Question

in october 1947, the gallup organization surveyed 1100 adults and asked, are you a total abstainer from, or do you on occasion consume, alcoholic beverages? of the 1100 adults surveyed, 407 indicated that they were total abstainers. in a portion of the results of a recent survey, the same question was asked of 800 adults and 240 indicated that they were total abstainers. complete parts (a) and (b).
a. the sample size is more than 5% of the population size for each sample
b. ( n_1hat{p}_1(1 - hat{p}_1)geq10 ) and ( n_2hat{p}_2(1 - hat{p}_2)geq10 )
c. the samples are independent.
d. the samples are dependent.
e. the sample size is less than 5% of the population size for each sample.
f. the data come from a population that is normally distributed.
identify the null and alternative hypotheses for this test. let ( p_1 ) represent the population proportion of 1947 adults who were total abstainers and ( p_2 ) represent the population proportion of recent adults who were total abstainers.
determine the null and alternative hypotheses.
( h_0:p_1 = p_2 )
( h_1:p_1
eq p_2 )
find the test statistic for this hypothesis test.
(round to two decimal places as needed)

Explanation:

Step1: Calculate sample proportions

For 1947 survey: \(n_1 = 1100\), \(x_1=407\), \(\hat{p}_1=\frac{x_1}{n_1}=\frac{407}{1100}\approx0.37\)
For recent survey: \(n_2 = 800\), \(x_2 = 240\), \(\hat{p}_2=\frac{x_2}{n_2}=\frac{240}{800}=0.3\)

Step2: Calculate pooled proportion

\(\hat{p}=\frac{x_1 + x_2}{n_1 + n_2}=\frac{407+240}{1100 + 800}=\frac{647}{1900}\approx0.3405\)

Step3: Calculate test - statistic

The formula for the test - statistic \(z\) in a two - proportion z - test is \(z=\frac{\hat{p}_1-\hat{p}_2}{\sqrt{\hat{p}(1 - \hat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}\)
Substitute the values:

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Answer:

\(3.18\)