Question

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

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

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%.

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%.

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.

Key concepts

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.