Question

The article "Fewer Parolees Land Back Behind Bars" (Associated Press, April 11,2006 ) includes the following statement: "Just over 38 percent 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 hypothesis test is not necessary because the observed proportion of felons who landed back in jail (just over 38%) is already less than the 40% stated in the null hypothesis. This directly contradicts the null hypothesis, thus we can reject it without needing to perform a hypothesis test.

Step-by-step solution

Understand the Null Hypothesis

In this context, the null hypothesis would express that the proportion of felons released in 2003, who ended back in prison by the end of the next year, is equal to or greater than 40%.

Analyzing the Given Data

The given data shows that just over 38 percent of all felons released in 2003 landed back in prison by the end of 2004. This result is already less than the 40% proportion mentioned in the null hypothesis.

Make the Conclusion

Since the proportion of felons (approximately 38%) who returned to prison is already less than 40%, we can directly reject the null hypothesis without needing to conduct a hypothesis test.

Key concepts

Null Hypothesis

In the realm of statistics, the null hypothesis plays a central role in hypothesis testing. It is a statement that there is no effect or no difference, and it sets the stage for statistical comparison. For example, in examining recidivism rates among parolees, a null hypothesis might state that the proportion of former inmates returning to prison does not differ from a historical rate, such as 40%. Essentially, it serves as a benchmark for judging whether the observed data show a statistically significant departure from what's expected under the null hypothesis.

To refute the null hypothesis, statisticians often use significance tests which can involve meticulous calculations. However, when straightforward observations—such as a proportion of 38% being lower than the comparison value of 40%—are evident, the need for complex testing diminishes. Rejecting the null hypothesis in such cases can be immediate and reflects a clear-cut finding that the observed proportion does indeed differ from the expected value.

Proportion Analysis

Proportion analysis is a statistical approach used to examine the ratio of a subset to the whole population. In context, the proportion in focus is the percentage of felons returning to prison. Through proportion analysis, statisticians can evaluate if an observed percentage significantly deviates from an expected or historic proportion. Such analyses are particularly potent when comprehending social trends or changes over time, like improvements in rehabilitation leading to fewer parolees reoffending.

While proportion analysis can employ complex statistical techniques for closer margins, instances with clear deviations might not necessitate such rigorous investigation. Therefore, when the proportion drops clearly below the benchmark (38% versus 40%), one could argue it may not require formal hypothesis testing to acknowledge a change, as the existing data is rather telling by itself.

Statistical Significance

The concept of statistical significance is pivotal in determining whether a result is not likely due to random chance. When statisticians set out to evaluate hypotheses, they're engaged in a quest to discern whether the observed effects stand out against the backdrop of variability that might arise simply by chance.

Statistical significance is quantified by a p-value, a probability that measures the strength of the evidence against the null hypothesis. A lower p-value suggests a greater statistical significance. Conventionally, a p-value lower than a threshold (often 0.05) would lead us to reject the null hypothesis, favoring the alternative.

In the provided example, since the proportion of 38% is notably less than the 40% mark cited in the null hypothesis, the difference is palpably significant without calculation—assuming the sample size is large enough. This intuitive assessment implies that full-fledged statistical testing may be unnecessary to recognize there's a significant decrease in the proportion of felons reoffending.