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Sagot :
Alright, let's walk through the problem step by step to determine whether the proportion of adults satisfied with the quality of the environment has declined from 2016 to 2018.
### Step 1: State the Null and Alternative Hypotheses
Our hypotheses are:
[tex]\[ \begin{array}{l:} H_0: p_1 = p_2 \quad \text{(The proportion of satisfied adults in 2016 is equal to that in 2018)} \\ H_A: p_1 > p_2 \quad \text{(The proportion of satisfied adults in 2016 is greater than that in 2018)} \end{array} \][/tex]
### Step 2: Calculate the Proportions
First, we need to find the proportions of satisfied adults for each year.
For 2016:
[tex]\[ p_1 = \frac{\text{satisfied\_2016}}{\text{satisfied\_2016} + \text{dissatisfied\_2016}} = \frac{552}{552 + 409} \approx 0.5744 \][/tex]
For 2018:
[tex]\[ p_2 = \frac{\text{satisfied\_2018}}{\text{satisfied\_2018} + \text{dissatisfied\_2018}} = \frac{450}{450 + 528} \approx 0.4601 \][/tex]
### Step 3: Calculate the Combined Proportion
We need the combined proportion for the standard error calculation:
[tex]\[ p_{\text{combined}} = \frac{\text{satisfied\_2016} + \text{satisfied\_2018}}{\text{satisfied\_2016} + \text{dissatisfied\_2016} + \text{satisfied\_2018} + \text{dissatisfied\_2018}} = \frac{552 + 450}{552 + 409 + 450 + 528} \approx 0.5168 \][/tex]
### Step 4: Calculate the Standard Error
The standard error (SE) can be calculated as follows:
[tex]\[ \text{SE} = \sqrt{p_{\text{combined}} \cdot (1 - p_{\text{combined}}) \cdot \left( \frac{1}{\text{total\_2016}} + \frac{1}{\text{total\_2018}} \right)} \][/tex]
Where:
[tex]\[ \text{total\_2016} = 552 + 409 = 961 \][/tex]
[tex]\[ \text{total\_2018} = 450 + 528 = 978 \][/tex]
Substitute the values:
[tex]\[ \text{SE} = \sqrt{0.5168 \cdot (1 - 0.5168) \cdot \left( \frac{1}{961} + \frac{1}{978} \right)} \approx 0.0227 \][/tex]
### Step 5: Calculate the Z-test Statistic
Calculate the Z-test statistic using:
[tex]\[ z = \frac{p_1 - p_2}{\text{SE}} = \frac{0.5744 - 0.4601}{0.0227} \approx 5.03 \][/tex]
### Step 6: Find the P-value
Using the Z-test statistic, we can find the p-value, which represents the probability that the observed difference is due to chance. For a one-tailed test, the p-value can be found as:
[tex]\[ p\text{-value} = 1 - \Phi(z) \quad \text{where} \quad \Phi \text{ is the cumulative distribution function of the standard normal distribution} \][/tex]
Given the Z-test statistic [tex]\( z \approx 5.03 \)[/tex]:
[tex]\[ \text{p-value} \approx 0.0 \][/tex]
### Conclusion
Since the p-value is smaller than our significance level of [tex]\(\alpha = 0.10\)[/tex], we reject the null hypothesis [tex]\(H_0\)[/tex]. Therefore, we have sufficient evidence to conclude that the proportion of adults who are satisfied with the quality of the environment has declined from 2016 to 2018.
Summary:
- Test statistic (z): 5.03
- P-value: 0.0 (rounded to three decimal places)
Thus, based on our test, the decline is statistically significant.
### Step 1: State the Null and Alternative Hypotheses
Our hypotheses are:
[tex]\[ \begin{array}{l:} H_0: p_1 = p_2 \quad \text{(The proportion of satisfied adults in 2016 is equal to that in 2018)} \\ H_A: p_1 > p_2 \quad \text{(The proportion of satisfied adults in 2016 is greater than that in 2018)} \end{array} \][/tex]
### Step 2: Calculate the Proportions
First, we need to find the proportions of satisfied adults for each year.
For 2016:
[tex]\[ p_1 = \frac{\text{satisfied\_2016}}{\text{satisfied\_2016} + \text{dissatisfied\_2016}} = \frac{552}{552 + 409} \approx 0.5744 \][/tex]
For 2018:
[tex]\[ p_2 = \frac{\text{satisfied\_2018}}{\text{satisfied\_2018} + \text{dissatisfied\_2018}} = \frac{450}{450 + 528} \approx 0.4601 \][/tex]
### Step 3: Calculate the Combined Proportion
We need the combined proportion for the standard error calculation:
[tex]\[ p_{\text{combined}} = \frac{\text{satisfied\_2016} + \text{satisfied\_2018}}{\text{satisfied\_2016} + \text{dissatisfied\_2016} + \text{satisfied\_2018} + \text{dissatisfied\_2018}} = \frac{552 + 450}{552 + 409 + 450 + 528} \approx 0.5168 \][/tex]
### Step 4: Calculate the Standard Error
The standard error (SE) can be calculated as follows:
[tex]\[ \text{SE} = \sqrt{p_{\text{combined}} \cdot (1 - p_{\text{combined}}) \cdot \left( \frac{1}{\text{total\_2016}} + \frac{1}{\text{total\_2018}} \right)} \][/tex]
Where:
[tex]\[ \text{total\_2016} = 552 + 409 = 961 \][/tex]
[tex]\[ \text{total\_2018} = 450 + 528 = 978 \][/tex]
Substitute the values:
[tex]\[ \text{SE} = \sqrt{0.5168 \cdot (1 - 0.5168) \cdot \left( \frac{1}{961} + \frac{1}{978} \right)} \approx 0.0227 \][/tex]
### Step 5: Calculate the Z-test Statistic
Calculate the Z-test statistic using:
[tex]\[ z = \frac{p_1 - p_2}{\text{SE}} = \frac{0.5744 - 0.4601}{0.0227} \approx 5.03 \][/tex]
### Step 6: Find the P-value
Using the Z-test statistic, we can find the p-value, which represents the probability that the observed difference is due to chance. For a one-tailed test, the p-value can be found as:
[tex]\[ p\text{-value} = 1 - \Phi(z) \quad \text{where} \quad \Phi \text{ is the cumulative distribution function of the standard normal distribution} \][/tex]
Given the Z-test statistic [tex]\( z \approx 5.03 \)[/tex]:
[tex]\[ \text{p-value} \approx 0.0 \][/tex]
### Conclusion
Since the p-value is smaller than our significance level of [tex]\(\alpha = 0.10\)[/tex], we reject the null hypothesis [tex]\(H_0\)[/tex]. Therefore, we have sufficient evidence to conclude that the proportion of adults who are satisfied with the quality of the environment has declined from 2016 to 2018.
Summary:
- Test statistic (z): 5.03
- P-value: 0.0 (rounded to three decimal places)
Thus, based on our test, the decline is statistically significant.
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