Question

A random sample of 113 felons convicted of nonviolent crimes in a state prison system completed a program designed to improve their employment possibilities before being released on parole. Fiftyeight eventually hecame repeat offenders. Is this recidivism rate significantly different from the rate for all offenders in that state \((57 \%) ?\) Summarize your conclusions in a sentence or two. (HINT: You must use the information given in the problem to compute a sample proportion. Remember to convert the population percentage to a proportion.)

Short answer

Calculate \(\hat{p}\ = 0.513\), \(SE = 0.047\), \(Z = -1.21\). The rate is not significantly different since \(|Z| < 1.96\).

Step-by-step solution

Compute Sample Proportion

First, find the sample proportion. The sample size is 113, and 58 of them became repeat offenders. The sample proportion \(\hat{p}\) is given by: \(\hat{p} = \frac{58}{113}\).

Convert Population Percentage to Proportion

Next, convert the given population recidivism rate to a proportion. The rate for all offenders in the state is 57%, which is equivalent to a proportion of \(p = 0.57\).

Calculate Standard Error

Calculate the standard error (SE) for the sample proportion using the formula: \(\text{SE} = \sqrt{\frac{p(1 - p)}{n}}\). Here, \(p = 0.57\) and \(n = 113\). Substitute these values to get: \(\text{SE} = \sqrt{\frac{0.57(1 - 0.57)}{113}}\).

Compute the Z-score

Compute the Z-score to determine how many standard deviations the sample proportion is from the population proportion. Use the formula: \(Z = \frac{\hat{p} - p}{\text{SE}}\).

Perform Hypothesis Testing

Perform a hypothesis test at the desired level of significance (commonly 0.05) to see if the observed recidivism rate is significantly different from the state's average. Compare your Z-score to the critical Z-value (e.g., \(\pm 1.96\) for a 95% confidence interval).

Draw a Conclusion

Based on the calculated Z-score and the critical value, determine whether to reject the null hypothesis (that the sample recidivism rate is equal to the state's rate).

Key concepts

Sample Proportion

The sample proportion \(\hat{p}\text\forall\) is a way to express the fraction of the sample that possesses a certain characteristic. Here, we want to know what proportion of the sampled felons became repeat offenders. To do this, we take the number of repeat offenders (58) and divide it by the total number of felons sampled (113). \[\hat{p}=\frac{58}{113}\]. This calculation gives us a proportion that we can compare to the population proportion.

Z-Score Computation

The Z-score is a measure of how many standard deviations our sample proportion is away from the population proportion. To compute it, we need both the sample proportion (\(\hat{p}\text\forall\)) and the population proportion (\(p\text\forall\)). The formula for the Z-score is \[Z = \frac{\hat{p} - p}{SE}\], where SE stands for the standard error. A positive Z-score indicates that the sample proportion is larger than the population proportion, whereas a negative score means it's smaller. By comparing this Z-score to a critical value (e.g., \pm 1.96\ in a 95% confidence interval), we can determine if our result is statistically significant.

Standard Error

The standard error (SE) of a sample proportion is a measure of the dispersion or variability of the proportion. It is calculated using the formula: \[SE = \sqrt{\frac{p (1 - p)}{n}}\], where \(p\text\forall\) is the population proportion and \(n\text\forall\) is the sample size. This formula accounts for the spread of the data within the sample and gives us a way to quantify the uncertainty in our sample proportion estimate. In our specific exercise, with a population proportion of 0.57 and a sample size of 113, the standard error informs us how much the sample proportion can vary simply due to the randomness of sampling.

Hypothesis Testing

Hypothesis testing is the process we use to make inferences about populations from samples. In this case, we want to test if the recidivism rate of our sample is significantly different from the state's rate. We start with the null hypothesis (H_0), which asserts that there is no difference between the sample and the population (i.e., the sample recidivism rate equals the state rate). We then compute the Z-score, compare it to a critical value (e.g., 1.96 for a 95% confidence level), and make a decision. If the Z-score lies outside the range of critical values, we reject the null hypothesis, indicating a significant difference.

Statistical Significance

Statistical significance helps us decide if our findings are likely due not to random chance. It involves determining whether the difference observed is large enough to be unlikely under the null hypothesis. In our exercise, after computing the Z-score, we compare it against a critical value (also known as a Z-critical). If the calculated Z-score lies beyond the critical value, we consider our result statistically significant. This simply means that the observed sample proportion is unlikely to have occurred if the true population proportion was 0.57. Therefore, any observed difference is probably not due to randomness but indicates a real discrepancy.