Question

2. APGAR scores are assigned to newborns between one and five minutes after birth and indicate the general state of the wellbeing of the baby. Scores of seven to ten are typical and indicate that the baby requires only routine post-natal care. Scores of four to six indicate that the baby may require assistance while scores of three or less indicate that the baby requires immediate assistance if life is to be sustained. Suppose a study is conducted to determine the consistency of APGAR scores and NNIT scores, which has the same scale and also indicates needed level of post-natal care but is set to come out next year. To this end, each baby has their current APGAR score and a NNIT score (table below). Use the following data to answer the below questions.

1. Calculate and interpret the Pearson correlation coefficient between APGAR and NNIT scores.
2. Please test if the correlation coefficient is different from zero.

Baby ID	APGAR score	NNIT score
1	9	7
2	8	9
3	7	8
4	8	8
5	6	7
6	4	4
7	9	8
8	7	7
9	2	3
10	8	6
11	7	8
12	5	5
13	6	9
14	4	5

Answer 1

a) -

Formula for the correlation coefficient is -

Observation table -

Baby ID	APGAR score(x)	NNIT score(y)	(x-x_bar)	(y-y_bar)	(x-x_bar)(y-y_bar)	(x-x_bar)^2	(y-y_bar)^2
1	9	7	2.5714	0.2857	0.7346	6.6121	0.0816
2	8	9	1.5714	2.2857	3.5917	2.4693	5.2244
3	7	8	0.5714	1.2857	0.7346	0.3265	1.653
4	8	8	1.5714	1.2857	2.0203	2.4693	1.653
5	6	7	-0.4286	0.2857	-0.1225	0.1837	0.0816
6	4	4	-2.4286	-2.7143	6.5919	5.8981	7.3674
7	9	8	2.5714	1.2857	3.306	6.6121	1.653
8	7	7	0.5714	0.2857	0.1632	0.3265	0.0816
9	2	3	-4.4286	-3.7143	16.4491	19.6125	13.796
10	8	6	1.5714	-0.7143	-1.1225	2.4693	0.5102
11	7	8	0.5714	1.2857	0.7346	0.3265	1.653
12	5	5	-1.4286	-1.7143	2.449	2.0409	2.9388
13	6	9	-0.4286	2.2857	-0.9797	0.1837	5.2244
14	4	5	-2.4286	-1.7143	4.1633	5.8981	2.9388
Total	90	94	-	-	38.7136	55.4286	44.8568

Calculations -

  

  

  

  

Hence, correlation between APGAR & NNIT score is 0.7764.

There is strong positive correlation between APGAR score & NNIT score. i.e. As APGAR score increases, there will be increase in NNIT score.

b) -

To test that correlation coefficient is different from zero, we have to use t-test for the significance of the correlation coefficient.

Hypothesis -

Null hypothesis - H₀ : There is zero correlation between APGAR score & NNIT score. i.e. = 0.

Alternative hypothesis - H₁ : There is correlation between APGAR score & NNIT score. i.e. 0.

Test statistic -

Test criterion -

We have to reject H₀ if t t_/2,n-2.

Calculations -

r = 0.7764, n = 14

So,

  

  

  

t

Critical value -

t_/2,n-2 = t_0.05/2,14-2 = t_0.025,12 = 2.1788

Result -

here, t (4.2677) > t_/2,n-2 (2.1788), hence, we will reject H₀.

Conclusion -

There is sufficient evidence that correlation coefficient is different from zero.