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Question: Problem 2 [One sample test of a proportion]: To ev...

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Problem 2 [One sample test of a proportion]: To evaluate the policy of routine vaccination of infants for whopping cough, adverse reactions were monitored in 339 infants who received their first injection of vaccine. Reactions were noted in 69 of the infants.

(i) What is the distribution of the number of reactions in the 339 infants?

(ii) Construct a 95% confidence interval for the probability of an adverse reaction to the vaccine.

(iii) Test if the true proportion of adverse events in the population is larger then 10% at type I error level of 2.5%. Step 1: Parameter: _____________________________________________ H0: ____________ Ha: _______________ Significance level:

Step 2: Verify necessary data conditions and compute an appropriate test statistic:

Step 3: Assuming the H0 is true, define decision rule:

Step 4: Decision [Circle One]: Reject H0 Fail to Reject H0

Step 5: Conclusion:

Explanation / Answer

i) This is a binomial distribution

ii) For 95% CI, z-value = 1.96
p = 69/339 = 0.2035
lower bound = p - z*sqrt(p*(1-p)/n) = 0.2035 - 1.96*sqrt(0.2035 *(1-0.2035)/339) = 0.1606

upper bound = p + z*sqrt(p*(1-p)/n) = 0.2035 + 1.96*sqrt(0.2035 *(1-0.2035)/339) = 0.2464

iii)
Below are the null and alternate hypothesis
H0: p <= 0.1
H1: p > 0.1

Test statistics,
z = (0.2035 - 0.1)/(sqrt(0.1 *(1-0.1)/339)
z = 6.3521

p-value = 0.000001 (almost zero)

As p-value is less than significance level of 0.025, we reject null hypotehsis.

This means there are enough evidence that the proportion of adverse events in the population is larger then 10%

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