Questions: In 1949, an 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, 374 indicated that they were total abstainers. In a recent survey, the same question was asked of 1100 adults and 352 indicated that they were total abstainers. Complete parts (a) and (b) below.
D. n1 hatp1(1-hatp1) geq 10 and n2 hatp2(1-hatp2) geq 10
E. The data come from a population that is normally distributed.
F. The sample size is more than 5% of the population size for each sample.
Identify the null and alternative hypotheses for this test. Let p1 represent the population proportion of 1949 adults who were total abstainers and p2 represent the population proportion of recent adults who were total abstainers.
Determine the null and alternative hypotheses.
H0: p1=p2
H1: p1 neq p2
Transcript text: In 1949, an 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, 374 indicated that they were total abstainers. In a recent survey, the same question was asked of 1100 adults and 352 indicated that they were total abstainers. Complete parts (a) and (b) below.
D. $n_{1} \hat{p}_{1}\left(1-\hat{p}_{1}\right) \geq 10$ and $n_{2} \hat{p}_{2}\left(1-\hat{p}_{2}\right) \geq 10$
E. The data come from a population that is normally distributed.
F. The sample size is more than $5 \%$ of the population size for each sample.
Identify the null and alternative hypotheses for this test. Let $p_{1}$ represent the population proportion of 1949 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.
\[
\begin{array}{l}
H_{0}: p_{1}=p_{2} \\
H_{1}: p_{1} \neq p_{2}
\end{array}
\]
Find the test statistic for this hypothesis test.
Solution
Solution Steps
Step 1: Sample Proportions
The sample proportions for the two surveys are calculated as follows:
\[
\hat{p}_1 = \frac{374}{1100} = 0.34
\]
\[
\hat{p}_2 = \frac{352}{1100} = 0.32
\]
Step 2: Hypotheses
The null and alternative hypotheses for this test are defined as:
\[
H_0: p_1 = p_2
\]
\[
H_1: p_1 \neq p_2
\]
Step 3: Pooled Sample Proportion
The pooled sample proportion is calculated using the formula:
\[
\hat{p}_{\text{pool}} = \frac{374 + 352}{1100 + 1100} = 0.33
\]
Step 4: Standard Error
The standard error (SE) is computed as:
\[
SE = \sqrt{\hat{p}_{\text{pool}} \left(1 - \hat{p}_{\text{pool}}\right) \left(\frac{1}{n_1} + \frac{1}{n_2}\right)} = \sqrt{0.33 \cdot 0.67 \left(\frac{1}{1100} + \frac{1}{1100}\right)} \approx 0.0200
\]
Step 5: Test Statistic
The test statistic \( z \) is calculated using the formula:
\[
z = \frac{\hat{p}_1 - \hat{p}_2}{SE} = \frac{0.34 - 0.32}{0.0200} \approx 0.9975
\]
Step 6: P-value
The p-value corresponding to the test statistic is found to be:
\[
P(Z > 0.9975) \approx 0.3184
\]