Question

A pediatrician wants to determine the relation that may exist between a child's height and head circumference. She randomly selects five 3-year-old children from her practice, measures their height and head circumference and obtains the data shown in the table:

height	27.75 24.5 25.5 26 25
head circumferences(in)	17.5 17.1 17.1 17.3 16.9

(a) if the pediatrician wants to use height to predict head circumference, determine which variable is the explanatory variable and which is the response variable:

(b)Draw a scatter diagram:

(c) compute the linear correlation coefficient,r, between the height and head circumference of a child:

r= n(summation xy)-(summation x)(summation y)/square root n(summation x^2)-(Summation x)^2 square root n(summation y^2)- (summation y)^2

(d) does a linear relation exist between height and head circumference? If it does, what kind?

Answer 1

Answer(a):

The pediatrician wants to use height to predict head circumference, which means head circumference is dependent on height. So height is the explanatory variable and head circumference is the response variable.

Answer(b):

From the scatter plot between Height and head circumference we can see that there is positive relationship between two variables as the head circumference seems to be increasing with the increase in the Height.

Answer(c):

	Height=x	Head Circumferences(in)=y
	27.75	17.5	2	0.32	0.64	4	0.1024
	24.5	17.1	-1.25	-0.08	0.1	1.5625	0.0064
	25.5	17.1	-0.25	-0.08	0.02	0.0625	0.0064
	26	17.3	0.25	0.12	0.03	0.0625	0.0144
	25	16.9	-0.75	-0.28	0.21	0.5625	0.0784
Total	128.75	85.9	0.00	0.00	1.00	6.25	0.21
Mean	25.75	17.18

The coefficient of correlation between two variables can be obtained by

From the value of correlation coefficient, we can interpret that there is strong positive correlation between Height and head circumference.

Answer(d):

To confirm the linear relation between height and head circumference, we need to check whether this correlation is significant or not and for this we have to conduct a t-test for significance of correlation coefficient.

We have to test the below null hypothesis at α=0.10

H0: ρ=0

HA: ρ>0

The test statistic to test this hypothesis is

The critical value of t at 0.05 level of significance with 3df is 2.35

The obtained t value is greater than the critical t value at α=0.05 for a right tailed test which suggests that we have enough evidence against H0 to reject it and we can conclude that the correlation is significant and positive.