Question

A psychologist noted that people have more difficulty sleeping in a bright room than in a dark room. She measured whether the intensity of the light could predict the time it took a sample of 4 participants to fall asleep. The data for this hypothetical study are listed in the following table.

Intensity of Light (in watts)	Time It Took to Sleep (in minutes)
X	Y
5	13
10	20
20	32
40	37

Compute an analysis of regression for this hypothetical study. (Round your answers to two decimal places.)

Source of Variation	SS	df	MS	Fobt
Regression		1
Residual (error)
Total

Answer 1

First we have to construct below table using give data in order to compute regression equation

	X	Y	X2	Y2	X*Y
	5	13	25	169	65
	10	20	100	400	200
	20	32	400	1024	640
	40	37	1600	1369	1480
Total	= 75	= 102	= 2125	= 2962	= 2385

We know that, the regression equation is

Where a is intercept and b is the slop and are calculated by using below formulae

Substitute these values in equation (i), we get

Now we have to construct below table using above equation (ii), we get

	X	Y	X2	Y2	X*Y		(Y - )^2	( - )^2	(Y - )^2
	5	13	25	169	65	16.46087	11.97762	81.70587902	156.25
	10	20	100	400	200	19.74783	0.063592	33.08750473	30.25
	20	32	400	1024	640	26.32174	32.24265	0.675255198	42.25
	40	37	1600	1369	1480	39.46957	6.098752	195.1487524	132.25
Total	75	102	2125	2962	2385	102	50.38261	310.6173913	361

Now we can find sum of squared regression (SSR), sum of squared error (SSE) and sum of squared total (SST) using below formulae and above table values

Where is the mean of predicted values () which is 25.50

Where is the mean of Y values, which is 25.5

Calculation of degrees of freedom:

Now we have to find the degrees sum of square for regression (dfreg), degrees sum of square for error or residual (dfres) and degrees sum of square for total (dftot) using below formulae, we get

dfreg = number of degrees of freedom for the regression sum of squares which is equal to the number of coefficients in the equation - 1 = 2 - 1 = 1

dftot = total number of observations - 1 = 4 - 1 = 3

dfres = dftot - dfreg = 3 - 1 = 2

Calculation of Mean sum square:

Now we have to find mean sum square for regression (MSreg) and mean sum square for error or residual (MSres) using below formulae

MSreg = SSR / dfreg = 310.6174 / 1 = 310.6174

MSres = SSE / dfres = 50.3826 / 2 = 25.1913

Calculation of F value:

F value is calculated using below formula, we get

F = MSreg / MSres = 310.6174 / 25.1913 = 12.3303

From the above values, we can construct below table

Source of Variation	SS	DF	MS	F
Regression	310.62	1	310.62	12.33
Residual (Error)	50.38	2	25.19
Total	361	3