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In statistics, the sample proportion is a key concept when dealing with data samples. It represents the ratio of the number of successes to the total number of trials or subjects in a sample. In the context of our exercise, the 'successes' refer to boot camp attendees who returned to prison, although it might sound a bit counterintuitive to label such an outcome a "success." Nevertheless, this is the terminology used in statistics.
To calculate the sample proportion, you use the formula:

• \(\hat{p} = \frac{x}{n}\)

where \(x\) is the number of successes, and \(n\) is the sample size. For the boot camp example, 44 out of 100 attendees returned to prison, making the sample proportion \(\hat{p} = \frac{44}{100} = 0.44\).
The sample proportion is used to get an estimate of the true proportion in the population. It's a foundation for hypothesis testing, which helps determine if the sample proportion is significantly different from a known or hypothesized proportion.

The null hypothesis is a fundamental part of statistical hypothesis testing. It represents a default position that indicates no effect or no difference. In simpler terms, it assumes that any kind of difference or significance observed in a set of data is purely due to noise or chance.
In our scenario, the null hypothesis states that there is no difference between the sample proportion of boot camp attendees who return to prison and the known recidivism rate for the entire state. This known recidivism rate is 40%, or as a decimal, \(p_{0} = 0.40\).
By evaluating statistical evidence against the null hypothesis, we determine if observed data is consistent or inconsistent with it. If we find a significant deviation, we might reject the null hypothesis, which implies there could be an underlying effect, like the boot camp having an influence on recidivism rates. It's essential to remember that failing to reject the null hypothesis doesn't necessarily mean it's true; it simply indicates there's not enough evidence against it.

The Z-Test Statistic is a measure that helps us determine if there is a significant difference between a sample proportion and a known population proportion under the null hypothesis. The z-test is particularly useful when sample sizes are large because the sampling distribution of the sample proportion will be approximately normal.
To calculate the z-test statistic, use the formula:

• \(z = \frac{(\hat{p} - p_{0})}{\sqrt{\frac{p_{0}(1 - p_{0})}{n}}}\)

Here, \(\hat{p}\) is the sample proportion, \(p_{0}\) is the proportion under the null hypothesis, and \(n\) is the sample size. For the boot camp study:

• \(\hat{p} = 0.44\)
• \(p_{0} = 0.40\)
• \(n = 100\)

Plugging these values into the formula gives:

• \(z = \frac{(0.44 - 0.40)}{\sqrt{\frac{0.40 \times 0.60}{100}}} = 1.00\)

The value of the z-test statistic allows us to determine the p-value, which indicates the probability of observing our results if the null hypothesis is true. A high z-score (either positive or negative) might suggest that the observed sample proportion is significantly different from the null hypothesis proportion.