Researchers are examining whether the presence of a comic‑relief character changes audience scream rates during a horror movie. Among 900 viewers at screenings without a comic‑relief character, 70 screamed during the movie. Among 950 viewers at screenings with a comic‑relief character, 62 screamed.

Definitions:
p1 = proportion of viewers who screamed without a comic‑relief character
p2 = proportion of viewers who screamed with a comic‑relief character

Hypotheses:
H0: p1 = p2 (Comic‑relief characters are not associated with scream rates)
H1: p1 ≠ p2 (Comic‑relief characters are associated with a difference in scream rates)

What is your decision, and what does it mean in the context of the study?

Question

Researchers are examining whether the presence of a comic‑relief character changes audience scream rates during a horror movie. Among 900 viewers at screenings without a comic‑relief character, 70 screamed during the movie. Among 950 viewers at screenings with a comic‑relief character, 62 screamed.

Definitions:
p1 = proportion of viewers who screamed without a comic‑relief character
p2 = proportion of viewers who screamed with a comic‑relief character

Hypotheses:
H0: p1 = p2 (Comic‑relief characters are not associated with scream rates)
H1: p1 ≠ p2 (Comic‑relief characters are associated with a difference in scream rates)

What is your decision, and what does it mean in the context of the study?

BlackTom AI · Accepted Answer

1. Relevant concepts and assumptions
Use a two‑proportion z‑test to compare p1 and p2. Assumptions: independent random samples and large counts so the sampling distribution of the difference in sample proportions is approximately normal (np and n(1−p) both ≥10). Here n1=900, x1=70; n2=950, x2=62, so counts are large enough.

2. Step‑by‑step calculations
- Sample proportions:
  p̂1 = 70/900 = 0.077777... ≈ 0.07778
  p̂2 = 62/950 = 0.065263... ≈ 0.06526
- Observed difference:
  p̂1 − p̂2 = 0.07778 − 0.06526 = 0.01252
- Pooled proportion under H0 (p1 = p2):
  p̂pool = (70 + 62) / (900 + 950) = 132 / 1850 ≈ 0.071351
- Standard error using pooled p:
  SE = sqrt[p̂pool(1 − p̂pool)(1/n1 + 1/n2)]
     = sqrt[0.071351*(0.928649)*(1/900 + 1/950)]
     = sqrt[0.071351*0.928649*(0.00111111 + 0.00105263)]
     = sqrt[0.066293*0.002163743] ≈ sqrt[0.00014349] ≈ 0.01198
- Z statistic:
  z = (p̂1 − p̂2) / SE = 0.01252 / 0.01198 ≈ 1.045
- Two‑sided p‑value:
  p = 2*(1 − Φ(1.045)) ≈ 2*(0.148) ≈ 0.296 ≈ 0.30

3. Why the correct decision is "fail to reject H0"
With p ≈ 0.30 which is much larger than α = 0.05, we do not have sufficient evidence to reject the null hypothesis. The observed difference (~1.25 percentage points) is small and not statistically significant given sampling variability.

4. Why other conclusions would be incorrect
Rejecting H0 would require a much smaller p‑value (≤0.05) or a larger z magnitude; neither is present. Claiming a practical effect without statistical significance risks attributing meaning to random variation.

5. Conclusion
Fail to reject H0; the data are consistent with no association between comic‑relief presence and scream rates. There is not enough evidence to conclude scream rates differ (p‑value ≈ 0.30). This matches the provided answer.

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