A movie critic conducts a hypothesis test because he has a theory that the proportion of adults who enjoy scary movies falls well below 0.45. After surveying a random sample of adults and conducting a hypothesis test, the movie critic reports a P-value of 0.01. Assuming the movie critic was using a sample size that was large enough to meet hypothesis testing conditions and that no calculation mistakes were made, what can we conclude based on this information?

Question

A movie critic conducts a hypothesis test because he has a theory that the proportion of adults who enjoy scary movies falls well below 0.45.   After surveying a random sample of adults and conducting a hypothesis test, the movie critic reports a P-value of 0.01.  Assuming the movie critic was using a sample size that was large enough to meet hypothesis testing conditions and that no calculation mistakes were made, what can we conclude based on this information?

BlackTom AI · Accepted Answer

A P-value of 0.01 means the observed sample result (or something more extreme in the direction of the alternative) has probability 0.01 under the null hypothesis. For a one-sided lower-tail test (Ha: p < 0.45), that corresponds to a z-score near -2.33, which is between two and three standard errors below the null proportion.

1) Relevant concepts
- P-value: probability, assuming H0 is true, of observing the test statistic as extreme as the sample result in the direction of Ha.
- For large samples, the sampling distribution of the sample proportion is approximately normal; the standardized test statistic is z = (p̂ − p0)/SE.
- Standard normal quantiles: P(Z ≤ z) = 0.01 → z ≈ −2.3263.

2) Step-by-step reasoning
- Null: p = 0.45. Alternative: p < 0.45 (one-sided lower tail).
- P-value = 0.01 means P(Z ≤ zobs) = 0.01 under H0.
- Use inverse-normal: zobs = Φ^(-1)(0.01) ≈ −2.3263.

3) Intermediate calculation
- zobs ≈ −2.3263.
- Relationship: zobs = (p̂ − 0.45)/SE, so (p̂ − 0.45) ≈ −2.3263 × SE.
- Thus the sample proportion p̂ lies about 2.33 standard errors below null proportion 0.45.

4) Why the correct answer is correct
- The statement "The sample statistic is between two and three standard errors away from 0.45." precisely captures that the standardized test statistic is about −2.33 (between −2 and −3), so the sample statistic is 2–3 SEs away from 0.45 in the direction of Ha.

5) Why the other choices are incorrect
- “There must be a large difference…”: False — a small difference can produce a small P-value if the sample size is large; P-value combines effect size and sample size.
- “If α is set at 0.01, the critic should fail to reject…”: Typically one rejects when p ≤ α; here p = 0.01 would lead to rejection (or at least be considered borderline), so the option’s statement is incorrect.
- “To obtain such a small P-value, the critic would need to have sampled well over 100 adults.”: Not deducible; required sample size depends on the true effect size, so this statement is not necessarily true.
- “None of the above”: incorrect because the first choice is correct.

6) Conclusion
The correct conclusion is that the sample statistic is between two and three standard errors away from 0.45, which matches the provided answer.

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