A researcher was interested in the relationship between the hours of sleep someo

Question

A researcher was interested in the relationship between the hours of sleep someone gets each night and their performance on a test of alertness. The following summary results were obtained:

                                 Hours of Sleep (X)           Score on Alertness Test (Y)
M                                     6.7                                    78.0
Sum                                670                                    780
SS                                   26                                    1310
Variance                           3                                        146
Standard Deviation        1.7                                       12

N=10

E{(X-Mx)(Y-My)} = 119

a. What is the covariance?
b. What is the correlation?
c. What is the prediction equation using score on the alertness test as the dependent variable?
d. Based on the prediction equation you calculated in part C, and with score as the dependent variable, substitute the value of 8 for the amount of hours spent sleeping. What would you predict as the score?

Explanation / Answer

a. The covariance is E{(X-Mx)(Y-My)} = 119

b. The correlation is = covariance(x,y) / (sd(x)sd(y)) = 119/(1.7*12) = 5.833333.

c. The prediction equation will be,

Y = My + byx(X - Mx)

Here byx = covariance(x,y)/variance(x) = 119/3 = 39.66667

So, Y = 78 + 39.66667 (X - 6.7)

=> Y =  -187.7667 + 39.66667 * X

d) For X = 8, we have Y = -187.7667 + 39.66667 * 8 = 129.5667 [Predicted].

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