A team of researchers is investigating the potential impact of anchoring bias in judicial sentencing. They hypothesize that the sentence length requested by prosecutors may serve as an anchor, influencing judges’ final sentencing decisions.
The researchers randomly selected 300 similar cases of first-time offenders convicted of grand theft. They divided these cases into three groups:
Group A (100 cases): Prosecutors requested a 3-year sentence.
Group B (100 cases): Prosecutors requested a 7-year sentence.
Group C (100 cases): Prosecutors made no specific sentence request.
Results:
Group A: Mean sentence = 2.8 years, Standard deviation = 0.6 years
Group B: Mean sentence = 5.2 years, Standard deviation = 0.9 years
Group C: Mean sentence = 3.5 years, Standard deviation = 1.2 years
1. Conduct a one-way ANOVA to determine if there are statistically significant differences in sentencing between the three groups. Use a significance level of 0.05.
2. Calculate the effect size (η² – eta squared) for the ANOVA result. Interpret the magnitude of the effect.
3. Perform post-hoc t-tests with Bonferroni correction to identify which specific groups differ significantly from each other.
4. Calculate Cohen’s d for the difference between Group A and Group B. Interpret the magnitude of this effect.
5. Construct 95% confidence intervals for the mean sentence in each group.
6. If the typical sentence range for grand theft is 1-10 years, calculate the probability of a sentence exceeding 8 years for each group, assuming normal distribution.
7. Discuss the potential implications of these findings for the justice system, considering both statistical significance and practical significance.
8. Propose a follow-up study to further investigate anchoring effects in judicial decision-making. Explain how this study would address any limitations of the current research.
1. One-way ANOVA:
H₀: μA = μB = μC
H₁: At least one group mean is different
Calculate:
SSB (Sum of Squares Between) = Σnᵢ(x̄ᵢ – x̄)² = 100[(2.8-3.83)² + (5.2-3.83)² + (3.5-3.83)²] = 307.23
SSW (Sum of Squares Within) = Σ(nᵢ-1)sᵢ² = 99(0.6² + 0.9² + 1.2²) = 250.47
SST (Sum of Squares Total) = SSB + SSW = 557.7
df(between) = 2, df(within) = 297, df(total) = 299
MSB = SSB/df(between) = 307.23/2 = 153.615
MSW = SSW/df(within) = 250.47/297 = 0.843
F = MSB/MSW = 153.615/0.843 = 182.22
Critical F(2,297) at α=0.05 ≈ 3.03
Since 182.22 > 3.03, we reject H₀. There are statistically significant differences between groups.
2. Effect size (η²):
η² = SSB/SST = 307.23/557.7 = 0.551
This indicates a large effect size, with about 55.1% of the variance in sentence length explained by the prosecutor’s request.
3. Post-hoc t-tests with Bonferroni correction:
New α = 0.05/3 = 0.0167 for each test
A vs B: t = (5.2-2.8)/√(0.6²/100 + 0.9²/100) = 22.36, p < 0.0001
A vs C: t = (3.5-2.8)/√(0.6²/100 + 1.2²/100) = 5.22, p < 0.0001
B vs C: t = (5.2-3.5)/√(0.9²/100 + 1.2²/100) = 11.57, p < 0.0001
All pairs are significantly different.
4. Cohen’s d for Group A vs B:
d = (5.2-2.8)/√((0.6² + 0.9²)/2) = 3.16
This indicates a very large effect size.
5. 95% Confidence Intervals:
Group A: 2.8 ± 1.96(0.6/√100) = (2.68, 2.92)
Group B: 5.2 ± 1.96(0.9/√100) = (5.02, 5.38)
Group C: 3.5 ± 1.96(1.2/√100) = (3.26, 3.74)
6. Probability of sentence > 8 years:
Group A: z = (8-2.8)/0.6 = 8.67, P(z>8.67) ≈ 0
Group B: z = (8-5.2)/0.9 = 3.11, P(z>3.11) ≈ 0.0009
Group C: z = (8-3.5)/1.2 = 3.75, P(z>3.75) ≈ 0.0001
7. Implications:
The results suggest a strong anchoring effect in judicial sentencing. The prosecutor’s request significantly influences the final sentence, potentially compromising fairness and consistency in the justice system. While statistically significant, the practical significance is also substantial, with mean differences of several years in sentencing based solely on the prosecutor’s initial request.
8. Follow-up study:
Propose a longitudinal study examining how judges’ sentencing patterns change over time with exposure to different anchors. This could involve tracking individual judges’ decisions across various cases and prosecutor requests, controlling for case specifics and judge characteristics. This approach would address limitations of the current study by accounting for individual differences and potential learning effects over time.
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