Chapter 12: Q. 12.33 (page 493)

Girls and Boys. One probability model for child gender is that a boy or a girl is equally likely to be born. If that model is correct, then, for a two-child family, the probabilities are 0.25,0.50, and 0.25 for two girls, one girl and one boy, and two boys, respectively. W. Stansfield and M. Carlton examined data collected in the National Health Interview Study on two-child families in the article "The Most Widely Published Gender Problem in Human Genetics" (Human Biology, Vol. 81, No. 1, pp. 3-11). Of 42,888 families with exactly two children, 9,523 had two girls, 22,031 had one girl and one boy, and 11,334 had two boys.
a) At the 1% significance level, do the data provide sufficient evidence to conclude that the distribution of genders in two-children families differs from the distribution predicted by the model described?
b) In view of your result from part (a), what conclusion can you draw?

Short Answer

Expert verified

Ans:

(a) Thus, the data provide sufficient evidence to conclude that gender distribution in families of two children differs from the predicted distribution of the model defined at the $1 %$ significance level.

(b) There is very strong evidence against the null hypothesis because the value of p is almost zero. This means that the null hypothesis is rejected at the value of $1 %$ . Therefore, it can be concluded that gender distribution in families of two children differs from the predicted propagation of the described model. That is, the model for child gender opportunities is that a boy or girl does not have an equal chance of being born.

Step by step solution

Step 1. Given information.

given,

One probability model for child gender is that a boy or a girl is equally likely to be born. If that model is correct, then, for a two-child family, the probabilities are 0.25,0.50, and 0.25 for two girls, one girl and one boy, and two boys, respectively.

Step 2. (a) Check whether the data provide sufficient evidence or not to conclude that gender distribution in families with two children differs from the predicted distribution of the model described at the 1% value level.

Null hypothesis:

$H_{0} : -$ Gender distribution in families with two children does not differ from the distribution predicted by the model described.

Another hypothesis:

$H_{1} : -$ The distribution of sex in families with two children differs from the predictive distribution of the model described.

Step 3. Use MINITAB to get test statistics and p-value.

MINITAB process:

Step 1: Select Number> Tables> Chi-Square-Goodness-of-Fit Test Test (One Variable)

Step 2: In the visual calculation, enter the "O" column.

Step 3: In the names of the categories, enter a column. "Gender".

Step 4: Under Test, select the "P" column for specific dimensions.

Step 5: Click OK.

Step 4. MINITAB output,

According to the output, the value of the test statics is $185.080$ and the p-value is 0.

Step 5. Conclusion,

Here, the value of p is smaller than the value level.

That is the value of $p (= 0) < α (= 0.01)$ .

Therefore, the null hypothesis was rejected at $1 %$ .

Thus, the data provide sufficient evidence to conclude that gender distribution in families of two children differs from the predicted distribution of the model defined at the $1 %$ significance level.

Step 6. (b) Conclusion:

There is very strong evidence against the null hypothesis because the value of p is almost zero. This means that the null hypothesis is rejected at the value of $1 %$ . Therefore, it can be concluded that gender distribution in families of two children differs from the predicted propagation of the described model. That is, the model for child gender opportunities is that a boy or girl does not have an equal chance of being born.