Question

8. (1+1+2+2+1+1=8 marks ) The medical researcher examined 80 families with four children. He has found that that 6 families did not have girls, 18 families had one girl, 33 families had two girls, 16 families had 3 girls, and 7 families had 4 girls. The results obtained by the researcher are given in the table below. Test the hypothesis that probability distribution of number of girls in the family with four child follows binomial distribution with n=4 and parameter p estimated from the sample data. Use α=0.05. c. Calculate the required probabilities and expected frequencies using appropriate formulas. In the expressions for expected frequencies leave two digits after decimal point. (2 marks). Place calculate frequencies into the last row of the table on the top of the page 10. Combine the cells with expected frequencies smaller than 5 (if required). d. Calculate observed value of test statistic. ( 2 mark) χ obs. 2 ​ = d. Determine critical value of the test. (1 mark) χ or.. 2 ​ = e. Determine whether H 0 ​ is rejected and make the conclusions. (1 mark)

Answer 1

This mathematics problem involves conducting a goodness-of-fit test to see if the number of girls in families with four children follows a binomial distribution, using the χ2 test statistic and an alpha level of 0.05.

Step-by-step explanation:

The medical researcher's data involves testing a hypothesis about the number of girls in families with four children. Given the data, we analyze whether it follows a binomial distribution with n=4 and an estimated parameter p from the sample. To conduct this goodness-of-fit test, we'll calculate the probabilities using the binomial formula along with the expected frequencies for each category.

We then combine categories with expected frequencies smaller than 5, if necessary. Next, we assess the observed χ2 statistic against the critical value determined from the χ2 distribution table at an alpha level of 0.05. The decision to reject the null hypothesis H0 depends on whether the computed χ2 statistic exceeds the critical value.

If χ2 is greater than the critical value, we reject H0, indicating the number of girls does not follow the presumed binomial distribution.