5) You are interested in the relationship between caffeine consumption and hours

Question

5) You are interested in the relationship between caffeine consumption and hours slept, so you collect information from a random sample of individuals on how many cups of coffee they drink per day (C) and the number of hours they sleep per night (S). Using this data, you calculate the following sample averages: Sample average of C Sample average of C2 Sample average of S Sample average of S2 2.1 7.8 7.2 60.1 Sample average of C'S 13.8 a) Given the sample averages above, what is the sample correlation coefficient between hours slept and cups of coffee consumed? b) Calculate the OLS estimates of ao and al for the following regression model and interpret the estimated coefficients. What do they tell you about the relationship between coffee and sleep? c) Using the information provided, is it possible to determine the R2 from this model? If so, interpret the meaning of this number d) Would you agree that coffee consumption impacts sleep, but that sleep has no impact on coffee consumption? Explain your reasoning

Explanation / Answer

(a) Sample correlation coefficient

r = [(CS) - (C)((S)] / sqrt [ ( (C2 ) - (C)2) ( (S2 ) - (S)2]

so here C and S have their notation which denotes average.

r = (13.8 - 2.1 * 7.2)/ sqrt [(7.8 - 2.12 ) * (60.1 - 7.22 )]

r = -1.32 / sqrt (3.39 * 8.26)

r = -0.25

(b) If linear line equation is

S = a + bC

so here a = [(S) (C2 ) - (C) (SC)]/ [ (C2 ) - (C)2 ]

a = [ 7.2 * 7.8 - 2.1 * 13.8] / [(7.8 - 2.12 )]

a = 27.18/ 3.39 = 8.02

b = [(SC) - (S)((C)]/ [ n (C2 ) - (C)2 ]

b = (13.8 - 2.1 * 7.2) / [(7.8 - 2.12 )

b = -1.32/ 3.39

b= -0.39

so Line equation

S = 8.02 - 0.39C

(c) R2 = r * r = (-0.25) * (-0.25) = 0.0625

we can see that R2 = 0.0625 , which is very small so we can say that the strength of the relationship is very weak.

Ans only 6.25% of the SLeep variable is expressed by the amount of coffee drink by participants.

(d) Yes, personally i feel that coffee consuption impacts sleep, but here data shows it is very weak relationship between both variables. The sleep doesn't impact coffee consumption because here independent variable is coffee consumption which impacts sleep. ANd causation doesn't imply correlation and viceversa too.

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