3. In a survey, 1200 adult Americans were asked, "Are you a total abstainer from, or...

3. In a survey, 1200 adult Americans were asked, "Are you a total abstainer from, or do you on occasion consume, alcoholic beverages?" Of the 1200 adults surveyed, 470 indicated that they were total abstainers. In a recent survey, the same question was asked of 1200 adult Americans and 400 indicated that they were total abstainers. Has the proportion of adult Americans who totally abstain from alcohol changed? Use the α=0.1 level of significance.

Expert Solution

Two-Proportion Z test

The following information is provided:
(a) Sample 1 - The sample size is N1 = 1200, the number of favorable cases is X1 = 470 and the sample proportion is p^1=X1/N1=470/1200=0.3917
(b) Sample 2 - The sample size is N2 = 1200, the number of favorable cases is X2 = 400 and the sample proportion is p^2=X2/N2=400/1200=0.3333

and the significance level is α=0.1

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 Two-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=0.1, therefore the critical value for this Two-tailed test is Zc=1.6449. This can be found by either using excel or the Z distribution table.

(2b) Rejection Region
The rejection region for this Two-tailed test is |Z|>1.6449 i.e. Z>1.6449 or Z<-1.6449

(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|>2.9723)=0.003

(5) The Decision about the null hypothesis
(a) Using traditional method
Since it is observed that |Z|=2.9723 > Zc=1.6449, 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=0.003, and since p=0.003≤0.1, 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 different than p2, at the 0.1 significance level.

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orchestra answered 2 years ago

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