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In statistics, hypothesis testing is a method used to make decisions or inferences about a population based on a sample of data. The main goal here is to determine whether there is enough evidence to reject a null hypothesis, which assumes there is no effect or association between the variables in question.
When performing a hypothesis test, you generally start with two hypotheses:

• The null hypothesis ( H_0 ), which in this case, states that there is no association between vaccination rates and race. This is what you assume to be true until proven otherwise.
• The alternative hypothesis ( H_a ), which suggests that there is an association. It's what you want to prove is true.

The hypothesis test involves collecting data and determining the probability of observing that data if the null hypothesis were true. If this probability is sufficiently low, you reject the null hypothesis. This process helps researchers understand if their findings reflect true effects rather than random variation.

The significance level, often denoted as \\( \alpha \)\, is a critical component of hypothesis testing. It represents the probability of rejecting the null hypothesis when it is actually true, also known as making a Type I error. Common significance levels are 0.05, 0.01, and 0.10.
In this exercise, a significance level of 0.05 was used. This means there's a 5% risk that the results are due to random chance rather than a true effect. A lower significance level would mean you're being more conservative, requiring stronger evidence to reject the null hypothesis.This threshold acts as a decision rule and helps statisticians and researchers maintain control over the likelihood of drawing incorrect conclusions from their studies. Moreover, it provides a clear guideline for interpreting results, particularly when comparing observed statistical measures to critical values from appropriate distributions.

Expected frequencies play a vital role in the Chi-Square Test. They are calculations of how the data might look if the null hypothesis were true. In other words, they are the frequencies you would expect to find in each cell of a contingency table if there were no association between the variables.
The formula for calculating expected frequency is: \\( E = \frac{(\text{Row Total} \times \text{Column Total})}{\text{Sample Size}} \).
In our solution, the expected frequencies for different cells such as 'AAPI, Yes', and 'White, No' were calculated. This involves using the row and column totals of the table, along with the overall sample size, which gives you a theoretical model to compare against the actual observed data.These expected frequencies are crucial for computing the Chi-Square statistic, as they serve as a benchmark to assess the differences between the observed and expected outcomes. The closer the observed frequencies are to the expected frequencies, the less likely it is that the variables are associated.

Observed frequencies refer to the actual counts you get from your data collection. They represent the reality captured in your study—what really happened in your sample. In our problem, these are the numbers of vaccinated and non-vaccinated individuals belonging to different racial groups.
In a Chi-Square Test, observed frequencies are compared against expected frequencies to see if there are significant differences. For instance, in the exercise, '136 AAPI who completed vaccinations' and '1170 White who completed vaccinations' are examples of observed frequencies.
The core idea is to determine if the observed distribution of frequencies differs significantly from what was expected under the null hypothesis. By examining this difference, you can discern whether the association between the categories is statistically significant or if it's likely due to chance.The sum of the squared differences between observed and expected frequencies divided by the expected frequencies gives the Chi-Square statistic \\( X^2 = \sum \frac{(O-E)^2}{E} \). This measure helps statistically validate or refute the initial hypothesis assumptions.