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The sample proportion is a key concept used in inferential statistics to make estimations about a population. In this context, it helps us determine the likelihood of convicts leading crime-free lives after rehabilitation. The sample proportion is denoted by \(\hat{p}\), and it is calculated using the formula:

• \(\hat{p} = \frac{x}{n}\)

Here, \(x\) represents the number of successful outcomes, and \(n\) is the total number of observations. In our example, 79 out of 121 convicts maintained a crime-free life for 3 years. So, \(\hat{p} = \frac{79}{121} = 0.6529\). This indicates that approximately 65.29% of the sample succeeded in living crime-free. Understanding the sample proportion is important because it acts as the foundation for further statistical analysis, like creating confidence intervals.

Standard error helps us measure the variability or spread of a sample statistic, such as the sample proportion, around the population parameter. In essence, it shows how much the sample proportion is expected to "sway" or deviate across different samples. The formula for the standard error of a sample proportion is:

• \(SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\)

Plugging in the values from our exercise, we get:This means the expected variability of our sample proportion about the true population proportion is about 4.29%. Understanding this variability is essential, especially when making predictions or constructing your confidence intervals.

The Z-score is a statistical measurement that indicates the number of standard deviations a data point is from the sample mean. When constructing a confidence interval, the Z-score helps adjust for the desired confidence level. For commonly used confidence intervals, like 95%, specific Z-scores are associated. A 95% confidence interval corresponds to a Z-score of 1.96.
This Z-score indicates that 95% of the data falls within 1.96 standard deviations of the mean for a normal distribution. It is integral in setting the width of the confidence interval, allowing us to estimate the range of plausible values for the population proportion. The inclusion of Z-scores ensures that confidence intervals more accurately reflect the level of certainty desired by statisticians and researchers.

Probability estimation is the process used to predict the likelihood of a particular outcome happening. Given our exercise, it is the probability of convicts leading crime-free lives after undergoing the rehabilitation technique.
Confidence intervals help provide a range in which this true probability is likely to lie. Using the calculated sample proportion, standard error, and Z-score, we can estimate that the true probability or proportion, \(p\), is likely between 56.9% and 73.7% according to our 95% confidence interval.

• The formula used is:
  • \(CI = \hat{p} \pm z\cdot SE\)

This interval offers insight into the reliability of the rehabilitation method, giving stakeholders a clearer picture of its effectiveness. Essentially, probability estimation allows us to make informed conclusions about how likely it is for similar future instances to follow the same pattern.