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California's controversial "three-strikes law" requires judges to sentence anyone convicted of three felony offenses to life in prison. Supporters say that this decreases crime both because it is a strong deterrent and because career criminals are removed from the streets. Opponents argue (among other things) that people serving life sentences have nothing to lose, so violence within the prison system increases. To test the opponents' claim, researchers examined data starting from the mid- 1990 s from the California Department of Corrections. "Three Strikes: Yes" means the person had committed three or more felony offenses and was probably serving a life sentence. "Three Strikes: No" means the person had committed no more than two offenses. "Misconduct" includes serious offenses (such as assaulting an officer) and minor offenses (such as not standing for a count). "No Misconduct" means the offender had not committed any offenses in prison. a. Compare the proportions of misconduct in these samples. Which proportion is higher, the proportion of misconduct for those who had three strikes or that for those who did not have three strikes? Explain. b. Treat this as though it were a random sample, and determine whether those with three strikes tend to have more offenses than those who do not. Use a \(0.05\) significance level. $$\begin{array}{|l|l|l|}\hline & \multicolumn{2}{c|} {\text { Three Strikes }} \\\\\hline & \text { Yes } & \text { No } \\ \hline \text { Misconduct } & 163 & 974 \\\\\hline \text { No Misconduct } & 571 & 2214 \\ \hline \text { Totals } & 734 & 3188 \\\\\hline\end{array}$$

Step by step solution

01

Comparing proportions of misconduct

First, calculate the ratio of people with misconduct to total people for both those who committed three or more felony offenses and those who committed no more than two offenses. For those with three strikes: \(\frac{163}{734} = 0.222\), and for those without three strikes: \(\frac{974}{3188} = 0.305\). Clearly, the proportion of misconduct for those who did not have three strikes is higher.

02

Formulating the hypothesis

This is a hypothesis test for comparing two proportions. The null hypothesis, denoted \(H_0\), is that the two population proportions are equal, i.e. those with three strikes do not tend to have more offenses. The alternative hypothesis, \(H_a\), is that the proportion of offenses is higher for those with three strikes. Mathematically, \(H_0: p_{3 strikes} = p_{no 3 strikes}\) and \(H_a: p_{3 strikes} > p_{no 3 strikes}\).

03

Calculating the Test Statistic

To perform the Z-test, first, calculate the pooled sample proportion (p_hat): \(p_{hat} = \frac{x1+x2}{n1+n2} = \frac{163+974}{734+3188} = 0.291\) Next, calculate the standard error: \(SE = \sqrt{ p_{hat} * ( 1 - p_{hat}) * [ (1/n1) + (1/n2) ] } = \sqrt{0.291*(1-0.291)*[ (1/734) + (1/3188) ] } = 0.0253\) Where \(x1, x2\) are the numbers of successes in each group and \(n1, n2\) are the numbers of observations in each group. The test statistic (Z) is then given by the formula: \(Z = (p1 - p2) / SE\) Here \(p1, p2\) are the observed sample proportions. Substituting the given values we get: \(Z = (0.222 - 0.305) / 0.0253 = -3.28\)

04

Comparing the test statistic to critical value

With a 0.05 significance level, our critical value(Z_crit) for a one-sided test from the standard normal distribution table is approx. -1.645(since it's a left-tailed test). Since our calculated Z value is less than the critical value, we reject the null hypothesis.

05

Interpretation

We conclude that given the data, at a 0.05 significance level, it appears that prisoners without three-strikes tend to have a higher proportion of offences, contrary to the opponents' claim in the three-strikes law.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Three-Strikes Law

The "three-strikes law" is a legal provision that significantly impacts sentencing for repeat offenders. Originating in California during the early 1990s, this law mandates a life sentence for individuals convicted of three separate felony offenses. Supporters claim it serves as both a deterrent to potential criminals and a method to incapacitate repeat offenders by removing them permanently from society. On the other hand, critics argue that it leads to more violence within the prison system as inmates with life sentences might feel they have nothing to lose. This context provides the basis for examining whether there is any relation between the "three-strikes law" and misconduct rates in prisons.

Comparing Proportions

Comparing proportions is a critical statistical method used to determine if there are significant differences between groups. In the case of the "three-strikes law," researchers wanted to see if prisoners with three or more felony offenses had higher rates of misconduct than those with fewer infractions. To compare these proportions, we calculate the number of misconduct occurrences over the total number of prisoners for each group. For prisoners with three strikes, the proportion is calculated as \( \frac{163}{734} = 0.222 \). For those without three strikes, it's \( \frac{974}{3188} = 0.305 \). By calculating these proportions, we determine that misconduct is higher in prisoners without three strikes, which is contrary to what might initially be assumed.

Z-test

A Z-test is a statistical test used to determine if there is a significant difference between the means of two groups, often applied to test proportions. In this exercise, the Z-test is used to compare the proportion of misconduct between prisoners who have the "three-strikes strike" and those who don't. The process involves the following steps:

• Calculate the pooled sample proportion, \( \hat{p} = \frac{163+974}{734+3188} = 0.291 \).
• Determine the standard error (SE): \( SE = \sqrt{0.291(1-0.291)\left( \frac{1}{734} + \frac{1}{3188} \right)} = 0.0253 \).
• Compute the test statistic (Z): \( Z = \frac{0.222 - 0.305}{0.0253} = -3.28 \).

The Z-test result provides a way to statistically assess if the differences in proportions are due to chance or are indeed significant.

Null Hypothesis

The null hypothesis is a fundamental concept in hypothesis testing, representing a statement of no effect or no difference. In the exercise related to the three-strikes law, the null hypothesis posits that there is no difference in the proportion of misconduct between prisoners who have three strikes and those without. Mathematically, this is represented as \( H_0: p_{3 \text{ strikes}} = p_{\text{no 3 strikes}} \). The alternative hypothesis \( H_a \), suggests that the proportion of offenses for prisoners with three strikes is greater than for those without. Using the calculated Z-statistic and comparing it to the critical value of -1.645 at a 0.05 significance level, we find that the null hypothesis should be rejected. This implies there is a significant difference contrary to the expected outcome, where prisoners without three strikes have higher misconduct rates.