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Chapter 10: Problem 41

The article "Fewer Parolees Land Back Behind Bars" (Associated Press, April II, 2006 ) includes the following statement: "Just over \(38 \%\) of all felons who were released from prison in 2003 landed back behind bars by the end of the following year, the lowest rate since \(1979 . "\) Explain why it would not be necessary to carry out a hypothesis test to determine if the proportion of felons released in 2003 was less than .40 .

Short Answer

Expert verified
The current recidivism rate for 2003, which is 'just over 38%', is less than 40%. Therefore, it already substantiates the claim, eliminating the need for a hypothesis test.

Step by step solution

01

Understanding The Statement

The statement provides us with a proportion: 'Just over 38% of all felons who were released from prison in 2003 landed back behind bars by the end of the following year'. Here, the proportion of felons who returned to prison by the end of the next year after their release is just over 38%.
02

Analyzing The Hypothesis

A hypothesis test is used to determine to what degree a statement (or hypothesis) might be true. In this case, we might have been asked to test the hypothesis that the proportion of released felons who returned to prison is less than 40%.
03

Comparing The Proportion With Hypothesized Value

The given recidivism rate in question is just over 38%, which is less than 40%. As the observed proportion (38%) is already less than the value we would hypothetically test (.40 or 40%), there is no need for a hypothesis test. The data already provides clear evidence that supports the claim.

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Key Concepts

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

Understanding Recidivism Rate
The term recidivism rate refers to the proportion of individuals who return to criminal behavior after having been released from incarceration. In the context of the provided exercise, the recidivism rate signifies the percentage of felons who were re-incarcerated by the end of the year after their release in 2003. A lower recidivism rate is an indicator of successful reintegration into society, whereas a higher rate might suggest issues with the prison system or the rehabilitation process. Recidivism is a critical metric used by law enforcement agencies, policymakers, and researchers to evaluate the effectiveness of corrections programs and to shape future criminal justice policies.

Understanding and analyzing recidivism rates can help in identifying patterns and factors that contribute to repeat offenses, which is crucial for developing programs aimed at reducing this phenomenon. It is important to not only look at the rate itself but also at the context in which it occurs such as socioeconomic factors, support systems in place, and the type of crimes being committed.
Grasping Statistical Hypothesis
A statistical hypothesis is an assumption about a population parameter. This assumption may or may not be true. Hypothesis testing is a formal process used in statistics to test the validity of the assumed hypothesis. In the exercise mentioned, the assumption would be that the proportion of felons released in 2003 who returned to prison by the end of the following year is less than 40%.

To frame this statistically, the null hypothesis (\(H_{0}\)) is typically a statement of no effect or no difference; it could be that the recidivism rate is equal to 40%. The alternative hypothesis (\(H_{1}\) or \(H_{a}\)), on the other hand, would assert that the recidivism rate is indeed less than 40%. The purpose of hypothesis testing is to determine which hypothesis the data supports, based on an established level of significance and using appropriate statistical tests.
Proportion Analysis in Hypothesis Testing
When it comes to the proportion analysis in the realm of hypothesis testing, we are concerned with the analysis of categorical data -- in other words, data that can be divided into distinct groups, such as 'returned to prison' or 'did not return to prison'. In the given exercise, the proportion of interest is the percentage of felons who were back in prison by the end of the year following their release.

Statistically, a proportion is a type of mean for categorical variable, representing the fraction of the total sample that has a particular characteristic. To assess whether there is statistically significant evidence that the actual proportion differs from what's hypothesized, statisticians utilize tests such as the Z-test for proportions. These tests compare the observed proportion to the hypothesized value, considering the variability in the proportion that could arise simply by chance. Since the observed recidivism rate is clearly below the threshold in question, 40%, such statistical analysis would be redundant in this scenario as the real-world data has already presented a conclusion.

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Most popular questions from this chapter

The poll referenced in the previous exercise (鈥淢ilitary Draft Study," AP- lpsos, June 2005 ) also included the following question: "If the military draft were reinstated, would you favor or oppose drafting women as well as men?" Forty-three percent of the 1000 people responding said that they would favor drafting women if the draft were reinstated. Using a .05 significance level, carry out a test to determine if there is convincing evidence that fewer than half of adult Americans would favor the drafting of women.

A certain pen has been designed so that true average writing lifetime under controlled conditions (involving the use of a writing machine) is at least 10 hours. A random sample of 18 pens is selected, the writing lifetime of each is determined, and a normal probability plot of the resulting data supports the use of a one-sample \(t\) test. The relevant hypotheses are \(H_{0}: \mu=10\) versus \(H_{a}: \mu<10 .\) a. If \(t=-2.3\) and \(\alpha=.05\) is selected, what conclusion is appropriate? b. If \(t=-1.83\) and \(\alpha=.01\) is selected, what conclusion is appropriate? c. If \(t=0.47,\) what conclusion is appropriate?

The paper "MRI Evaluation of the Contralateral Breast in Women with Recently Diagnosed Breast Cancer" (New England Journal of Medicine \([2007]: 1295-1303)\) describes a study of the use of MRI (Magnetic Resonance Imaging) exams in the diagnosis of breast cancer. The purpose of the study was to determine if MRI exams do a better job than mammograms of determining if women who have recently been diagnosed with cancer in one breast have cancer in the other breast. The study participants were 969 women who had been diagnosed with cancer in one breast and for whom a mammogram did not detect cancer in the other breast. These women had an MRI exam of the other breast, and 121 of those exams indicated possible cancer. After undergoing biopsies, it was determined that 30 of the 121 did in fact have cancer in the other breast, whereas 91 did not. The women were all followed for one year, and three of the women for whom the MRI exam did not indicate cancer in the other breast were subsequently diagnosed with cancer that the MRI did not detect. The accompanying table summarizes this information. Suppose that for women recently diagnosed with cancer in only one breast, the MRI is used to decide between the two "hypotheses" \(H_{0}\) : woman has cancer in the other breast \(H_{a}:\) woman does not have cancer in the other breast (Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) a. One possible error would be deciding that a woman who does have cancer in the other breast is cancerfree. Is this a Type I or a Type II error? Use the information in the table to approximate the probability of this type of error. b. There is a second type of error that is possible in this setting. Describe this error and use the information in the given table to approximate the probability of this type of error.

A television manufacturer claims that (at least) \(90 \%\) of its TV sets will need no service during the first 3 years of operation. A consumer agency wishes to check this claim, so it obtains a random sample of \(n=100\) purchasers and asks each whether the set purchased needed repair during the first 3 years after purchase. Let \(\hat{p}\) be the sample proportion of responses indicating no repair (so that no repair is identified with a success). Let \(p\) denote the actual proportion of successes for all sets made by this manufacturer. The agency does not want to claim false advertising unless sample evidence strongly suggests that \(p<.9 .\) The appropriate hypotheses are then \(H_{0}: p=.9\) versus \(H_{a}: p<.9\). a. In the context of this problem, describe Type I and Type II errors, and discuss the possible consequences of each. b. Would you recommend a test procedure that uses \(\alpha=.10\) or one that uses \(\alpha=.01 ?\) Explain.

The mean length of long-distance telephone calls placed with a particular phone company was known to be 7.3 minutes under an old rate structure. In an attempt to be more competitive with other long-distance carriers, the phone company lowered long-distance rates, thinking that its customers would be encouraged to make longer calls and thus that there would not be a big loss in revenue. Let \(\mu\) denote the mean length of long-distance calls after the rate reduction. What hypotheses should the phone company test to determine whether the mean length of long-distance calls increased with the lower rates?
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