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: Formulate 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. Since the gym owner claims equal distribution, divide the total number of members by the number of age groups to find the expected frequency for each group. Use the formula: $E=\frac{N}{k}$, where N is the total number of members and k is the number of age groups.
3. Step 3: Compute the chi-square statistic using the formula: $x^2=\sum _i^k\frac{{(}^O}{E}$, where O is the observed frequency and E is the expected frequency for each age group.
4. Step 4: Compare the calculated chi-square statistic to the critical value from the chi-square distribution table at the given significance level (α = 0.05) and degrees of freedom (df = k - 1, where k is the number of age groups). Alternatively, use the provided P-value (P = 0.92) for comparison.
5. Step 5: Make a decision based on the comparison. If P-value > α, fail to reject H0, meaning there is not enough evidence to suggest the gym has unequal numbers of members across age groups. If P-value < α, reject H0, meaning there is enough evidence to suggest the gym has unequal numbers of members across age groups.