Question

TDaP is a booster shot that prevents Diphtheria, Tetanus, and Pertussis in adults and adolescents. It should be administered every 8 years for it to remain effective. A random sample of 550 people living in a town that experienced a pertussis outbreak this year were divided into two groups. Group 1 was made up of 145 individuals who had not had the TDaP booster in the past 8 years, and Group 2 consisted of 355 individuals who had. In Group 1, 18 individuals caught pertussis during the outbreak, and in Group 2, 13 individuals caught pertussis. Is there evidence to suggest that the proportion of individuals who caught pertussis and were not up to date on their booster shot is higher than those that were? Test at the 0.05 level of significance. Enter the p-value - round to 5 decimal places.

Answer 1

Answer:

Yes. There is enough evidence to support the claim that the proportion of individuals who caught pertussis and were not up to date on their booster shot is higher than those that were.

P-value = 0.00013.

Step-by-step explanation:

This is a hypothesis test for the difference between proportions.

The claim is that the proportion of individuals who caught pertussis and were not up to date on their booster shot is higher than those that were.

Then, the null and alternative hypothesis are:

$H_0: \pi_1-\pi_2=0\\\\H_a:\pi_1-\pi_2> 0$

The significance level is 0.05.

The sample 1 (Group 1), of size n1=145 has a proportion of p1=0.124.

$p_1=X_1/n_1=18/145=0.124$

The sample 2 (Group 2), of size n2=355 has a proportion of p2=0.037.

$p_2=X_2/n_2=13/355=0.037$

The difference between proportions is (p1-p2)=0.088.

$p_d=p_1-p_2=0.124-0.037=0.088$

The pooled proportion, needed to calculate the standard error, is:

$p=\dfrac{X_1+X_2}{n_1+n_2}=\dfrac{18+13.135}{145+355}=\dfrac{31}{500}=0.062$

The estimated standard error of the difference between means is computed using the formula:

$s_{p1-p2}=\sqrt{\dfrac{p(1-p)}{n_1}+\dfrac{p(1-p)}{n_2}}=\sqrt{\dfrac{0.062*0.938}{145}+\dfrac{0.062*0.938}{355}}\\\\\\s_{p1-p2}=\sqrt{0+0}=\sqrt{0.001}=0.024$

Then, we can calculate the z-statistic as:

$z=\dfrac{p_d-(\pi_1-\pi_2)}{s_{p1-p2}}=\dfrac{0.088-0}{0.024}=\dfrac{0.088}{0.024}=3.68$

This test is a right-tailed test, so the P-value for this test is calculated as (using a z-table):

$\text{P-value}=P(z>3.68)=0.00013$

As the P-value (0.00013) is smaller than the significance level (0.05), the effect is significant.

The null hypothesis is rejected.

There is enough evidence to support the claim that the proportion of individuals who caught pertussis and were not up to date on their booster shot is higher than those that were.