variation in the number of employees is explained by the regression model chosen.
(e) If the company receives 15 pounds of mail, how many employees should be assigned mail duty?
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Part (a)

Table

Weight of Mail (pounds)	Number of Employees
11	6
20	10
16	9
6	5
12	8
18	14
23	13
25	16

Scatter Plot

There's an upward trend, so the data is positively correlated. As x (weight of mail) goes up, y (number of employees) goes up as well. This makes sense because you need more workers to process the larger volume of mail.
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Part (b)

Linear

Quadratic

Cubic

Exponential

Logarithmic

Power

For each chart, f(x) is the approximate regression line and R^2 is the approximate value of the Coefficient of Determination. The closer R^2 is to 1, the better the fit. If R^2 = 1 exactly, then we have an exact fit and that means all points are on the function curve.

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Part (c)

From the charts in part (b), we see that R^2 = 0.875 approximately for the exponential regression. This is the largest R^2 value of the six charts. Therefore, the exponential regression model is the best fit among the choices.

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Part (d)

Roughly 87.5% of variation in the number of employees is explained by the variation in the weight of mail. This figure is directly tied to the R^2 value.

======================================================================
Part (e)

Plug x = 15 into the exponential function model (from part (b)). We pick this function due to it being the best fit.

f(x) = 3.55314210119616e^(0.059651797827595x)
f(15) = 3.55314210119616e^(0.059651797827595*15)
f(15) = 8.69379261400984
f(15) = 9
I'm rounding to the nearest whole number since every y value in the original table is a whole number.

If the company receives x = 15 pounds of mail, then we predict or estimate there will about y = 9 employees assigned to mail duty.

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