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Answer :
To test the hypothesis that the proportion of white collar criminals serving prison time differs from 69%, we can perform a hypothesis test for a population proportion. Here's a step-by-step breakdown:
Define the Hypotheses:
- Null Hypothesis ([tex]H_0[/tex]): The proportion of white collar criminals serving prison time is equal to 69%, or [tex]p = 0.69[/tex].
- Alternative Hypothesis ([tex]H_a[/tex]): The proportion differs from 69%, or [tex]p \neq 0.69[/tex].
Determine the Significance Level ([tex]\alpha[/tex]):
- We're given [tex]\alpha = 0.05[/tex].
Calculate the Test Statistic:
The test statistic for a population proportion is calculated using the formula:
[tex]z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}[/tex]
where:\- [tex]\hat{p}[/tex] is the sample proportion, $\hat{p} = \frac{116}{161} \
- [tex]p_0[/tex] is the population proportion, [tex]p_0 = 0.69[/tex] \
- [tex]n[/tex] is the sample size, [tex]n = 161[/tex]
First, calculate [tex]\hat{p}[/tex]:\
[tex]\hat{p} = \frac{116}{161} \approx 0.7205[/tex]Now, plug the values into the formula:\
[tex]z = \frac{0.7205 - 0.69}{\sqrt{\frac{0.69 \times 0.31}{161}}}[/tex]
[tex]z = \frac{0.0305}{\sqrt{\frac{0.2139}{161}}}[/tex]
[tex]z = \frac{0.0305}{0.0366} \approx 0.833[/tex]Determine the Critical Value and Make a Decision:
For [tex]\alpha = 0.05[/tex] in a two-tailed test, the critical values for the standard normal distribution are approximately [tex]\pm1.96[/tex].- Decision Rule: Reject [tex]H_0[/tex] if [tex]|z| > 1.96[/tex].
In this case, [tex]|0.833| \not> 1.96[/tex], so we fail to reject the null hypothesis.
Construct a Confidence Interval:
To construct a 95% confidence interval for the true proportion, use the formula:
[tex]\hat{p} \pm z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}[/tex]The critical value [tex]z_{\alpha/2} = 1.96[/tex] for a 95% confidence interval.
[tex]0.7205 \pm 1.96 \times \sqrt{\frac{0.7205 \times (1-0.7205)}{161}}[/tex]
[tex]0.7205 \pm 1.96 \times \sqrt{\frac{0.201}}{161}[/tex]
[tex]0.7205 \pm 1.96 \times 0.0352[/tex]
[tex]0.7205 \pm 0.069[/tex]
[tex](0.6515, 0.7895)[/tex]Interpretation:
The 95% confidence interval is approximately (0.6515, 0.7895). Since this interval includes the hypothesized proportion of 0.69, we have additional support for not rejecting the null hypothesis. Therefore, there is no significant evidence to conclude that the actual proportion of white collar criminals serving prison time is different from 69%.
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