A fitness center claims that 54% of its customers prefer being able to work out before 7 a.m. An employee at this fitness center believes this claimed percentage is too low. Data is subsequently gathered from a random sample of customers, and upon conducting a hypothesis test, a P-value of 0.04 is reported. If the significance level for this hypothesis test was set at 0.05, which one of the following conclusions is most accurate?

Question

A fitness center claims that 54% of its customers prefer being able to work out before 7 a.m.  An employee at this fitness center believes this claimed percentage is too low.  Data is subsequently gathered from a random sample of customers, and upon conducting a hypothesis test, a P-value of 0.04 is reported.  If the significance level for this hypothesis test was set at 0.05, which one of the following conclusions is most accurate?

BlackTom AI · Accepted Answer

Because the P-value (0.04) is less than the significance level (0.05), we reject the null hypothesis H0: p = 0.54 in favor of the employee’s one-sided alternative H1: p > 0.54. 1. Relevant knowledge points and concepts - Hypothesis test for a population proportion: set H0 (null) and H1 (alternative). Here H0: p = 0.54 and H1: p > 0.54 (employee believes the true proportion is larger). - P-value: probability, assuming H0 is true, of obtaining a sample statistic at least as extreme as the observed one. - Significance level (α): threshold for rejecting H0. If P-value ≤ α, reject H0; if P-value > α, fail to reject H0. - Statistical significance does not imply absolute proof; it indicates evidence against H0. 2. Step-by-step reasoning - State hypotheses: H0: p = 0.54, H1: p > 0.54. - Compare P-value to α: observed P-value = 0.04 and α = 0.05. - Since 0.04 < 0.05, the result is statistically significant at the 5% level, so reject H0 in favor of H1. 3. Calculations and intermediate steps - Numerical comparison: 0.04 < 0.05 → decision rule triggers rejection of H0. - Interpretation of P-value: P(observed or more extreme data | p = 0.54) = 0.04. 4. Why the correct answer is correct - Rejecting H0 indicates the sample provides statistically significant evidence that the true proportion exceeds 0.54. That conclusion matches the employee’s belief and is properly stated as the sample statistic being consistent with the employee’s theory (i.e., evidence supporting p > 0.54). 5. Why the other choices are incorrect - Option claiming “proven” is wrong: statistical tests give evidence, not proof. - Option stating the sample percentage ended up smaller than 54% misinterprets the P-value; P-value does not indicate direction of the sample proportion by itself. - Option claiming a 1% difference between claimed and sample percentage misreads the P-value as a difference in percentages. - Option about “0.04 lies far below 1” is meaningless and incorrectly interprets magnitude of evidence. 6. Conclusion The correct conclusion is that the resulting sample statistic is consistent with the employee’s theory that more than 54% prefer working out before 7 a.m., which matches the provided answer.

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