Question

Consider a large population of families in which each family has exactly three children. If the genders of the three children in any family are independent of one another, the number of male children in a randomly selected family will have a binomial distribution based on three trials.

a. Suppose a random sample of 160 families yields the following results. Test the relevant hypotheses by proceeding as in Example 14.5.
b. Suppose a random sample of families in a nonhuman population resulted in observed frequencies of 15, 20, 12, and 3, respectively. Would the chi-squared test be based on the same number of degrees of freedom as the test in part (a)? Explain.

Answer 1

A chi-square goodness-of-fit test is used to compare observed and expected frequencies in a hypothesis test. Degrees of freedom in a chi-square test are always one less than the number of categories being considered.

Step-by-step explanation:

The question relates to the application of the chi-square distribution to conduct a goodness-of-fit hypothesis test and to determine the degrees of freedom in chi-square tests.

For part a, you would conduct a chi-square goodness-of-fit test by comparing the observed number of male children in each family against the expected number based on the binomial distribution. However, without the specific observed and expected frequencies, we cannot calculate the chi-square statistic or the p-value. Generally, you would calculate the expected frequency for 0, 1, 2, and 3 male children using the binomial probabilities and compare these with the observed frequencies using the chi-square formula. With the degrees of freedom being one less than the number of possible outcomes (n-1)

For part b, the degrees of freedom would depend on the number of categories minus one. If there are four categories (like the observed frequencies given), then the degrees of freedom for the chi-square test would be 3 (4-1). This is the same as in part a if part a also had four observed frequency categories for the number of male children.