Answered

A poll asked random samples of adults in 2016 and 2018 if they were satisfied with the quality of the environment. In 2016, 546 were satisfied with the quality of the environment and 418 were dissatisfied. In 2018, 464 were satisfied and 516 were dissatisfied. Determine whether the proportion of adults who are satisfied with the quality of the environment has declined. Use a 0.01 significance level.

Let [tex]p_1[/tex] represent the population proportion of adults in 2016 who were satisfied with the quality of the environment and let [tex]p_2[/tex] represent the population proportion of adults in 2018 who were satisfied with the quality of the environment. Write the hypotheses for the test.
[tex]\[
\begin{array}{l}
H_0: p_1 = p_2 \\
H_a: p_1 \ \textgreater \ p_2
\end{array}
\][/tex]

Find the test statistic for this test.
[tex]\[
z = 4.10 \quad \text{(Round to two decimal places as needed.)}
\][/tex]

Find the p-value for this test.
[tex]\[ p \text{-value} = \square \quad \text{(Round to three decimal places as needed.)} \][/tex]



Answer :

GinnyAnswer
Certainly! Let's go through the problem step by step.

### Step 1: Define the Hypotheses
We want to test if the proportion of adults satisfied with the quality of the environment has declined from 2016 to 2018.

- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\( p_1 = p_2 \)[/tex]

This means that the proportion of satisfied adults in 2016 ([tex]\(p_1\)[/tex]) is equal to the proportion in 2018 ([tex]\(p_2\)[/tex]).

- Alternative Hypothesis ([tex]\(H_a\)[/tex]): [tex]\( p_1 > p_2 \)[/tex]

This means that the proportion of satisfied adults in 2016 ([tex]\(p_1\)[/tex]) is greater than the proportion in 2018 ([tex]\(p_2\)[/tex]).

### Step 2: Sample Sizes and Observed Proportions
First, let's find the sample sizes and observed proportions for each year.

In 2016:
- Number satisfied: 546
- Number dissatisfied: 418
- Total sample size [tex]\(n_1\)[/tex] = 546 + 418 = 964
- Proportion satisfied [tex]\(p_1\)[/tex] = 546 / 964 ≈ 0.5664

In 2018:
- Number satisfied: 464
- Number dissatisfied: 516
- Total sample size [tex]\(n_2\)[/tex] = 464 + 516 = 980
- Proportion satisfied [tex]\(p_2\)[/tex] = 464 / 980 ≈ 0.4735

### Step 3: Combined Proportion and Standard Error
Let's find the combined proportion and the standard error.

- Combined proportion [tex]\(p_{\text{combined}}\)[/tex] = (Number satisfied in 2016 + Number satisfied in 2018) / (Total number of adults in 2016 + Total number of adults in 2018)

So, [tex]\( p_{\text{combined}} = \frac{546 + 464}{964 + 980} = \frac{1010}{1944} ≈ 0.5195\)[/tex]

- Standard Error ([tex]\(SE\)[/tex]) is calculated as:

[tex]\( SE = \sqrt{ p_{\text{combined}} \times (1 - p_{\text{combined}}) \times \left( \frac{1}{n_1} + \frac{1}{n_2} \right) }\)[/tex]

Substituting the values, we get:

[tex]\( SE ≈ \sqrt{0.5195 \times (1 - 0.5195) \times \left( \frac{1}{964} + \frac{1}{980} \right) } ≈ 0.0227 \)[/tex]

### Step 4: Test Statistic
The test statistic [tex]\( z \)[/tex] is:
[tex]\[ z = \frac{p_1 - p_2}{SE} \][/tex]

So:
[tex]\[ z = \frac{0.5664 - 0.4735}{0.0227} ≈ 4.10 \][/tex]

### Step 5: Find the p-value
To find the p-value, we use the cumulative distribution function (CDF) of the standard normal distribution for the calculated [tex]\( z \)[/tex]-score.

Given:
[tex]\[ z = 4.10 \][/tex]

The p-value for this test is the probability that a standard normal variable is less than 4.10:
[tex]\[ \text{p-value} ≈ 0.999979 \][/tex]

### Step 6: Conclusion
Given the p-value = 0.999979, which is much greater than the significance level of 0.01, we fail to reject the null hypothesis. Therefore, we do not 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
- Hypotheses:
[tex]\[ \begin{aligned} & H_0: p_1 = p_2 \\ & H_a: p_1 > p_2 \end{aligned} \][/tex]

- Calculated Test Statistic: [tex]\[ z ≈ 4.10 \][/tex]

- Calculated p-value: [tex]\[ \text{p-value} ≈ 0.999 \][/tex]

Since the p-value is greater than 0.01, we conclude that there is not enough evidence to support the claim that the proportion of satisfied adults has declined.