Understanding Chi-Squared Tests

Chi-squared tests are helpful tools that A-Level students can use to see how well their data matches what they expect to see. This is really important in two main ways: the Goodness of Fit test and when looking at contingency tables. These tests help us find out if there is a big difference between what we expected and what we actually observed in the data we collected.

What is a Chi-Squared Test?

At the heart of the chi-squared test is a comparison. We look at what we saw in our data and what we would expect to see if a certain idea (called the null hypothesis) was true.

In a Goodness of Fit test, the null hypothesis usually says that the data we saw follows a specific pattern, like a normal or binomial distribution.

Here’s how the process works in simple steps:

1. Write Down the Hypotheses:

   • Null Hypothesis (H₀): The data we observed match the expected pattern.
   • Alternative Hypothesis (Hₐ): The data we observed do not match the expected pattern.

2. Collect Your Data: Start by gathering the data you need, which should be in categories.

3. Find Expected Frequencies: Based on the null hypothesis, figure out what counts you would expect in each category. For example, if you roll a six-sided die 60 times, you would expect each number (1 to 6) to appear about 10 times if the die is fair.

4. Calculate the Chi-Squared Statistic: Use this formula:

   $$\chi^2 = \sum \frac{(O - E)^2}{E}$$

   Here, (O) is the number you observed, and (E) is the number you expected. You do this for all the categories.

5. Determine Degrees of Freedom: For a Goodness of Fit test, you find the degrees of freedom (df) by subtracting 1 from the number of categories: (df = k - 1), where (k) is the number of categories.

6. Look at the Chi-Squared Distribution: Using the chi-squared value you calculated and the degrees of freedom, you look it up in a chi-squared distribution table to find the critical value. This tells you what number you need to reach to say your results are significant.

7. Make a Decision:

   • If your chi-squared value is bigger than the critical value, you reject the null hypothesis.
   • If it is not, you fail to reject the null hypothesis.

Example: Goodness of Fit Test

Let’s say you want to check if a six-sided die is fair. You roll it 60 times and record the results:

• 1: 12
• 2: 8
• 3: 10
• 4: 7
• 5: 11
• 6: 12

If the die were fair, you would expect to see each number about 10 times (because (60 / 6 = 10)).

Now, let’s calculate the chi-squared statistic:

$$\chi^2 = \frac{(12-10)^2}{10} + \frac{(8-10)^2}{10} + \frac{(10-10)^2}{10} + \frac{(7-10)^2}{10} + \frac{(11-10)^2}{10} + \frac{(12-10)^2}{10}$$

Doing the math gives you (\chi^2 \approx 2.8). You have 5 degrees of freedom (because (6 - 1 = 5)).

Now, you check a chi-squared distribution table to find the critical value (around (11.07) at a significance level of (0.05)). Since (2.8 < 11.07), you do not reject the null hypothesis, suggesting that the die is likely fair.

Where Chi-Squared Tests are Used

Besides checking if data fits a pattern, chi-squared tests are also great for looking at contingency tables. These tables help us understand the relationship between two categories. For example, you might explore how people of different ages prefer tea or coffee based on their survey answers.

In summary, chi-squared tests give a clear way to analyze categorical data. They help students build a strong foundation in understanding and interpreting statistics. Happy studying!