To do a Chi-Square Test for Goodness-of-Fit, just follow these simple steps:

1. Develop Your Hypotheses

• Null Hypothesis ($H_0$): This means that the results you see are what you expected to see.
• Alternative Hypothesis ($H_a$): This means that the results you see are different from what you expected.

2. Gather Your Data

Choose a way to collect your data:

• Random Sampling: Everyone has the same chance to be picked. This reduces bias (unfairness).
• Stratified Sampling: Split your group into smaller parts and take samples from each. This helps in getting a good mix.
• Systematic Sampling: Pick every n-th person from a list. This means you’re choosing with regular spacing.

3. Find Expected Frequencies

Here’s the formula you’ll use: $E_i = n \cdot p_i$

• $E_i$ is the expected frequency for a category.
• $n$ is the total number of observations (how many times you looked).
• $p_i$ is the chance (probability) of that category happening.

4. Calculate the Chi-Square Statistic

Use this formula: $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$

• $O_i$ is what you actually observed.
• $E_i$ is what you expected to see.

5. Find the Degrees of Freedom

You can calculate degrees of freedom (df) like this: $df = k - 1$

• $k$ is the number of categories you have.

6. Understand Your Results

• Check your calculated $\chi^2$ value against a critical value from a table, using a common significance level (like $\alpha = 0.05$).
• If your calculated value is greater than the critical value, that means you can reject $H_0$.

7. Make Your Conclusions

Decide if there is enough evidence to say that the results you observed are different from what you expected.