27. The distribution of marks scored by a large class in an exam can be represented as a normal
distribution with means 𝜇 and standard deviation 𝜎. In a follow–up exam in the same class, everyone
scored 2 marks more than their respective score in the earlier exam. For this follow–up exam, the
distribution of marks can be represented as a normal distribution with means 𝜇2 and standard deviation
𝜎2. Which one of the following is correct?
(a) 𝜇 = 𝜇2; 𝜎 > 𝜎2. (b) 𝜇 < 𝜇2; 𝜎 > 𝜎2.
(c) 𝜇 > 𝜇2; 𝜎 < 𝜎2. (d) 𝜇 < 𝜇2; 𝜎 = 𝜎2.
Adding 2 marks to every student’s score in a normally distributed exam shifts the mean rightward by exactly 2 while the spread, measured by standard deviation, remains identical. This preserves the bell-shaped curve’s shape, just translated horizontally. The correct choice is (d) μ < μ₂; σ = σ₂.
Core Properties
Normal distributions depend solely on mean μ (center) and standard deviation σ (spread). Shifting all values by +2 adds directly to μ, yielding μ₂ = μ + 2, so μ < μ₂. Variance σ²—and thus σ—stays constant since relative differences between scores are unchanged.
Option Breakdown
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(a) μ = μ₂; σ > σ₂: Wrong—mean must increase; spread unchanged, not reduced.
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(b) μ < μ₂; σ > σ₂: Mean shift correct, but σ = σ₂; no compression occurs.
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(c) μ > μ₂; σ < σ₂: Mean direction reversed; spread unaffected.
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(d) μ < μ₂; σ = σ₂: Matches exactly—universal +2 shift boosts center, keeps variability.
Visual Insight
Imagine the original curve centered at μ with tails extending ±σ. Post-shift, it centers at μ+2, tails still ±σ wide—68% within 1σ, 95% within 2σ of new mean. Mathematically, if X ~ N(μ, σ²), then X+2 ~ N(μ+2, σ²).
