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.

What is the p-value?

Verify np(1-p) =

Answer 1

Hi Emma,

This is a one-proportion z-test. Formula for those is:

z = (p^ - p0) / sqrt(p0(1-p0)/n)

Before we do that,

n= 75

p= 0.106

np = 75*.106 7.95

n(1-p) = 75 * .894 = 67.05

It depends on your criteria for inference whether or not we can proceed--some books say np>5, some say np>10. We'll proceed as if it's the lower value.

p^ = sample proportion

p0 = estimated population proportion

n= sample size

Now, for the null and alternative hypotheses, null means no difference between 2006 and now, so:

H0: p = 0.106

The sociologist thinks current birth rate is lower, so:

Ha: p < 0.106

For your problem:

p^ = 4/46 = 0.087

p0 = 0.106

n = 46

z = (p^ - p0) / sqrt(p0(1-p0)/n)

z= (0.087 - 0.106) / sqrt((0.106*0.894)/46)

z = -0.42

Remember we are interested in the less than alternative, so, from z-table:

P(Z < -0.42) = 0.3372

Recall that alpha ,aka significance level, is 0.05.

0.3372 > 0.05

We fail to reject H0 and conclude that current population proportion of births to mothers under 20 does not differ significantly from 0.106. I hope this helps.