Question

A community wishes to investigate the proportion of their population that has been vaccinated against measles. In particular, they are interested in whether this proportion is less than 90%, the minimum required to achieve herd immunity. They randomly select 300 individuals from the community and question them regarding their vaccination status. Of these individuals, 259 had been vaccinated. They choose a significance level of 5%.

a. What are the null and alternative hypotheses?

b. (1 mark) What is the value of the test statistic?

c. (1 mark) What is the p-value?

d. State your conclusions in the language of the problem.

e. Give a 95% confidence interval for the probability of being vaccinated against measles in this community.

Answer 1

(a):

H0: Null Hypothesis: P 0.90

HA: Alternative Hypothesis: P<0.90

= Sample Proportion = 259/300 = 0.8633

(b)

Test Statistic is given by:

(c)

Table of Area Under Standard Normal Curve gives area = 0.4830

So,

p - value = 0.5 - 0.4830 = 0.0170

(d)

Since p - value = 0.0170 is less than = 0.05, the difference is significant. Reject null hypothesis.

Conclusion:

The data support the claim that the proportion of their population that has been vaccinated against meas is less than 90%, the minimum required

(e)

= 0.05

From Table, critical values of Z = 1.96

Confidence Interval:

0.8633 (1.96 X 0.01732051)

= 0.8633 0.0339

= (0.8294    ,0.8972)

So,

Confidence Interval:

0.8294    < P < 0.8972