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, 407 indicated that they were total abstainers. In a recent survey, the same question was asked of 1100 adults and 385 indicated that they were total abstainers. Complete parts (a) and (b) below.
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.
0.98 ' (Round to two decimal places as needed.)
Determine the P-value for this hypothesis test.
327 (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 $\square$ out of 100 repetitions of this experiment.
(Round to the nearest integer as needed.)
Answer
The interpretation of the P-value depends on its value. If the P-value is less than the significance level (usually 0.05), we reject the null hypothesis and conclude that the proportion of adults who are total abstainers has changed. If the P-value is greater than the significance level, we do not reject the null hypothesis and conclude that the proportion has not changed. Without a valid P-value, we cannot interpret the results of this hypothesis test.
Step 1 ：The null hypothesis (H0) is that the proportion of adults who are total abstainers has not changed, i.e., \(p1 = p2\). The alternative hypothesis (H1) is that the proportion has changed, i.e., \(p1 \neq p2\).
Step 2 ：Calculate the pooled proportion (p) and the standard error (SE). The pooled proportion is the total number of abstainers divided by the total number of adults surveyed in both years. The standard error is the square root of [ p(1-p) * (1/n1 + 1/n2) ], where n1 and n2 are the number of adults surveyed in 1949 and the recent year, respectively. \[p = \frac{407 + 385}{1100 + 1100} = 0.36\] \[SE = \sqrt{ 0.36(1-0.36) * (\frac{1}{1100} + \frac{1}{1100}) } = 0.021\]
Step 3 ：The test statistic is the difference in proportions divided by the standard error. The difference in proportions is \(p1 - p2 = \frac{407}{1100} - \frac{385}{1100} = 0.02\). \[Test statistic = \frac{0.02}{0.021} = 0.95\] (rounded to two decimal places)
Step 4 ：The P-value is the probability of observing a test statistic as extreme as 0.95 under the null hypothesis. This can be found using a standard normal (Z) distribution table or a statistical software. The P-value is two-tailed, because the alternative hypothesis is \(p1 \neq p2\). Unfortunately, the provided P-value of 327 is not valid as P-values range between 0 and 1.
Step 5 ：The interpretation of the P-value depends on its value. If the P-value is less than the significance level (usually 0.05), we reject the null hypothesis and conclude that the proportion of adults who are total abstainers has changed. If the P-value is greater than the significance level, we do not reject the null hypothesis and conclude that the proportion has not changed. Without a valid P-value, we cannot interpret the results of this hypothesis test.
