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The Chi-square test is a statistical method used to determine whether there is a significant difference between the expected and observed frequencies in categorical data.
In the case of the church raffle, we set up a chi-square test to verify if the selection of winning tickets was indeed random.
To perform the test, we first divide our data into categories. Here, we place all 50 raffle numbers into two bins: numbers less than or equal to 60, and numbers greater than 60.
This categorization helps us compare observed frequencies with expected ones, assuming a fair raffle.

Once categories are defined, we calculate the chi-square statistic using the formula: \[\chi^2 = \sum \frac{(O - E)^2}{E}\]where \(O\) represents observed frequencies and \(E\) represents expected frequencies.
For instance, if we find 30 numbers in one category and 20 in the other, while expecting 25 in each, we input these values into the formula to compute the chi-square statistic.
The chi-square test is crucial for evaluating the fairness of random draws and other similar experiments.

Randomness refers to the unpredictability and lack of pattern in the selection process.
In the context of the raffle, it signifies that each ticket had an equal chance of being picked.
If the raffle's results show unexpected patterns, such as clustering of numbers around specific areas, it may suggest some bias or error in the drawing process.

Random draws imply that the process is free from human intervention aiming to manipulate outcomes.
This is a cornerstone in ensuring fairness and integrity.

• Unbias: Each ticket number should be independent of others.
• Equity: All numbers have equal chances.
• Consistency: Results should reflect a diverse spread across possible outcomes.

To test for randomness in our exercise, we use a chi-square goodness-of-fit test.
If the test indicates a departure from expected randomness, it suggests potential issues like mechanical or procedural biases during the draw.

The significance level, often denoted as \(\alpha\), is a threshold for determining whether to reject the null hypothesis.
It's a way to measure the strength of the evidence against the null hypothesis.
In our raffle problem, we set the significance level at 0.05, a common choice for statistical tests.

Setting \(\alpha = 0.05\) means that we are willing to accept a 5% probability that we will mistakenly reject the null hypothesis when it is actually true.
This is called a Type I error.
This level balances the risk of being too lenient or too strict, making it a preferred choice in hypothesis testing for moderate assurance.

• If the calculated chi-square statistic exceeds the critical value at this significance level, we reject the null hypothesis.
• If it is below, we do not reject the hypothesis, suggesting adequate randomness in the draw.

It's important to consider this threshold carefully, as it impacts the decision-making process across various statistical tests.