Question

3. Individuals who enrolled in Affordable Care Act Heath Insurance Marketplace plans represented about 4.1% of all Americans with insurance in 2015. The Department of Health & Human Service wanted to determine whether the rate of enrollment in these marketplace plans in 2016 grew as compared to the 2015 rate of 4.1%. It selected a random sample of 500 insured Americans in 2016 and found that 29 of them enrolled in ACA marketplace plans.

a) Could the Department of Health & Human Service be reasonably sure that the enrollment rate increased? Calculate a p-value for the observed proportion and interpret it.

b) If the significance level is set at .01, is this change in enrollment rate statistically significant? State your hypotheses and conduct a test.

Answer 1

Solution: For the given question we construct our null and alternative hypotheses as:

H0: p = 0.041 vs Ha: p >0.041 [Since we are to test whether the enrollment rate has increased or not. So, it is a upper tailed test]

[Here p= unknown rate of enrollment in the population of marketplace]

The test statistic for this is Z = (p_hat - p0)/sqrt(p0*(1-p0)/n) ~ N(0,1) under H0.

here p0 = hypothetical value of the unknown population proportion, n= sample size, p_hat = sample proportion.

Here p0= 0.041, p_hat= 29/500, n = 500

We reject H0 if--

a) p-value for the observed test statistic is less than the level of significance.where p-value is the probability of observing a value atleast as extreme as the observed test statistic assuming the null hypothesis is true.

OR

b) The observed test statistic, Z(observed) > tau(alpha) where tau(alpha) is the upper alpha point of a standard normal distribution.

Here Z(observed) = 1.917047, p-value = 0.027616 and tau(alpha) = 2.326348 [Here alpha = 0.01]
.

So, we see that p-value > alpha and Z(observed) < tau(alpha), hence

a) We fail to reject H0 as p-value > alpha and say that  the Department of Health & Human Service cannot reasonably be sure that the enrollment rate has increased. The interpretation of p-value is, if the null hypothesis is true, there is a 2.76% chance that the enrollment rate is greater than 0.041 due to random noise.

b) Also,  If the significance level is set at .01, we fail to reject H0 as Z(observed) < tau(alpha) and conclude at a 5% level of significance on the basis of the given sample measures that there is enough evidence to say that change in enrollment rate is not statistically significant.

[The values are obtained using R-software. Code and output are attached below for verification.]