Question

5. The following sample of 9 values were obtained in a study of the relationship between the weight and chest size of infants at birth:

Weight (kg)	2.75	2.15	4.41	5.52	3.21	4.32	2.31	4.30	3.71
Chest size (cm)	29.5	26.3	32.2	36.5	27.2	27.7	28.3	30.3	28.7

Obtain a scatter plot of the data and comment on the correlation between the two variables. Compute the correlation coefficient (r) and test whether it differs from 0 (use level of significance of 5%). Finally, obtain the 95% confidence interval for the correlation coefficient, ρ. Is the confidence interval consistent with the results of the hypothesis test?

Answer 1

there seems to be positive correlation between two variables

b)

x	y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
2.75	29.5	0.78	0.02	0.12
2.15	26.3	2.19	11.11	4.94
4.41	32.2	0.61	6.59	2.00
5.52	36.5	3.57	47.15	12.97
3.21	27.2	0.18	5.92	1.02
4.32	27.7	0.47	3.74	-1.33
2.31	28.3	1.75	1.78	1.76
4.3	30.3	0.45	0.44	0.45
3.71	28.7	0.01	0.87	-0.07

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	32.68	266.7	9.995488889	77.6	21.85
mean	3.63	29.63	SSxx	SSyy	SSxy

correlation coefficient ,    r = Sxy/√(Sx.Sy) =   0.7845

correlation hypothesis test              
Ho:   ρ = 0       tail=   2
Ha:   ρ ╪ 0          
n=   9          
alpha,α =    0.05          
correlation , r=   0.7845          
t-test statistic = r*√(n-2)/√(1-r²) =        3.347      
DF=n-2 =   7          
p-value =    0.0123          
Decison:   p value < α , So, Reject Ho      

   

CI for correlation

Convert r to z’ using Fisher’s z’ transform:

confidence intervals using the resulting z’ value:

LOWER TAIL = 0.25131

UPPER TAIL =0.952412

CI does not contain 0 , so reject null hypothesis

and it is same as hypothesis test