Question

A newborn baby is considered to be premature if its gestational age is less than 37 weeks. A doctor wants to determine whether more than 10% of the newborn babies in some city last year were premature. In order to do so, he collects a simple random sample of 100 babies born last year in this city. (a) State the null hypothesis and the test statistic for this problem. (b) State the decision rule at significance level 5%. What is a false positive in this context? (c) What is the probability of rejecting the null hypothesis if the population proportion of premature babies in the city last year was 15%? (d) After collecting the sample, the doctor found that the gestational age was less than 34 weeks for 5 babies in the sample, between 34 and 36 weeks for 10 babies, and equal to or exceeding 37 weeks for 85 of them. What is the the p-value? (e) Based on the data in (d), is the null hypothesis rejected when α = 1%? α = 5%? α = 20%? (f) Based on the data in (d), obtain a 95% confidence interval for the population proportion of premature babies in the city last year.

Answer 1

(a)

Here we have

(b)

Test is right tailed so critical value for significance level 5% is 1.645. The rejection region:

If z >= 1.645, reject H0

The false positive is the probability of rejecting the true null hypothesis.

(c)

The probability of rejecting the null hypothesis if the population proportion of premature babies in the city last year was 15%, that is power of the test, is 0.5120.

(d)

The number of  newborn baby with gestational age is less than 37 weeks in the sample is 5+10 = 15.

(e)

For α = 1%:

Since p-value is greater than 0.01 so we fail to reject the null hypothesis.

For α = 5%:

Since p-value is not greater than 0.05 so we reject the null hypothesis.

For α = 20%:

Since p-value is not greater than 0.20 so we reject the null hypothesis.

(f)