Question

In: Statistics and Probability

4. Among 198 smokers who underwent a “sustained care” program, 51 were no longer smoking after...

4. Among 198 smokers who underwent a “sustained care” program, 51 were no longer smoking after 6 months. Among 199 smokers who underwent an “standard care” program 30 were no longer smoking after six months. Use a 0.01 significance level to test the claim the rate of success for smoking cessation is greater with the sustained care program.

Solutions

Expert Solution

Two-Proportion Z test

The following information is provided:
(a) Sample 1 - The sample size is N1 = 198, the number of favorable cases is X1 = 51 and the sample proportion is p^1​=X1/N1​=51/198​=0.2576
(b) Sample 2 - The sample size is N2 = 199, the number of favorable cases is X2 = 30 and the sample proportion is p^2​=X2/N2​=30/199​=0.1508

and the significance level is α=0.01

Pooled Proportion
The value of the pooled proportion is computed as


(1) Null and Alternative Hypotheses
The following null and alternative hypotheses need to be tested:
Ho: p1 = p2
Ha: p1 > p2
This corresponds to a Right-tailed test, for which a z-test for two population proportions needs to be conducted.

(2a) Critical Value
Based on the information provided, the significance level is α=0.01, therefore the critical value for this Right-tailed test is Zc​=2.3263. This can be found by either using excel or the Z distribution table.

(2b) Rejection Region
The rejection region for this Right-tailed test is Z>2.3263

(3) Test Statistics
The z-statistic is computed as follows:

(4) The p-value
The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true. In this case,
the p-value is p =P(Z>2.6408)=0.0041

(5) The Decision about the null hypothesis
(a) Using traditional method
Since it is observed that Z=2.6408 > Zc​=2.3263, it is then concluded that the null hypothesis is rejected.

(b) Using p-value method
Using the P-value approach: The p-value is p=0.0041, and since p=0.0041≤0.01, it is concluded that the null hypothesis is rejected.

(6) Conclusion
It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population proportion p1 is greater than p2, at the 0.01 significance level.

Let me know in the comments if anything is not clear. I will reply ASAP! Please do upvote if satisfied!

orchestra answered 2 years ago
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