Question

chi-squared test ( χ2 test) problem
We're interested in the weight of babies born not prematurely but after a pregnancy shorter than average. Of a sample of 300 children born in April, the found as follows:

. born after 37 weeks: 30 slightly underweight; 20 normal weight.

. born after 38 weeks: 40 slightly underweight; 60 normal weight.

. born after 39 weeks: 50 born slightly underweight; 100 normal weight.

(a) Identify the two variables studied in the sample and the values of the two variables they can take, then build a full frequency table.

b) Calculate the value χ2 from the table obtained in a). Explain why we can be almost certain that there is an interdependence between the number of weeks of pregnancy and birth weight.

(c) As it has been found that there is an interdependence between the two, an attempt will be made to establish a numerical relationship between the number of weeks of pregnancy and the proportion of children born underweight. What will our variables be here and what does our study tell us?

d) Calculate the equation for the regression line obtained from the points found in c).

Answer 1

a)

	Observed Frequencies
	0
0	underweight	normal weight					Total
37 weeks	30	20					50
38 weeks	40	60					100
39 weeks	50	100					150
Total	120	180					300

b)

	Expected frequency of a cell = sum of row*sum of column / total sum
	Expected Frequencies
	underweight	normal weight					Total
37 weeks	120*50/300=20	180*50/300=30					50
38 weeks	120*100/300=40	180*100/300=60					100
39 weeks	120*150/300=60	180*150/300=90					150

	(fo-fe)^2/fe
37 weeks	5.000	3.333
38 weeks	0.000	0.000
39 weeks	1.6667	1.1111

Ho: given two variable are independent
H1: Given two variables are not independent

Chi-Square Test Statistic,χ² = Σ(fo-fe)^2/fe =   11.111                          
                              
Level of Significance =   0.05                          
Number of Rows =   3                          
Number of Columns =   2                          
Degrees of Freedom=(#row - 1)(#column -1) = (3- 1 ) * ( 2- 1 ) =   2                          
                              
p-Value =   0.0039   [Excel function: =CHISQ.DIST.RT(χ²,df) ]                      
Decision:    p-value < α , Reject Ho         

It means we can be almost certain that there is an interdependence between the number of weeks of pregnancy and birth weight.

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