Regression and correlation NOTE: this is hand calculation assignment. DO NOT use
ID: 3316414 • Letter: R
Question
Regression and correlation
NOTE: this is hand calculation assignment. DO NOT use SPSS. Columns must be filled as guided in the lecture note
Investigate the relationship between crying and IQ
Infants who cry easily may be more easily stimulated than others. This may be a sign of higher IQ. Child development researchers explored the relationship between the crying of infants four to ten days old and their later IQ test scores. The researchers recorded the crying of babies and measured its intensity by the number of peaks in the most active 20 seconds. They later measured the children’s IQ at age three years using the Stanford-Binet IQ test. The follow table contains data on 10 infants (the sample size reduced for this homework). Do children with higher crying counts tend to have higher IQ?
NOTE: use all columns to calculate components of regression and correlation. Add column if is needed
IQ (X)
Crying (Y)
87
10
97
12
103
9
106
16
109
18
114
15
119
12
132
20
136
16
159
33
Find the mean of each variable
Write the regression equation. Write the formula of a & b and calculate the slope (b) and the y-intercept (a). plug in a & b in the equation
What the regression equation tells us about the relationship between IQ and Crying?
Predict the intensity of crying if the child IQ would be 112
Write correlation formula and calculate the correlation between crying and IQ. Determine the strength and the direction of the of relationship between variables
What percentage of variance (or change) in the dependent variable crying is explained by the IQ variable?
IQ (X)
Crying (Y)
87
10
97
12
103
9
106
16
109
18
114
15
119
12
132
20
136
16
159
33
Explanation / Answer
Line of Regression Y on X i.e Y = bo + b1 X
calculation procedure for regression
mean of X = X / n = 116.2
mean of Y = Y / n = 16.1
(Xi - Mean)^2 = 4037.6
(Yi - Mean)^2 = 426.9
(Xi-Mean)*(Yi-Mean) = 1111.8
b1 = (Xi-Mean)*(Yi-Mean) / (Xi - Mean)^2
= 1111.8 / 4037.6
= 0.275
bo = Y / n - b1 * X / n
bo = 16.1 - 0.275*116.2 = -15.897
value of regression equation is, Y = bo + b1 X
Y'=-15.897+0.275* X
intensity of crying if the child IQ would be 112
Y'=-15.897+0.275* 112
= 14.903
calculation procedure for correlation
sum of (x) = x = 1162
sum of (y) = y = 161
sum of (x^2)= x^2 = 139062
sum of (y^2)= y^2 = 3019
sum of (x*y)= x*y = 19820
to caluclate value of r( x,y) = covariance ( x,y ) / sd (x) * sd (y)
covariance ( x,y ) = [ x*y - N *(x/N) * (y/N) ]/n-1
= 19820 - [ 10 * (1162/10) * (161/10) ]/10- 1
= 111.18
and now to calculate r( x,y) = 111.18/ (SQRT(1/10*19820-(1/10*1162)^2) ) * ( SQRT(1/10*19820-(1/10*161)^2)
=111.18 / (20.094*6.534)
=0.847
value of correlation is =0.847
percentage of variance (or change) in the dependent variable crying is explained by the IQ variable = coeffcient of determination = r^2 = 0.717
87 10 852.64 37.21 178.12 97 12 368.64 16.81 78.72 103 9 174.24 50.41 93.72 106 16 104.04 0.01 1.02 109 18 51.84 3.61 -13.68 114 15 4.84 1.21 2.42 119 12 7.84 16.81 -11.48 132 20 249.64 15.21 61.62 136 16 392.04 0.01 -1.98 159 33 1831.84 285.61 723.32