Which of The Following Is Closest to The Calculated Chi-Square

What is Chi-Square?

The chi-square (χ²) test is a statistical method used to examine the differences between categorical variables in one or more populations. It's commonly used in hypothesis testing to determine whether there's a significant association between two categorical variables.

The chi-square statistic measures the discrepancy between the observed frequencies in the data and the expected frequencies under a specified hypothetical model. A higher chi-square value indicates a greater discrepancy between observed and expected values.

How to Calculate Chi-Square

The basic formula for calculating the chi-square statistic is:

To calculate chi-square:

1. Determine the observed frequencies from your data
2. Calculate the expected frequencies under your null hypothesis
3. For each category, subtract the expected frequency from the observed frequency, square the result, and divide by the expected frequency
4. Sum these values across all categories to get the chi-square statistic

For large samples, the chi-square statistic follows a chi-square distribution with degrees of freedom equal to (number of categories - 1).

Interpreting Chi-Square Results

After calculating your chi-square statistic, you'll want to compare it to critical values from a chi-square distribution table to determine statistical significance. The critical value depends on:

• The degrees of freedom (df) of your test
• The significance level (α) you've chosen (typically 0.05 or 0.01)

If your calculated chi-square value is greater than the critical value, you reject the null hypothesis and conclude there's a statistically significant association between the variables. If it's less than the critical value, you fail to reject the null hypothesis.

Example Calculation

Let's look at an example to see how to determine which value is closest to the calculated chi-square.

Suppose you're testing whether there's a relationship between eye color and gender in a sample of 100 people. Your observed frequencies are:

Your expected frequencies would be calculated based on the overall proportions of each category. For example, the expected number of blue-eyed males would be (25/90) × 50 ≈ 13.89.

After performing the calculations, you find your chi-square statistic is approximately 3.45. You look up the chi-square distribution table with df = 2 (since there are 3 eye colors × 2 genders - 1 = 5 categories, but typically we use (rows-1) × (columns-1) = 2 × 1 = 2 for this type of test) and α = 0.05, and find the critical value is 5.99.

Since 3.45 is less than 5.99, you fail to reject the null hypothesis and conclude there's no significant association between eye color and gender in this sample.

Common Mistakes

When determining which value is closest to the calculated chi-square, be aware of these common pitfalls:

• Using the wrong degrees of freedom - remember it's (number of categories - 1) for a goodness-of-fit test or (rows-1) × (columns-1) for a test of independence
• Misinterpreting the critical value - it's the threshold for significance, not the exact value your statistic should match
• Ignoring sample size requirements - chi-square tests work best with larger samples (typically n > 20 or 30)
• Assuming a significant result means the null hypothesis is false - it only means you have enough evidence to reject it