Are two proportions significantly different?

Question

everyone! The questions below I need help from are from 3-6. Here it is: Risk Factors for Low Birth Weight:

Rates of infant mortality, birth defect, and premature labor are high for babies with low birth weight. There are many factors that may contribute to low birth weight. In this activity, we use data from a random sample of women who participated in a study in 1986 at the Baystate Medical Center in Springfield, MA. (Source: Hosmer and Lemeshow (2000), Applied Logistic Regression: Second Edition.)

For the 30 women in the study with a history of premature labor, a proportion of 18/30 = 0.60 (60%) had babies with low birth weight. For the remaining 159 women, a proportion of 41/159 = 0.26 (26%) had babies with low birth weight. We now investigate the following research question: do the data provide evidence that the proportion of babies born with low birth weight is higher for women with a history of premature labor? This question is answered with a hypothesis test. To conduct the test we use a 1% level of significance.

Question 3: We will test the claim that the proportion of women with low birth weight babies is higher among women with a history of premature labor. What are the null and alternative hypotheses?

Question 4: Are the criteria for approximate normality satisfied?

Question 5: State the test statistic and P-value. Interpret these values.

Question 6: Give a conclusion in context, and discuss whether a causal conclusion is appropriate.

Answer 1

The null hypothesis is that the proportion of low birth weight babies is the same in the two groups. The alternative is that the proportion is higher among mothers with a history of premature births.

There are several methods of finding a Z-statistic that can be used to do the test. There is no way for me to know which one(s) are advocated in your text or class notes. Here is output (including a P-value) from Minitab statistical software (slightly abridged to omit irrelevant results).

Test for Two Proportions 

Sample   X    N  Sample p
1       18   30  0.600000
2       41  159  0.257862

Difference = p (1) - p (2)
Estimate for difference:  0.342138
Test for difference = 0 (vs > 0):  Z = 3.57  P-Value = 0.000

The very small P-value (< 0.0005) gives strong evidence for rejecting the null hypothesis. It is unlikely that any of the slight variations on this method would fail to reject. (You should use whatever formula for $Z$ is recommended in your text or notes. I suppose you will get $Z$ and P-values about the same as shown in the Minitab printout.)

The sample sizes seem large enough to assume that the $Z$ statistic is approximately normally distributed (especially so, because the group with the smaller sample size has a rate of premature births that is not very far from $1/2$).

This is an observational study. It is unclear what precautions may have been taken to ensure that the two groups of women were substantially alike in attributes other than their histories of premature births. (There was no randomization of subjects into groups; indeed, it is difficult to see how a randomized study on this topic could have been organized.) Therefore, it would be overreaching to say that a history of premature labor is a cause of low birth weights at subsequent births.

However, the results of this study may be useful in practice. It might be reasonable for pediatricians at this hospital to monitor women with a history of premature labor and take reasonable extra measures to try to prevent low birth weight babies and to be especially ready to respond if low birth weights occur.