13. Chi-Square Tests & Goodness of Fit
Goodness of Fit Test
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A gym owner wants to know if the gym has similar numbers of members across different age groups. The table shows the distribution of ages for members from a random survey. Find the x2 statistic to test the claim that the gym has equal numbers of members of all age ranges.
A
0.92
B
0.46
C
0.08
D
0.54
Verified step by step guidance1
Step 1: Understand the problem. The gym owner wants to test if the number of members is evenly distributed across different age groups. This is a chi-square goodness-of-fit test, which compares observed frequencies to expected frequencies.
Step 2: Calculate the expected frequency for each age group. Since the claim is that the gym has equal numbers of members across all age groups, divide the total number of members by the number of age groups. Use the formula: Expected Frequency = Total Members / Number of Age Groups.
Step 3: Compute the chi-square statistic using the formula: χ² = Σ((Observed - Expected)² / Expected). For each age group, subtract the expected frequency from the observed frequency, square the result, and divide by the expected frequency.
Step 4: Sum up the values obtained for each age group to get the chi-square statistic. This value represents the overall deviation of the observed frequencies from the expected frequencies.
Step 5: Compare the calculated chi-square statistic to the critical value from the chi-square distribution table (based on the degrees of freedom and significance level) to determine whether to reject or fail to reject the null hypothesis.
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