Question

The article "Fewer Parolees Land Back Behind Bars" (Associated Press, April 11,2006 ) 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 who landed back behind bars by the end of the following year was less than 0.40.

Short answer

It would not be necessary to carry out a hypothesis test to determine if the proportion of felons released in 2003 who landed back behind bars by the end of the following year was less than 0.40 as the given percentage (38 percent) is already less than 40 percent.

Step-by-step solution

Understating the Problem

To start with, we need to digest what the exercise statement is asking. Here, we are given a clear percentage of the proportion of felons who have landed back in prison upon their release in 2003, which is exactly 38 percent.

Comparing the given proportion

The next step is to compare this given proportion to the question 'is this number less than 0.40?' Since 38 percent is clearly less than 40 percent, or in decimal form 0.38 is less than 0.40, we can directly conclude this.

Explain the Unnecessity of Hypothesis Testing

The explanation for the unnecessity of a hypothesis test here is due to the clear given information. Hypothesis testing is a statistical method which allows us to make inferences or draw conclusions about a population based on a sample of data. But since we are directly given the actual proportion of the year 2003 there is no need for such inferential testing.

Key concepts

Understanding the Felon Recidivism Rate

The term 'recidivism' often appears in studies pertaining to criminal justice, referring to the act of a person's relapse into criminal behavior, often after the person receives sanctions or undergoes intervention for a previous crime. In the context of statistics, felon recidivism rate is a crucial measure used to understand the effectiveness of correctional programs and policies.

Specifically, it represents the proportion of individuals who are rearrested, reconvicted, or reincarcerated after their release from custody. When assessing data such as a 38% recidivism rate, what is essential to note is that it gives us direct insight into how many felons have returned to prison within a particular period, in this case, by the end of the year following 2003.

This rate can be helpful for policymakers, researchers, and the general public to gauge trends over time and the outcomes of the criminal justice system. It essentially aids in determining whether initiatives aimed at reducing recidivism are achieving their goals or if there's a need for different strategies.

Proportion Comparison in Statistics

Moving beyond the realm of recidivism, the proportion comparison is at the core of many statistical analyses. Generally, this process involves comparing the proportion of a particular characteristic in a sample to a known proportion or to another sample's proportion. It helps in determining whether there is a statistically significant difference between the two.

For example, when comparing the recidivism rate of 38% to a specified benchmark of 40%, we are simply contrasting two different proportions to see if they are notably different from one another.

Simple Versus Advanced Comparison

In some cases, a simple comparison, just like subtracting one percentage from another or evaluating them side by side suffices. In more complex cases, statistical tests like z-tests or chi-square tests may be necessary to account for sample sizes and variances to draw valid conclusions.

However, statistical tests are only relevant when there's uncertainty involved. When you have clear, definitive figures, such as in our example with the recidivism rate, advanced statistical comparisons might be entirely unnecessary.

The Role of Statistical Inference in Hypothesis Testing

Lastly, let's consider the overarching context of statistical inference, which encompasses the process of making educated guesses about a population's characteristics based on a sample. It's a cornerstone of statistics that includes methods like estimation and hypothesis testing.

Hypothesis Testing Explained

Hypothesis testing is a method that statisticians use to decide whether to accept or reject a particular hypothesis about a population parameter, based on sample data. It allows us to draw conclusions even when full population data isn't accessible — we infer properties of the whole group from the part we can see.

This typically involves choosing a significance level, calculating a test statistic, and determining the p-value to decide whether the observed data are unlikely under the null hypothesis. If the data highly contradict the null hypothesis, we might favor the alternative hypothesis.

Nonetheless, in some instances, like the one mentioned where the proportion of felons who returned to prison by the end of the following year was less than 40%, statistical inference through hypothesis testing is not warranted since the answer is clearly evident from the data provided. In such cases, further testing not only consumes unnecessary time but also fails to offer any additional insight.