Question

The drug Prevnar is a vaccine meant to prevent meningitis. It is typically administered to infants. In clinical trials, the vaccine was administered to 710 randomly sampled infants between 12 and 15 months of age. Of the 710 infants, 121 experienced a loss of appetite. Is this significant evidence to conclude that the proportion of infants who receive Prevnar and experience a loss of appetite is different than 0.135 which is the proportion of children who experience a loss of appetite with competing medications? Use an α of 0.05.

Calculate a 95% confidence interval for the proportion of infants on Prevnar who experience a loss of appetite. Interpret the confidence interval in context of the problem.

Does your confidence interval agree with the results of the hypothesis test? Explain.

Answer 1

Hypothesis test

The following information is provided: The sample size is N=710, the number of favorable cases is X=121, and the sample proportion is , and the significance level is α=0.05

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

This corresponds to a two-tailed test, for which a z-test for one population proportion needs to be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a two-tailed test is Zc​=1.96.

The rejection region for this two-tailed test is R={z:∣z∣>1.96}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that ∣z∣=2.762>zc​=1.96, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0057, and since p=0.0057<0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population proportion p is different than p0​, at the α=0.05 significance level.

Graphically

Confidence Interval

Therefore, based on the data provided, the 95% confidence interval for the population proportion is 0.143<p<0.198, which indicates that we are 95% confident that the true population proportion p of infants who receive Prevnar and experience a loss of appetite is contained by the interval (0.143, 0.198)

Yes, confidence interval agrees with the hypothesis test since 0.135 is not contained in the confidence interval,

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