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) has the proportion of adults who totally abstain from alcohol changed? use the α = 0.05 level of significance
first verify the model requirements. select all that apply
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) dropdown ( p_2 )
( h_1:p_1) dropdown ( p_2 )

Explanation:

Step1: Calculate sample proportions

For the 1947 sample: $\hat{p}_1=\frac{407}{1100}\approx0.37$
For the recent sample: $\hat{p}_2=\frac{240}{800} = 0.3$

Step2: Calculate the pooled proportion

$\hat{p}=\frac{407 + 240}{1100+800}=\frac{647}{1900}\approx0.3405$

Step3: Calculate the test statistic

$z=\frac{\hat{p}_1-\hat{p}_2}{\sqrt{\hat{p}(1 - \hat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}$
$=\frac{0.37 - 0.3}{\sqrt{0.3405\times(1 - 0.3405)\times(\frac{1}{1100}+\frac{1}{800})}}$
$=\frac{0.07}{\sqrt{0.3405\times0.6595\times(\frac{800 + 1100}{1100\times800})}}$
$=\frac{0.07}{\sqrt{0.2247\times\frac{1900}{880000}}}$
$=\frac{0.07}{\sqrt{\frac{426.93}{880000}}}$
$=\frac{0.07}{\sqrt{0.00048515}}$
$=\frac{0.07}{0.022}$
$\approx3.18$

Step4: Determine the critical value

For $\alpha = 0.05$, the critical values are $z_{\alpha/2}=\pm1.96$ (two - tailed test)

Step5: Make a decision

Since $|z| = 3.18>1.96$, we reject the null hypothesis.

For the null and alternative hypotheses:
The null hypothesis $H_0:p_1 = p_2$ (the proportion of total abstainers has not changed)
The alternative hypothesis $H_1:p_1
eq p_2$ (the proportion of total abstainers has changed)

Answer:

$H_0:p_1 = p_2$; $H_1:p_1
eq p_2$