The distribution of IQ scores of 12-year-olds is normal with a mean of 100 and a standard deviation of 16. The distribution of IQ scores for 20-year-olds is also unimodal and symmetric but with a mean of 120 and a standard deviation of 20. (a) Sketch the distribution of IQ score for 12-year-olds. Be sure to use the correct shape, center, and spread. [15 pts] 0 20 40 60 80 100 120 140 160 180 200 IQ Scores (b) Jessica is a 12-year-old with an IQ of 124. Jessica’s 20-year-old brother, Mark, has an IQ score of 155. Relative to their respective age groups, who had the higher IQ score? Show all your work clearly and justify your answer with a complete sentence. 20 pts Tommy is 20 years old and has a z-score of 1.2. Explain what this score means in this context and find Tommy’s IQ score? [15 pts
The Correct Answer and Explanation is:
(a) Sketch of the Distribution for 12-Year-Olds’ IQ Scores
The distribution of IQ scores for 12-year-olds is normal, which means it has a bell-shaped curve. The center of the distribution is at a mean of 100, and the spread is described by a standard deviation of 16. This means:
- Approximately 68% of the data lies between 84 and 116 (100 ± 16)
- About 95% lies between 68 and 132 (100 ± 2×16)
- Nearly all scores lie between 52 and 148 (100 ± 3×16)
The x-axis (IQ Scores) should be labeled from 0 to 200, with tick marks highlighting the mean (100) and intervals of 16 above and below the mean (84, 116, etc.).
(b) Comparing Jessica and Mark’s IQ Scores Relative to Their Groups
To compare IQ scores across different age groups with different means and standard deviations, we compute z-scores:
Jessica (12 years old)
- Mean = 100
- Standard deviation = 16
- IQ = 124
z=124−10016=2416=1.5z = \frac{124 – 100}{16} = \frac{24}{16} = 1.5z=16124−100=1624=1.5
Mark (20 years old)
- Mean = 120
- Standard deviation = 20
- IQ = 155
z=155−12020=3520=1.75z = \frac{155 – 120}{20} = \frac{35}{20} = 1.75z=20155−120=2035=1.75
Conclusion: Mark has a z-score of 1.75 while Jessica has a z-score of 1.5. This means Mark’s IQ is further above average compared to others in his age group than Jessica’s IQ is in hers. Therefore, Mark had the higher IQ score relative to his age group.
(c) Tommy’s Z-score and IQ Score
Tommy is 20 years old and has a z-score of 1.2. This means that his IQ score is 1.2 standard deviations above the mean for 20-year-olds.
- Mean = 120
- Standard deviation = 20
- z = 1.2
IQ=μ+z⋅σ=120+(1.2⋅20)=120+24=144\text{IQ} = \mu + z \cdot \sigma = 120 + (1.2 \cdot 20) = 120 + 24 = 144IQ=μ+z⋅σ=120+(1.2⋅20)=120+24=144
Interpretation: Tommy’s IQ is 144, which means his IQ is 1.2 standard deviations above the average IQ of 20-year-olds, indicating he performs better than most people in his age group.
