Answer :
To determine whether the proportion of adults who are satisfied with the quality of the environment has declined from 2016 to 2018, we will conduct a hypothesis test for the difference in proportions.
Firstly, we need to establish our null and alternative hypotheses:
- Null Hypothesis ([tex]\( H_0 \)[/tex]): The proportion of adults who are satisfied with the quality of the environment in 2016 is equal to the proportion in 2018. This can be written as [tex]\( H_0: P_1 = P_2 \)[/tex].
- Alternative Hypothesis ([tex]\( H_3 \)[/tex]): The proportion of adults who are satisfied with the quality of the environment in 2016 is greater than the proportion in 2018. This can be written as [tex]\( H_3: P_1 \geq P_2 \)[/tex].
Given the data:
- Number satisfied in 2016: [tex]\( 546 \)[/tex]
- Number dissatisfied in 2016: [tex]\( 418 \)[/tex]
- Number satisfied in 2018: [tex]\( 484 \)[/tex]
- Number dissatisfied in 2018: [tex]\( 548 \)[/tex]
We first compute the total sample sizes for each year:
- Total sample in 2016 ([tex]\( n1 \)[/tex]): [tex]\( 546 + 418 = 964 \)[/tex]
- Total sample in 2018 ([tex]\( n2 \)[/tex]): [tex]\( 484 + 548 = 1032 \)[/tex]
Next, we calculate the sample proportions:
- Proportion satisfied in 2016 ([tex]\( \hat{p1} \)[/tex]): [tex]\( \frac{546}{964} \)[/tex]
- Proportion satisfied in 2018 ([tex]\( \hat{p2} \)[/tex]): [tex]\( \frac{484}{1032} \)[/tex]
Now, we need the combined proportion so we can find the pooled estimate of the proportion:
- Combined proportion ([tex]\( \hat{p} \)[/tex]): [tex]\( \frac{546 + 484}{964 + 1032} \)[/tex]
Using the combined proportion, we can calculate the standard error (SE):
[tex]\[ \text{SE} = \sqrt{\hat{p} \left(1 - \hat{p}\right) \left( \frac{1}{n1} + \frac{1}{n2} \right) } \][/tex]
Finally, we compute the test statistic, which is a z-score calculated by comparing the difference in sample proportions to the standard error:
[tex]\[ z = \frac{\hat{p1} - \hat{p2}}{\text{SE}} \][/tex]
After performing these calculations, the test statistic [tex]\(z\)[/tex] is found to be:
[tex]\[ z = 4.35 \ (rounded \ to \ two \ decimal \ places) \][/tex]
Therefore, the test statistic for this hypothesis test is:
[tex]\[ z = 4.35 \][/tex]
This completes the hypothesis test to determine whether the proportion of adults who are satisfied with the quality of the environment has declined from 2016 to 2018 at a significance level of 0.01.
Firstly, we need to establish our null and alternative hypotheses:
- Null Hypothesis ([tex]\( H_0 \)[/tex]): The proportion of adults who are satisfied with the quality of the environment in 2016 is equal to the proportion in 2018. This can be written as [tex]\( H_0: P_1 = P_2 \)[/tex].
- Alternative Hypothesis ([tex]\( H_3 \)[/tex]): The proportion of adults who are satisfied with the quality of the environment in 2016 is greater than the proportion in 2018. This can be written as [tex]\( H_3: P_1 \geq P_2 \)[/tex].
Given the data:
- Number satisfied in 2016: [tex]\( 546 \)[/tex]
- Number dissatisfied in 2016: [tex]\( 418 \)[/tex]
- Number satisfied in 2018: [tex]\( 484 \)[/tex]
- Number dissatisfied in 2018: [tex]\( 548 \)[/tex]
We first compute the total sample sizes for each year:
- Total sample in 2016 ([tex]\( n1 \)[/tex]): [tex]\( 546 + 418 = 964 \)[/tex]
- Total sample in 2018 ([tex]\( n2 \)[/tex]): [tex]\( 484 + 548 = 1032 \)[/tex]
Next, we calculate the sample proportions:
- Proportion satisfied in 2016 ([tex]\( \hat{p1} \)[/tex]): [tex]\( \frac{546}{964} \)[/tex]
- Proportion satisfied in 2018 ([tex]\( \hat{p2} \)[/tex]): [tex]\( \frac{484}{1032} \)[/tex]
Now, we need the combined proportion so we can find the pooled estimate of the proportion:
- Combined proportion ([tex]\( \hat{p} \)[/tex]): [tex]\( \frac{546 + 484}{964 + 1032} \)[/tex]
Using the combined proportion, we can calculate the standard error (SE):
[tex]\[ \text{SE} = \sqrt{\hat{p} \left(1 - \hat{p}\right) \left( \frac{1}{n1} + \frac{1}{n2} \right) } \][/tex]
Finally, we compute the test statistic, which is a z-score calculated by comparing the difference in sample proportions to the standard error:
[tex]\[ z = \frac{\hat{p1} - \hat{p2}}{\text{SE}} \][/tex]
After performing these calculations, the test statistic [tex]\(z\)[/tex] is found to be:
[tex]\[ z = 4.35 \ (rounded \ to \ two \ decimal \ places) \][/tex]
Therefore, the test statistic for this hypothesis test is:
[tex]\[ z = 4.35 \][/tex]
This completes the hypothesis test to determine whether the proportion of adults who are satisfied with the quality of the environment has declined from 2016 to 2018 at a significance level of 0.01.