Question

2. A new flu vaccine was developed to protect elderly people against viral infection during the winter season. In 120 patients, 70 of the patients who were vaccinated. Of the 70 patients, 56 of the patients had no flu symptoms. 50 of the patients who were not vaccinated, 28 patients have no flu symptoms.

a) Determine the confidence interval (CI) at 95% (z*=1.96) level for the probability that a patient had no flu symptoms when a new vaccine is used during the winter season. What is the sample proportion (p̂ ) of patients with no symptoms when vaccine is used?

b) Carry out a statistical test at the α=0.05 level for whether the probability of vaccinated patients with no flu symptoms is higher than the proportion of people who are not vaccinated. Determine the null and alternative hypothesis. What is the pooled sample proportion (p̂ )?

Answer 1

a)

We need to construct the 95% confidence interval for the population proportion.

We have been provided with the following information about the number of favorable cases:

Favorable Cases X=	56
Sample Size N =	70

The sample proportion is computed as follows, based on the sample size N = 70 and the number of favorable cases X = 56:

The critical value for \alpha = 0.05 is

The corresponding confidence interval is computed as shown below:

Therefore, based on the data provided, the 95% confidence interval for the population proportion 0.706<p<0.894, which indicates that we are 95% confident that the true population proportion pp is contained by the interval (0.706, 0.894)

b)

For sample 1, we have that the sample size is N1​=70, the number of favorable cases is X_1 = 56, so then the sample proportion is

For sample 2, we have that the sample size is N2​=50, the number of favorable cases is X_2 = 28, so then the sample proportion is

The value of the pooled proportion is computed as

Also, the given 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 right-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 α=0.05, and the critical value for a right-tailed test is zc​=1.64.

The rejection region for this right-tailed test is

R={z:z>1.64}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that z=2.828>zc​=1.64,

it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value is p = 0.0023, and since p=0.0023<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 population proportion p1​ is greater than p2​, at the 0.05 significance level.

please like