Question

only 13% of registered voters voted in the last election. will voter participation decline for the upcoming election? of the 387 randomly selected registered voters surveyed, 43 of them will vote in the upcoming election. what can be concluded at the \\( \alpha=0.05 \\) level of significance?
a. for this study, we should use z - test for a population proportion
b. the null and alternative hypotheses would be:
\\( h_{0}: p=0.13 \\) (please enter a decimal)
\\( h_{1}: p<0.13 \\) (please enter a decimal)
c. the test statistic? = (please show your answer to 3 decimal places.)
d. the p - value = (please show your answer to 4 decimal places.)
e. the p - value is? \\( \alpha \\)
f. based on this, we should select an answer the null hypothesis.
g. thus, the final conclusion is that...
the data suggest the population proportion is not significantly lower than 13% at \\( \alpha = 0.05 \\), so there is statistically significant evidence to conclude that the percentage of registered voters who will vote in the upcoming election will be equal to 13%.
the data suggest the populaton proportion is significantly lower than 13% at \\( \alpha = 0.05 \\), so there is statistically significant evidence to conclude that the the percentage of all registered voters who will vote in the upcoming election will be lower than 13%.
the data suggest the population proportion is not significantly lower than 13% at \\( \alpha = 0.05 \\), so there is statistically insignificant evidence to conclude that the percentage of registered voters who will vote in the upcoming election will be lower than 13%.

Explanation:

Step1: Calculate sample proportion

The sample proportion $\hat{p}=\frac{x}{n}$, where $x = 43$ and $n=387$. So, $\hat{p}=\frac{43}{387}\approx0.111$.

Step2: Calculate test - statistic

The formula for the test - statistic $z$ in a proportion test is $z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}$. Here, $p = 0.13$, $\hat{p}=0.111$, and $n = 387$.

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Step3: Calculate p - value

Since this is a left - tailed test ($H_1:p<0.13$), the p - value is $P(Z

Answer:

c. The test statistic $z=-1.111$
d. The p - value $=0.1335$
e. The p - value is $> \alpha$ (since $0.1335>0.05$)
f. Based on this, we should fail to reject the null hypothesis.
g. The data suggest the population proportion is not significantly lower than $13\%$ at $\alpha = 0.05$, so there is statistically insignificant evidence to conclude that the percentage of registered voters who will vote in the upcoming election will be lower than $13\%$.