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The Poisson Distribution is a fundamental concept in probability theory and statistics. It's often used to model the number of times an event occurs in a fixed interval of time or space. For this distribution, two parameters are important:

• Mean (\( \lambda \)): which represents the average number of occurrences in the interval.
• Variance (\( \lambda \)): which is the same as the mean for a Poisson distribution.

The unique property here is that both the mean and variance are equal. This distribution is typically skewed if the mean is small, but it becomes more symmetrical as the mean increases.
The Poisson model is particularly useful for count data and modeling rare events, making it ideal for our semiconductor wafer example where each wafer has a small probability of defects. As the sample size increases, like with our 500 wafers, the Poisson distribution begins to approximate a normal distribution.

The Standard Normal Distribution is a special kind of normal distribution. It has a mean of zero and a standard deviation of one. This is a key concept for statisticians as it simplifies the process of hypothesis testing and data analysis.
When data is normally distributed, you can assess probabilities and critical values using the standard normal distribution, often through standardized test scores known as Z-scores. By converting various distributions to fit a standard normal, you eliminate the initial complexity and only deal with the standardized form of the data.

• Z-score: This score tells us how many standard deviations an element is from the mean. It's calculated using:\[Z = \frac{X - \mu}{\sigma}\]with \( \mu \) being the mean and \( \sigma \) the standard deviation of the population.

This is particularly useful as we transition from a Poisson to a normal distribution model in larger samples, allowing us to perform hypothesis testing effectively, like in our semiconductor wafer example.

A Test Statistic is a standardized value that is calculated from sample data during a hypothesis test. It is used to decide whether to reject the null hypothesis. In the problem above, the Z-test statistic calculated from the sample data of the wafer's open circuits is used.
The test statistic provides a means to evaluate the discrepancy between the sample data and the null hypothesis. In our example, \( Z\) was determined using:\[Z = \frac{\bar{X} - \lambda_0}{\sqrt{\frac{\lambda_0}{n}}}\]This particular Z-score formula lets us measure how far our sample mean (\(\bar{X}\)) is from the hypothetical population mean (\(\lambda_0\)) while accounting for sample size (\(n\)).
Once the test statistic is computed, we compare it with a critical value from the standard normal distribution to make a decision regarding the null hypothesis. This comparison helps us conclude if the observed effect is statistically significant, as was done in the open circuits test case.