The sex of consecutive children is thought to be independent in humans. That is,

Question

The sex of consecutive children is thought to be independent in humans. That is, whether a couple's first child is a boy should have no bearing on whether the second child is a boy. In the absence of any complications, the number of boys in 2-child families should match a binomial distribution with n equal to 2 and p equal to the probability of having a male child. Consider the following data from a real, random sample of 2444 families with 2 children. Observed number of families obs proP exp pro p Epet rels 01216 0123 o.quh Number of boys 530 1332 582 2444 0 Total Recall that for binomial random variables And that the chi squared statistic is (observed-Expected) Expected

Explanation / Answer

Solution:

Here, we have to use Chi square test for goodness of fit.

Null hypothesis: H0: Given data follows binomial distribution.

Alternative hypothesis: Ha: Given data do not follow binomial distribution.

We consider level of significance as = 0.05

The test statistic formula is given as below:

Chi square = [(O – E)^2/E]

Where, O is observed frequencies and E is expected frequencies.

Expected frequencies are calculated by using expected proportion.

Expected proportion is calculated by using binomial distribution.

We are given 2-child family, this means n = 2 and probability for boy is 0.5.

P(X=x) = [n!/x!(n – x)!]*p^x*(1 – p)^(n – x)

So, for X = 0 boys,

P(X=0) = [2!/(0!*(2 – 0)!)]*0.5^0*(1 – 0.5)^(2 – 0) = 0.25

For X = 1

P(X=1) = [2!/(1!*(2 – 1)!)]*0.5^1*(1 – 0.5)^(2 – 1) = 0.50

For X = 2

P(X=2) = [2!/(2!*(2 – 2)!)]*0.5^2*(1 – 0.5)^(2 – 2) = 0.25

Expected frequencies are given as below:

Total number of families = n = 2444

For X = 1

E = n*p = 2444*0.25 = 611

For X = 2

E = 2444*0.50 = 1222

For X = 3

E = 2444*0.25 = 611

Calculation table is given as below:

Number of boys

O

Exp. Prop.

E

(O - E)

(O - E)^2

(O - E)^2/E

0

530

0.25

611

-81

6561

10.73813421

1

1332

0.5

1222

110

12100

9.901800327

2

582

0.25

611

-29

841

1.376432079

Total

2444

1

2444

22.01636661

Chi square = [(O – E)^2/E] = 22.01636661

N = 3

DF = 3 – 1 = 2

P-value = 0.0000166

= 0.05

P-value < = 0.05

So, we reject the null hypothesis that the given data follows binomial distribution.

There is sufficient evidence to conclude that given data do not follow binomial distribution. From the given analysis, there is insufficient evidence to conclude that sex of consecutive children is independent.

Number of boys

O

Exp. Prop.

E

(O - E)

(O - E)^2

(O - E)^2/E

0

530

0.25

611

-81

6561

10.73813421

1

1332

0.5

1222

110

12100

9.901800327

2

582

0.25

611

-29

841

1.376432079

Total

2444

1

2444

22.01636661

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