1. **Problem Statement:** We analyze teacher performance using percentiles, z-scores, and stanines based on normally distributed scores with given means and standard deviations. --- ### Part A: Percentile Analysis 1. Calculate percentile ranks for teachers Ama (80), Kwame (56), Abena (74) with $\mu=68$, $\sigma=12$. - Compute z-scores: $z=\frac{X-\mu}{\sigma}$ - Ama: $z=\frac{80-68}{12}=1$ - Kwame: $z=\frac{56-68}{12}=-1$ - Abena: $z=\frac{74-68}{12}=0.5$ - Use standard normal table or calculator: - $P(Z<1)=0.8413$ (84.13 percentile) - $P(Z<-1)=0.1587$ (15.87 percentile) - $P(Z<0.5)=0.6915$ (69.15 percentile) 2. Interpretation of Ama's percentile (84.13): - Ama scored better than approximately 84% of teachers. - GES can identify Ama as a high performer for advanced training or leadership roles. --- ### Part B: Standard Scores and Comparative Analysis 3. Calculate z-scores for Ama (current year: $\mu=68$, $\sigma=12$, score=80) and last year’s teacher (score=82, $\mu=72$, $\sigma=10$): - Ama: $z=\frac{80-68}{12}=1$ - Last year teacher: $z=\frac{82-72}{10}=1$ - Both have the same z-score of 1, so they performed equally well relative to their cohorts. 4. Find minimum raw score for 90th percentile this year ($\mu=68$, $\sigma=12$): - $z$ for 90th percentile is approximately 1.28. - Raw score: $X=\mu + z\sigma = 68 + 1.28 \times 12 = 68 + 15.36 = 83.36$ - Minimum score to qualify: 84 (rounding up). --- ### Part C: Stanine Classification 5. Stanine percentages and score ranges: - Stanine 9: 4% (Very High) - Stanine 8: 7% (High) - Stanine 7: 12% (Above Average) Calculate missing percentages and score ranges: - Stanine distribution percentages: 4%, 7%, 12%, 17%, 20%, 17%, 12%, 7%, 4% (total 100%) - Score ranges correspond to z-score intervals: - Stanine 9: top 4% $z > 1.75$ - Stanine 8: next 7% $1.25 < z \leq 1.75$ - Stanine 7: next 12% $0.75 < z \leq 1.25$ - Stanine 6: next 17% $0.25 < z \leq 0.75$ - Stanine 5: middle 20% $-0.25 < z \leq 0.25$ - Stanine 4: next 17% $-0.75 < z \leq -0.25$ - Stanine 3: next 12% $-1.25 < z \leq -0.75$ - Stanine 2: next 7% $-1.75 < z \leq -1.25$ - Stanine 1: bottom 4% $z \leq -1.75$ - Convert z to raw scores: $X=\mu + z\sigma$ Example for Stanine 8: - $z$ range: 1.25 to 1.75 - Scores: $68 + 1.25 \times 12 = 83$ to $68 + 1.75 \times 12 = 89$ 6. Teacher Abena's score 74: - $z=\frac{74-68}{12}=0.5$ - Falls in Stanine 6 ($0.25 < z \leq 0.75$) - Stanines simplify resource allocation by grouping teachers into performance bands rather than raw scores, enabling targeted interventions. --- ### Task 2: Normal Distribution in Educational Decision-Making #### Part A: Normal Distribution Properties 1. Sketch normal curve with $\mu=65$, $\sigma=15$ showing: - Mean at 65 - Mark $\mu \pm \sigma$ (50, 80), $\mu \pm 2\sigma$ (35, 95), $\mu \pm 3\sigma$ (20, 110) - Percentages: 68% within $\pm 1\sigma$, 95% within $\pm 2\sigma$, 99.7% within $\pm 3\sigma$ 2. Two key features: - Symmetry: simplifies interpretation and comparison. - Empirical rule: allows quick estimation of probabilities and cutoffs. #### Part B: Critical Decision-Making 3. Minimum Competency Standard: (a) Cutoff at 25th percentile: - $z$ for 25th percentile is approximately -0.674 - Cutoff score: $65 + (-0.674) \times 15 = 65 - 10.11 = 54.89$ (b) Percentage needing remedial: 25% - This is a significant portion; may strain resources given teacher shortages. 4. Excellence Recognition (top 10%): - $z$ for 90th percentile is 1.28 - Minimum score: $65 + 1.28 \times 15 = 65 + 19.2 = 84.2$ --- #### Part C: Synthesis and Application 5. (a) Critique of fixed raw score cutoffs: - Ignores cohort variability in difficulty and ability. - May unfairly pass or fail due to distribution shifts. - Does not reflect relative performance or maintain standards over time. (b) Alternative: Use norm-referenced cutoffs based on percentiles or z-scores. - Adjusts for yearly variations. - Ensures fairness by comparing within cohort. - Maintains consistent standards relative to population. --- **Final answers:** - Percentiles: Ama 84.13, Kwame 15.87, Abena 69.15 - Ama and last year teacher z=1, equal relative performance - 90th percentile cutoff this year: 84 - Stanine 8 percentage: 7%, score range approx 83-89 - Abena in Stanine 6 - Remedial cutoff: 54.89, 25% need remedial - Distinction cutoff: 84.2 - Fixed raw cutoffs problematic; norm-referenced better.