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) determine the sample proportion for each sample
the proportions of the adults who took the 1947 survey and the recent survey who were total abstainers are 0.370 and 0.300, respectively
(round to three decimal places as needed.)
(b) has the proportion of adults who totally abstain from alcohol changed? use the \\( \alpha = 0.05 \\) level of significance
first verify the model requirements. select all that apply
\\( \square \\) a. the sample size is more than 5% of the population size for each sample
\\( \square \\) b. \\( n _ { 1 } \hat { p } _ { 1 } \left( 1 - \hat { p } _ { 1 } \
ight) \geq 10 \\) and \\( n _ { 2 } \hat { p } _ { 2 } \left( 1 - \hat { p } _ { 2 } \
ight) \geq 10 \\)
\\( \square \\) c. the samples are independent
\\( \square \\) d. the samples are dependent
\\( \square \\) e. the sample size is less than 5% of the population size for each sample
\\( \square \\) f. the data come from a population that is normally distributed

Explanation:

Step1: Check sample independence

Since the two samples (1947 survey and recent survey) are of different groups of adults, they are independent. So, option C is correct.

Step2: Check sample - size condition

For the 1947 sample: \(n_1 = 1100\), \(\hat{p}_1=\frac{407}{1100}\approx0.37\). Then \(n_1\hat{p}_1(1 - \hat{p}_1)=1100\times0.37\times(1 - 0.37)=1100\times0.37\times0.63 = 1100\times0.2331=256.41\geq10\)
For the recent sample: \(n_2 = 800\), \(\hat{p}_2=\frac{240}{800}=0.3\). Then \(n_2\hat{p}_2(1 - \hat{p}_2)=800\times0.3\times(1 - 0.3)=800\times0.3\times0.7 = 168\geq10\). So, option B is correct.

Step3: Check sample - to - population ratio

In most cases (unless the population of adults is known to be very small, which is not indicated here), a sample size less than \(5\%\) of the population size is a reasonable assumption. We assume that the samples are less than \(5\%\) of the population of all adults. So, option E is correct.

Answer:

B. \(n_1\hat{p}_1(1-\hat{p}_1)\geq10\) and \(n_2\hat{p}_2(1 - \hat{p}_2)\geq10\), C. The samples are independent, E. The sample size is less than \(5\%\) of the population size for each sample.