5. There is some indication in medical literature that doctors may have become more aggressive in
inducing labor or doing preterm cesarean sections when a woman is carrying twins. Records at a large hospital show that of the 43 sets of twins born in 1990, 20 were delivered before the 37th week of pregnancy. In 2000, 26 of 48 sets of twins were born preterm. Does this indicate an increase in the incidence of early births of twins? Test an appropriate hypothesis and state your conclusion. Calculate and interpret a 90% confidence interval.
To determine if there is an increase in the incidence of early births of twins, we can perform a hypothesis test and calculate a confidence interval.
Hypothesis Test:
The null hypothesis (H0) states that there is no increase in the incidence of early births of twins. The alternative hypothesis (Ha) states that there is an increase in the incidence of early births of twins.
To test this hypothesis, we can use a chi-square test for independence. This test compares the observed frequencies (1990 and 2000) to the expected frequencies if there is no relationship between the year of birth and the incidence of early births of twins.
1. First, we calculate the expected frequencies for each year. Since there is no indication of a specific pattern, we assume that the proportions of early births in the two years are the same.
Expected frequency = (Total number of twins born in a year * Proportion of early births in the entire dataset)
2. Next, we calculate the chi-square test statistic using the formula:
χ2 = ∑ [(Observed frequency - Expected frequency)² / Expected frequency]
3. With the calculated chi-square statistic, we can compare it to the critical value from the chi-square distribution at a significance level of 0.05 (or any other desired level). If the calculated chi-square value is greater than the critical value, we reject the null hypothesis in favor of the alternative hypothesis.
Confidence Interval:
To calculate a 90% confidence interval for the proportion of early births in each year, we can use the formula for proportions:
Confidence interval = p̂ ± z * sqrt(p̂(1-p̂) / n)
Where p̂ is the proportion of early births in each year, z is the z-score corresponding to the desired confidence level (90% corresponds to z = 1.645 for a two-tailed test), and n is the total number of twins born in each year.
Interpretation:
If the null hypothesis is rejected, we can conclude that there is evidence of an increase in the incidence of early births of twins. Additionally, by comparing the confidence intervals of each year, we can determine the range within which the true proportion of early births lies with 90% confidence. If the confidence intervals do not overlap, it provides further evidence for an increase in early births.