91Ó°ÊÓ

Hypothesis testing is a statistical method that enables us to make inferences or educated decisions about a population, using sample data. In the context of this exercise, we are testing the mean number of open circuits on semiconductor wafers. The first step in hypothesis testing is to formulate the null and alternative hypotheses. These are structured as follows:

• Null Hypothesis (\(H_0\)): This assumes there is no effect or difference. For our case, \(H_0: \lambda = 2\), indicating that the mean number of open circuits per wafer is 2.
• Alternative Hypothesis (\(H_1\)): This suggests that there is an effect or difference. Here, \(H_1: \lambda > 2\) is posed, suggesting that the mean exceeds 2.

The aim of hypothesis testing is to determine if the sample data provide enough evidence to reject the null hypothesis, in favor of the alternative. This is achieved by examining how likely the observed sample results are, assuming that the null hypothesis is true.
A key part of this process involves choosing a significance level (\(\alpha\)), which defines the probability of rejecting the null hypothesis when it is actually true (Type I error). In this exercise, \(\alpha = 0.05\).

In situations where data follow a Poisson distribution, as in this exercise with the semiconductor wafers, exact calculations can become complex with large sample sizes. Fortunately, when you have a sufficiently large sample, the Central Limit Theorem provides a way to simplify the process by using a normal approximation.
The Poisson distribution is defined by a single parameter (\(\lambda\)), which is both the mean and the variance. When we have a large number of observations, the distribution of the sample mean approaches a normal distribution with mean \(\lambda\) and variance \(\lambda/n\), where \(n\) represents the sample size. This means:

• The estimation of the mean number of opens (\(\bar{X}\)) will also follow a normal distribution.
• The test statistic can then be approximated using a normal distribution, facilitating the use of standard tests like the Z-test.

This approximation is especially useful because it allows us to apply widely known properties of normal distributions, such as calculating Z-values, to make further decisions within this hypothesis testing framework.

The Z-test is a statistical method used to determine whether there is a significant difference between the sample mean and the population mean. It's particularly useful when applying a normal approximation, as seen in this exercise.
The Z-test involves calculating a Z-value, which is an approximation of how far (in standard deviations) the sample mean is from the population mean. The formula used here is:\[ Z = \frac{\bar{X} - \lambda_0}{\sqrt{\lambda_0 / n}} \]
Where:

• \(\bar{X}\): sample mean,
• \(\lambda_0\): hypothesized population mean under \(H_0\),
• \(n\): sample size.

The calculated Z-value is then compared against critical values from the standard normal distribution. These critical values are determined by the chosen significance level (\(\alpha\)).
If the Z-value exceeds the critical value in a one-tailed test, the null hypothesis is rejected. In this exercise, for an \(\alpha\) = 0.05 one-tailed test, the critical Z-value is approximately 1.645. However, the computed Z-value was 1.20, which does not exceed 1.645—instructing us not to reject the null hypothesis.
Ultimately, this step helps confirm whether the observed sample mean is significantly higher than the hypothesized population mean, using the threshold provided by the critical Z-value.