Question

a) State the null and alternative hypotheses, b) Verify that the requirements that allow use of the normal model to test the hypothesis are satisfied, c) Show work for calculating the test statistic, d) Compare the critical value with the test statistic, and e) State the conclusion.

5% of children under 18 years of age have asthma. In a random sample of 250 children under 18 years of age, 8% say they have asthma. Is there enough evidence to conclude that the proportion of children under 18 years of age is not 5%? Use an α = 0.05 level of significance.

Answer 1

The following information is provided: The sample size is N = 250 , the number of favorable cases is X = 20 , 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:

Ho: p = 0.05

The proportion is equal to 5% or 0.05

Ha: p≠​0.05

The proportion is not equal to 5% or 0.05

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 z_c = 1.96(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that |z| = 2.176 > zc​=1.96, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0.0295 , and since p = 0.0295 <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.