Statistics please help

Question

A particular professor has noticed that the number of people, P, who complain about his attitude is dependent on the number of cups of coffee, n, he drinks. From eight days of tracking he compiled the following data:

People (P)	11	11	9	10	8	7	7	4
Cups of coffee (n)	1	1	2	3	3	4	4	5

Unless otherwise stated, you can round values to two decimal places.

a) Using regression to find a linear equation for P(n)

P(n)P(n) =    

b) Interpret the meaning of the slope of your formula in the context of the problem

 

c) Interpret the meaning of the P intercept in the context of the problem

 

d) Use your model to predict the number of people that will complain about his attitude if he drinks 9 cups of coffee.

   

e) Is the answer to part f reasonable? Why or why not?

 

f) How many cups of coffee should he drink so that no one will complain about his attitude? It is ok to round to one decimal place.

  

Answer 1

First of all, we will assume this is a SAMPLE of the total population. (we use the "n - 1" formula)

Using a TI-84 this is easy. Just plug the data into a list and push the button.

Manually, create a table with "n" in the first column and P(n) in the second column:

n P(n)

1.00 11.00

1.00 11.00

2.00 9.00

3.00 10.00

3.00 8.00

4.00 7.00

4.00 7.00

5.00 4.00

Calculate the mean (average of each)

mean of the x's (aks x-bar) = 2.88

mean of the y's (aks y-bar) = 8.38

Create a third column by subtracting x and x-bar. Then create a fourth column by subtracting y and y-bar.

n P(n) x-(x-bar) y-(y-bar)

1.00 11.00 -1.88 2.63

1.00 11.00 -1.88 2.63

2.00 9.00 -0.88 0.63

3.00 10.00 0.13 1.63

3.00 8.00 0.13 -0.38

4.00 7.00 1.13 -1.38

4.00 7.00 1.13 -1.38

5.00 4.00 2.13 -4.38

Then create a 5th column (x-(x-bar))•(y-(y-bar)) or column 3 times column 4 and sum the 5th column:

n P(n) x-(x-bar) y-(y-bar) (col 4)•(col 5)

1.00 11.00 -1.88 2.63 -4.92

1.00 11.00 -1.88 2.63 -4.92

2.00 9.00 -0.88 0.63 -0.55

3.00 10.00 0.13 1.63 0.20

3.00 8.00 0.13 -0.38 -0.05

4.00 7.00 1.13 -1.38 -1.55

4.00 7.00 1.13 -1.38 -1.55

5.00 4.00 2.13 -4.38 -9.30

Sum: -22.63

Then create a 6th column which is the square of the 3rd column (x - x-bar)² and a 7th column which is the 4th column squared (y - y-bar)² and sum those columns:

n P(n) x-(x-bar) y-(y-bar) (col 4)•(col 5) (col 3)² (col 4)²

1.00 11.00 -1.88 2.63 -4.92 3.52 6.89

1.00 11.00 -1.88 2.63 -4.92 3.52 6.89

2.00 9.00 -0.88 0.63 -0.55 0.77 0.39

3.00 10.00 0.13 1.63 0.20 0.02 2.64

3.00 8.00 0.13 -0.38 -0.05 0.02 0.14

4.00 7.00 1.13 -1.38 -1.55 1.27 1.89

4.00 7.00 1.13 -1.38 -1.55 1.27 1.89

5.00 4.00 2.13 -4.38 -9.30 4.52 19.14

Sum: -22.63 14.88 39.88

r = (sum of column 5)/squareroot(sum of column 6 times the sum of column 7)

r = (-22.63)/√(14.88•39.88)

r = -0.92899

Sx = squareroot(sum of column 6/(n - 1))

Sx = √14.88/(8 - 1) = 1.457738

Sy = squareroot(sum of column 7/(n - 1))

Sx = √39.88/(8 - 1) = 2.386719

The slope of the line (m) = r•Sy/Sx = (-0.92899)•(2.386719)/(1.457738) = -1.52101

The y-intercept of the line = y-bar - slope•x-bar = 8.38 - (-1.52101)(2.88) = 12.7479

So P(n) = -1.52n + 12.75

The slope represent the change in the number of people complaining to the number of cups of coffee he drank.

P(9) = -1.52(9) + 12.75 = -0.93 (not reasonable because the fewest that could complain is zero)

To find the number of cups resulting in zero people complaining, make P(n) = 0

0 = -1.52n + 12.75

1.52n = 12.75

n = 12.75/1.52 = 8.4 cups of coffee