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) 12 11 10 6 5 6 4 4 Cups of coffee (x) 1 2 3 4 4 5 5 1 Unless otherwise stated, you can round values to two decimal places. a) Using regression to find a linear equation for y, Write in the form: a + bx Round a and b to three decimal places. b) Find the correlation coefficient. Round to three or four decimals. T. c) Does the correlation coefficient indicate a strong Linear trend, a weak linear trend, or no linear trend? Ostrong linear trend weak linear trend no linear trend d) Interpret the meaning of the slope of your formula in the context of the problem e) Interpret the meaning of the y intercept in the context of the problem

Answer 1

The necessary calculations done below, where x: Cups of coffee, y=People

Obs 1 2 3 4 5 6 7 8 Cups of coffee People(y) s1=(x-xbar)^2 s2ly-ybar) 2 s3=(x-xbar)*y- 1 12 4.5156 22.5625 -10.09375 1 11 4.5156 14.0625 -7.96875 2 10 1.2656 7.5625 -3.09375 3 6 0.0156 1.5625 0.15625 4 5 0.7656 5.0625 -1.96875 4 6 0.7656 1.5625 -1.09375 5 4 3.5156 10.5625 -6.09375 5 4 3.5156 10.5625 -6.09375 25 18.8748 73.5 -36.25 3.125 7.25 58 sum mean

a) b= sum(s3)/sum(s1) = Sxy/Sxx= -36.25/18.8748 = -1.920

here xbar= mean(x) = 3.125 , ybar=mean(y)=7.25

a= ybar-b*xbar = 7.25 -(-1.920)*(3.125) = 13.252

Therefore linear eqaution of yhat is

yhat= 13.252 -1.920*x

b) Correlation coefficient(r)

r= sum((x-xbar)*(y-ybar))/ sqrt(sum((x-xbar)^2)*sum((y-ybar)^2) = sum(s3)/sqrt(sum(s1)*(sum(s2))

= -36.25/ sqrt(18.8748*73.5)

r=-0.9732

c) r= -0.9732, suggest that there is strong linear trend.

d) slope=b= -1.920 , When unit change in number of coffee cups we expect the people that complain about profeesor's attitude will decreases by 1.92.

e) intercept= a= 13.252, When there no coffee cups then we expect the on average there are 13.252 people that complain about profeesor's attitude.

f) The number of coffee cups is the 8 then the predicted people is

yhat= 13.252 -1.920*x

yhat= 13.252 -1.920*8

       =-2.1126

The number of coffee cups is the 8 then the predicted people that complain about profeesor's attitude is -2.1126

g) The prediction is not resonable, beacuse value of number of people is never negative.