A gym offers a popular Pilates class (a form of core exercise with postures similar to yoga) at numerous locations each week. 5% of attendees are men. Knowing the exercise to be great for everyone, the gym ran a month long promotion encouraging men to try the class. The next month the marketing director sampled 100 of the classes and found an average of 9% of attendees were men. At 5% significance, can she conclude the promotion worked?

Question

A gym offers a popular Pilates class (a form of core exercise with postures similar to yoga) at numerous locations each week.  5% of attendees are men.  Knowing the exercise to be great for everyone, the gym ran a month long promotion encouraging men to try the class.  The next month the marketing director sampled 100 of the classes and found an average of 9% of attendees were men.  At 5% significance, can she conclude the promotion worked?

BlackTom AI · Accepted Answer

Question restatement: A gym runs a promotion and collects data to test if the share of male attendees increased from 5% to 9% in a sample of 100 classes. At 5% significance, evaluate the claim. Answer options:
1) At 5% significance the promotion is a success by the critical value approach, but not by the p-value approach.
2) At 5% significance the director may conclude from the sample that the promotion was a  success.
3) At 5% significance the promotion is a success by the p-value approach, but not by the critical value approach.
4) At 5% significance the promotion is a success by the critical value approach, but not by the p-value approach.

Option 1 analysis: This option asserts that the conclusion depends on using the critical value method rather than the p-value method, indicating a mismatch between the two. In a standard one‑sample proportion test with p0 = 0.05, n = 100, and observed p̂ = 0.09, you compute the test statistic using either approach. The z statistic for H0: p = 0.05 against p > 0.05 is z = (0.09 − 0.05) / sqrt(0.05 × 0.95 / 100) ≈ 1.835. The critical value at alpha = 0.05 for a one-sided test is zα ≈ 1.645. Since 1.835 > 1.645, the critical-value rule leads to rejecting H0, i.e., concluding a significant increase under that criterion. On the p-value side, the p-value is P(Z > 1.835) ≈ 0.033, which is also less than 0.05, so the p-value approach would also lead to rejecting H0. Thus, this option’s claim that it’s a success by the critical-value approach but not by the p-value approach is inconsistent with the calculation above. The idea that only the critical-value method yields significance while the p-value method does not is not supported by these numbers; both methods agree in this case. Still, the option correctly identifies that a one-sided test under a 5% significance level uses a threshold around 1.645, and exceeding it would lead to rejecting the null hypothesis by that criterion.

Option 2 analysis: This option states that at 5% significance the director may conclude the promotion increased the percentage of male attendees. Given the calculations (z ≈ 1.835, p-value ≈ 0.033), the data provide evidence against H0 in favor of an increase, so the conclusion supporting an increase would be consistent with both the critical-value and p-value approaches. Therefore, this option aligns with the standard interpretation of the test result. A potential subtlety is whether the test is framed as one-sided or two-sided; here the hypothesis is clearly one-sided (increase), so rejecting H0 supports an increase. However, if one insisted on a strict dependence on a single method, this option would be judged by those criteria, but the numerical evidence supports a positive conclusion.

Option 3 analysis: This option claims significance only under the p-value approach and not under the critical-value approach. As shown earlier, z ≈ 1.835 exceeds the one-sided critical value of 1.645, so the critical-value method would also declare significance. Additionally, the p-value (≈ 0.033) is below 0.05. Therefore, this option is inconsistent with the actual calculations, since both methods indicate significance rather than only one.

Option 4 analysis: This option states significance under the critical-value approach but not under the p-value approach, which mirrors the claim in option 1 but with reversed emphasis. As discussed, both methods yield significance in this scenario (critical-value threshold met and p-value below 0.05). Thus this option also mischaracterizes the actual result, though it correctly references the critical-value criterion being involved. The mismatch between the two criteria is not present in the computed values for this data, since both criteria indicate significance.

Summary of option-wise reasoning flow: Option 1 presents a split outcome between methods that does not match the computed z and p-value for these data, where both approaches would indicate significance. Option 2 asserts an increase is supported by the test, which is consistent with rejecting H0 in favor of a higher proportion of men. Option 3 incorrectly separates the methods, since the p-value and critical-value approaches agree in this case. Option 4 similarly splits the methods but with the opposite claim to option 1, which again conflicts with the actual calculations. The core calculations involve the z statistic derived from the observed proportion, the standard error under H0, and the corresponding p-value and critical value for a 5% one-sided test. The numerical results (z ≈ 1.835; p-value ≈ 0.033; critical value ≈ 1.645) determine the agreement or disagreement between methods and the final interpretation.

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