Question

A particular professor has noticed that the number of people, y, who complain about his attitude is dependent on the number of cups of coffee, x, he drinks. From eight days of tracking he compiled the following data: People (y) 11 8 8 6 8 5 4 4 Cups of coffee (x) 1 2 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 ˆ y . Round to three decimal places. ˆ y = b) Find the correlation coefficient. Round to three or four decimals. r = c) Does the correlation coefficient indicate a strong linear trend, a weak linear trend, or no linear trend? strong linear trend weak linear trend no linear trend d) Use your model to predict the number of people that will complain about his attitude if he drinks 10 cups of coffee.

Answer 1

X	Y	XY	X²	Y²
1	11	11	1	121
2	8	16	4	64
2	8	16	4	64
3	6	18	9	36
3	8	24	9	64
4	5	20	16	25
4	4	16	16	16
5	4	20	25	16
Ʃx =	Ʃy =	Ʃxy =	Ʃx² =	Ʃy² =
24	54	141	84	406

Sample size, n =	8
x̅ = Ʃx/n = 24/8 =	3
y̅ = Ʃy/n = 54/8 =	6.75
SSxx = Ʃx² - (Ʃx)²/n = 84 - (24)²/8 =	12
SSyy = Ʃy² - (Ʃy)²/n = 406 - (54)²/8 =	41.5
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 141 - (24)(54)/8 =	-21

a)

Slope, b = SSxy/SSxx = -21/12 = -1.75

y-intercept, a = y̅ -b* x̅ = 6.75 - (-1.75)*3 = 12

Regression equation :

ŷ = 12 + (-1.75) x

-------

b)

Correlation coefficient, r = SSxy/√(SSxx*SSyy) = -21/√(12*41.5) = -0.9410

-------

c) The correlation coefficient indicate a strong linear trend.

-------

d)

Predicted value of y at x = 10

ŷ = 12 + (-1.75) * 10 = -5.5