Question

(Aside from providing the results of the simulation or statistical tests, be sure to answer the questions – in bold.)

A new vaccine was recently tested to see if it could prevent the painful and recurrent ear infections that many infants suffer from. The Lancet, a medical journal, reported a study in which babies about a year old were randomly divided into two groups. One group received vaccinations; the other did not. During the following year, only 333 of 2455 vaccinated children had ear infections, compared to 499 of 2452 unvaccinated children in the control group. Is there evidence that the vaccine is effective? Use simulation. Include your null and alternative hypotheses.

Answer 1

Let p1 be the proportion of children who received the vaccine but had the infection. p1 = 333/2455 = 0.14

Let p2 be the proportion of children who did not received the vaccine but had the infection. p2 = 499/2452 = 0.20

We have to test if p1-p2 significantly differs or p1<p2

Since the significance level is not given, we will take it 0.05

pooled proportion p = 333+499/ 2455+2452= 0.17

Standard error SE(p1-p2) = = 0.01

Now z = (p1-p2) / SE(p1-p2) = (0.14 - 0.20) / 0.01 = -6

Now z being very less, it is very unlikely for z to be less than 2.5. Also its p value is close to zero.

Clearly p < 0.05, We have strong evidence against the null hypothesis. Also we estimated p1-p2 < 0 which means p1 < p2 .

Hence, there is an evidence that vaccine is effective.