A vaccine to prevent a severe virus was glven to children within the first year

Question

A vaccine to prevent a severe virus was glven to children within the first year of life as part of a drug study. The study reported that of the 3233 children randomly assigned the vaccine, 45 got the virus. Of the 1663 children randomly assigned the placebo, 43 got the virus. a. Find the sample percentage of children who caught the virus in each group. Is the sample percentage lower for the vaccine group, as investigators hoped? b. Determine whether the vaccine is effective in reducing the chance of catching the virus, using a significance level of 0.10. The first few steps of the hypothesis-testing procedure are given. Complete the procedure Click the icon to view the first few steps of the hypothesis-testing procedure a. Find the sample percentage of children who caught the vius in the vaccine group. % (Round to two decimal places as needed.) Find the sample percentage of children who caught the virus in the placebo group. % (Round to two decimal places as needed.) Is the sample percentage lower for the vaccine group, as investigators hoped? O No O Yes b. Find the test statistic for this test. (Round to two decimal places as needed.) Find the p-value for this test. p-value(Round to three decimal places as needed.) What is the conclusion for this test? O A. Do not reject Hg. There is sufficient evidence to conclude that the vaccine is effective in reducing the chance of catching the virus at the significance level of 0.10. O B. Reject Ho- There is not sufficient evidence to conclude that the vaccine is effective in reducing the chance of catching the virus at the significance level of 0.10 O C. Reject Ho. There is sufficient evidence to condlude that the vaccine is effective in reducing the chance of catching the virus at the significance level of 0.10. O D Do not reject Ho . There is not sufficient evidence to conclude at the vaccine is efective in reducing the chance o catching the virus atthe significance level of0.1

Explanation / Answer

1) percentage of vaccine group= 45*100/3233= 1.39%

2) percentage of placebo group= 43*100/1663= 2.59%

3) Yes vaccine group percentage is lower

4) The following null and alternative hypotheses need to be tested:

Ho:p1=p2

Ha:p1<p2

This corresponds to a left-tailed test, for which a z-test for two population proportions needs to be conducted.

(2) Rejection Region

Based on the information provided, the significance level is =0.1, and the critical value for a left-tailed test is zc=1.28.

The rejection region for this left-tailed test is R={z:z<1.28}

(3) Test Statistics

The z-statistic is computed as follows:

z =p¯(1p¯)(1/n1+1/n2)p^1p^2=0.018(10.018)(1/3233+1/1663)0.01390.0259=2.978

(4) Decision about the null hypothesis

Since it is observed that z=2.978<zc=1.28, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0015= 0.001 , and since p=0.001<0.1, 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 population proportion p1 is less than p2, at the 0.1 significance level.

option C

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