Question

In the year 2002, approximately 22.5% of the U.S. population were currently smokers. We wonder if in Tennessee the proportion of current smokers was also 0.225 or if it was higher than that. Assume that in that year, a survey was conducted in Tennessee, and that a simple random sample of 2466 individuals were selected. Out of the 2466, 611 were classified as ’current smokers’.

We wish to conduct the test of hypothesis.

(a) State the appropriate null and alternative hypothesis.

(b) Calculate the test statistic using the Central Limit Theorem.

(c) Using the previous result, find the p-value.

(d) State your conclusion.

Answer 1

The sample size is N = 2466, the number of favorable cases is X = 611, and the sample proportion is , and as the significance level is not given. Therefore, we are taking as α = 0.05

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho : p = 0.225

Ha : p > 0.225

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

(2) Rejection Region

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 = 2.708 > zc ​= 1.64, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0.0034, and since p = 0.0034 < 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 greater than p0​, at the α=0.05 significance level.

Therefore, there is enough evidence to claim that the proportion of current smokers was higher than 0.225.