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).
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 square p_2 )
( h_1: p_1 square 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.
0.001 (round to three decimal places as needed.)
interpret the p - value.
if the population proportions are one would expect a sample difference proportion the one observed in about out of 1000 repetitions of this experiment. (round to the nearest integer as needed)

Explanation:

Step1: Calculate sample proportions

For 1947: $\hat{p}_{1}=\frac{407}{1100}\approx0.37$
For recent: $\hat{p}_{2}=\frac{240}{800} = 0.3$

Step2: Calculate pooled proportion

$\hat{p}=\frac{407 + 240}{1100+800}=\frac{647}{1900}\approx0.34$

Step3: Calculate test statistic

$z=\frac{\hat{p}_{1}-\hat{p}_{2}}{\sqrt{\hat{p}(1 - \hat{p})(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}=\frac{0.37 - 0.3}{\sqrt{0.34(1 - 0.34)(\frac{1}{1100}+\frac{1}{800})}}\approx3.18$

Step4: Calculate P - value

Since it's a two - tailed test, $P - value = 2\times(1-\Phi(|z|))$. Using standard normal table, for $z = 3.18$, $\Phi(3.18)\approx0.9993$, so $P - value=2\times(1 - 0.9993)=0.0014\approx0.001$

Step5: Interpret P - value

If the population proportions are equal, one would expect a sample difference proportion greater than or equal to the one observed in about 1 out of 1000 repetitions of this experiment.

Answer:

If the population proportions are equal, one would expect a sample difference proportion greater than or equal to the one observed in about 1 out of 1000 repetitions of this experiment.