In 2006, 10.6% of all live births in the United States were to mothers under 20 years of age.

Question

In 2006, 10.6% of all live births in the United States were to mothers under 20 years of age. A sociologist claims that births to mothers under 20 years of age is decreasing. The sociologist conducts a simple random sample of 46 births and finds that 4 of them were to mothers under 20 years of age. Test the sociologist's claim at the a=0.05 level of significance.

Answer 1

First, establish your null and alternative hypotheses. The null hypothesis is based on that first statement. The alternative hypothesis is based on the second sentence: if births are decreasing, then the proportion is less than 10.6%. Also, if necessary, check to make sure the assumptions for hypothesis testing are met: in this case, only having 4 of 46 be under 20 years old might cause our test to be inaccurate because we want to have at least 5 for the normal distribution assumption.

H₀: p = 0.106

H₁: p < 0.106

Second, you want to find your test statistic. Your sample had 4 of 46 births by mothers under 20 years of age, so phat (the p with the ^ on top; I don't have that symbol on wyzant) is 4/46 or ~0.087. Then, remember that your standard error of √[p•(1-p)/n] is based on p = 0.106 because we assume the null hypothesis to be true.

z = (0.087 - 0.106)/√(0.106•0.894/46) = -0.4186

You can then either (1) compare this test statistic to your critical value or (2) find the p-value for this test statistic and compare with α.

(1) crit.value = -1.645; since the test statistic is greater than our critical value, we are not in the rejection region. Therefore we fail to reject the null hypothesis.

(2) p-value = 0.3378 (calculator) or 0.3372 (table); since this p-value is more than α, we fail to reject the null hypothesis.

In either case we've failed to reject the null hypothesis, but now we need to translate this back into something the sociologist would easily understand. Here's an example of that phrasing: "There is not enough evidence to support the claim that the proportion of teen births (or births by mothers under 20 years of age) has reduced since 2006."

NOTE: if you are working with a TI-84 calculator, you can do all this work for the second step in the 1-PropZtest (under STAT→TESTS→5:1-PropZtest). In this case, p: 0.106, x: 4, n: 46, and prop: <.