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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, 462 were satisfied and 548 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]\[
z = \square \text{ (Round to two decimal places as needed.)}
\][/tex]

(Round to two decimal places as needed.)

Sagot :

GinnyAnsweridn
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.
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