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. Using x² = 0.92 & α = 0.05, test the claim that the gym has equal numbers of members of all age ranges.

• A. Because P-value > α, we REJECT H₀. There is ENOUGH evidence that at the # of members is significantly different for at least one of the age groups at this gym. So the claimed dist. IS NOT a good fit.
• B. Because P-value > α, we FAIL TO REJECT H₀. There is NOT ENOUGH evidence that at the # of members is significantly different for at least one of the age groups at this gym. So the claimed dist. IS a good fit.
• C. Because P-value < α, we REJECT H₀. There is NOT ENOUGH evidence that at the # of members is significantly different for at least one of the age groups at this gym. So the claimed dist. IS NOT a good fit.
• D. Because P-value < α, we FAIL TO REJECT H₀. There is NOT ENOUGH evidence that at the # of members is significantly different for at least one of the age groups at this gym. So the claimed dist. IS a good fit.

Answer: (B)

Verified step by step guidance

1. Step 1: Define the null hypothesis (H0) and the alternative hypothesis (H1). H0: The gym has equal numbers of members across all age groups. H1: The gym does not have equal numbers of members across all age groups.
2. Step 2: Calculate the expected frequency for each age group under the assumption of equal distribution. Divide the total number of members by the number of age groups to find the expected frequency for each group.
3. Step 3: Use the chi-square test formula to calculate the test statistic. The formula is: χ² = Σ((Observed - Expected)² / Expected), where 'Observed' is the actual number of members in each age group and 'Expected' is the calculated frequency from Step 2.
4. Step 4: Compare the calculated chi-square test statistic (χ² = 0.92, as given in the problem) to the critical value from the chi-square distribution table at α = 0.05 and degrees of freedom = (number of age groups - 1).
5. Step 5: Determine the P-value and compare it to α = 0.05. If P-value > α, fail to reject H0, meaning there is not enough evidence to conclude that the distribution of members is significantly different across age groups. If P-value < α, reject H0.