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In hypothesis testing, the null hypothesis is typically your starting point. It is a general statement or default position that there is no relationship between two measured phenomena. In our scenario, the null hypothesis (\(H_0\)) asserts that the true proportion of white collar criminals serving prison time is 69.4%. This is represented mathematically as:

• \(H_0: p = 0.694\)

Setting up a null hypothesis helps in determining whether any observed effect or difference in the sample proportion is due to random chance or if it is statistically significant. The null hypothesis is always presumed to be true until we find enough evidence to support the alternative hypothesis.
Hence, the objective is often to challenge or reject the null hypothesis to demonstrate that an effect exists.

The alternative hypothesis is crucial as it describes the specific condition you are testing against the null hypothesis. In this exercise, the alternative hypothesis (\(H_a\)) posits that the actual proportion of white collar criminals serving time is different from 69.4%.

• \(H_a: p eq 0.694\)

Investigating the alternative hypothesis allows researchers to determine if a real effect is present in the data. It is what you hope to support through your analysis. For our exercise, we employ a two-tailed test to consider both above and below the hypothesized 69.4% proportion. This approach is useful when deviations in both directions are of concern.

A significance level is a threshold for determining whether to reject the null hypothesis. Symbolized as \(\alpha\), it represents the probability of rejecting the null hypothesis when it is actually true, known as a Type I error. In our example, the significance level is set at 0.05.

• \(\alpha = 0.05\)

This value suggests that there is a 5% risk of concluding that a deviation exists when there isn't one in reality. The significance level is determined before the test and guides the decision-making process when comparing the p-value. Selecting an appropriate \(\alpha\) helps balance the need to detect effects and minimize errors.

The sample proportion is an estimate of the true population parameter, calculated from the sample data. It gives a preliminary idea about the proportion of the sample exhibiting a certain characteristic—in this case, those serving prison time.

• Calculated as: \(\hat{p} = \frac{x}{n} \)

Here, \(x\) is the number who served prison time, and \(n\) is the total number of sample observations. For the exercise with 120 criminals serving time out of 165, the sample proportion is:

• \(\hat{p} \approx 0.7273\)

The sample proportion provides insight into the data extracted from the sample and facilitates the computation of the test statistic. It is vital for determining how far the sample's observations are from the hypothesized population proportion.