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)
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
3.18 (round to two decimal places as needed.)
determine the p - value for this hypothesis test
(round to three decimal places as needed.)

Explanation:

Step1: Calculate sample proportions

For 1947: $\hat{p}_{1}=\frac{407}{1100}\approx0.37$
For recent: $\hat{p}_{2}=\frac{240}{800} = 0.3$
Pooled proportion $\hat{p}=\frac{407 + 240}{1100+800}=\frac{647}{1900}\approx0.3405$

Step2: Calculate the 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:

$$
LATEXBLOCK0
$$

Step3: Calculate the P - value

Since this is a two - tailed test ($H_{1}:p_{1}
eq p_{2}$), the P - value is $2\times(1 - P(Z<|z|))$
Using a standard normal table or calculator, $P(Z < 3.18)\approx0.9993$
So, $P - value=2\times(1 - 0.9993)=2\times0.0007 = 0.0014\approx0.001$

Answer:

The P - value is $0.001$