Question

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) (X-X ?) (Y-Y ?) (X-X ?)2 (Y-Y ?)2 (X-X ?) (Y-Y ?) 87 10 97 12 103 9 106 16 109 18 114 15 119 12 132 20 136 16 159 33 136 26 1298 187 0 0 0 0 1290 Find the mean of each independent and dependent variable X= 1298 = 118 11 Y= 187 = 17 11 Obtain the slope (b) and the y-intercept (a) of the least square regression. Describe what is b and what is a in this least square regression equation. Write the regression equation for the crying (Y) and IQ (X) Predict the intensity of crying if the child IQ would be 112 Calculate the correlation between crying and IQ Calculate R2. What does R2 tell us about the relationship between the independent and dependent variables

Answer 1

cov(x,y)=sum((x-x-)(y-y-))/n=117.3

var(x)=sum((x-x-)²/n=399.5, sd(x)=sqrt(var(x))=sqrt(399.5)=19.97

var(y)=sum(y-y-)²/n=46.9, sd(y)=sqrt(var(y))=sqrt(46.9)=6.85

correlation(x,y)=r=cov(x,y)/(sd(x)sd(y))=117.3/(19.97*6.85)=0.8575

regression equation y=a+bx=-17.73+0.2944*x

b=r*sd(y)/sd(x)=0.8585*6.85/19.97=0.2944

a=mean(y)-b*mean(x)=17-0.2944*118=-17.7392

R²=r*r=0.8575*0.8575=0.7353,

R² is proportion of variation explained by the inpdependent variable x in dependent variable y

	X	Y	X-X-	Y-Y-	(X-X-)2	(Y-Y-)2	(X-X-)(Y-Y-)
	87	10	-31	-7	961	49	217
	97	12	-21	-5	441	25	105
	103	9	-15	-8	225	64	120
	106	16	-12	-1	144	1	12
	109	18	-9	1	81	1	-9
	114	15	-4	-2	16	4	8
	119	12	1	-5	1	25	-5
	132	20	14	3	196	9	42
	136	16	18	-1	324	1	-18
	159	33	41	16	1681	256	656
	136	26	18	9	324	81	162
sum	1298	187	0	0	4394	516	1290
mean	118	17.0	0.0	0.0	399.5	46.9	117.3
n	11	11	11	11	11	11	11