To address the question of whether the proportion of adults satisfied with the quality of the environment has declined from 2016 to 2018, we can perform a hypothesis test for the difference in proportions. Given the data, we can follow these steps:
1. Define the Hypotheses:
- Null Hypothesis [tex]\(H_0: P_1 = P_2 \)[/tex] (The proportion of satisfied adults has not changed from 2016 to 2018).
- Alternative Hypothesis [tex]\(H_A: P_1 > P_2 \)[/tex] (The proportion of satisfied adults has declined from 2016 to 2018).
2. Calculate the Sample Proportions:
- For 2016, there were 552 satisfied out of a total of [tex]\(552 + 409 = 961\)[/tex] adults:
[tex]\[
p_1 = \frac{552}{961} \approx 0.5744
\][/tex]
- For 2018, there were 450 satisfied out of a total of [tex]\(450 + 528 = 978\)[/tex] adults:
[tex]\[
p_2 = \frac{450}{978} \approx 0.4601
\][/tex]
3. Calculate the Combined Proportion:
[tex]\[
p_{\text{combined}} = \frac{552 + 450}{961 + 978} = \frac{1002}{1939} \approx 0.5168
\][/tex]
4. Calculate the Standard Error:
[tex]\[
\text{SE} = \sqrt{p_{\text{combined}} \cdot (1 - p_{\text{combined}}) \left( \frac{1}{961} + \frac{1}{978} \right)}
\][/tex]
[tex]\[
\text{SE} \approx \sqrt{0.5168 \cdot 0.4832 \left( \frac{1}{961} + \frac{1}{978} \right)} \approx 0.0227
\][/tex]
5. Calculate the Z-Statistic:
[tex]\[
z = \frac{p_1 - p_2}{\text{SE}} = \frac{0.5744 - 0.4601}{0.0227} \approx 5.03
\][/tex]
6. Conclusion:
Given the computed Z-statistic of 5.03, which is significantly larger than the critical value for a one-tailed test at [tex]\(\alpha = 0.10\)[/tex] (which is approximately 1.28), we reject the null hypothesis [tex]\(H_0\)[/tex].
Thus, we have statistically significant evidence at the 0.10 significance level to conclude that the proportion of adults satisfied with the quality of the environment has declined from 2016 to 2018.
The test statistic is:
[tex]\[
z = 5.03
\][/tex]
Rounded to two decimal places.
