Answer: ($54.27, $175.91)

## Explanation Given: - Initial price = $80 - Price after one year: P = 80 * exp(i) - i ~ N(μ=0.2, σ=0.3) - Need 95% confidence interval for P **Step 1: Find 95% CI for i** For a normal distribution, the 95% confidence interval is: CI = μ ± Z_{α/2} * σ where α = 0.05 Z_{0.025} = 1.96 CI for i = 0.2 ± 1.96 * 0.3 = 0.2 ± 0.588 = (-0.388, 0.788) **Step 2: Transform to CI for P** P = 80 * exp(i) Lower bound: P_lower = 80 * exp(-0.388) = 80 * 0.6784 ≈ $54.27 Upper bound: P_upper = 80 * exp(0.788) = 80 * 2.1989 ≈ $175.91 **Step 3: Final CI** 95% CI for P = ($54.27, $175.91) **Key Points:** 1. The rate of return i is normally distributed, so we first construct the CI for i 2. We use the exponential transformation to get the CI for the price 3. The correct answer is C: ($54.27, $175.91) **Why other options are incorrect:** - A: Only includes lower bound up to initial price - B: Starts at initial price, missing lower bound below $80 - D: Upper bound is too low ($140 vs correct $175.91)

Author: Nikitesh Somanthe