Explanation

Given:

• Mean (μ) = 80
• Standard deviation (σ) = 24
• We want P(32 ≤ X ≤ 116)

Standardize the values:

• Lower bound: z₁ = (32 - 80) / 24 = -48 / 24 = -2
• Upper bound: z₂ = (116 - 80) / 24 = 36 / 24 = 1.5

Using standard normal distribution:

• P(Z ≤ 1.5) = 0.9332
• P(Z ≤ -2) = 0.0228
• P(-2 ≤ Z ≤ 1.5) = 0.9332 - 0.0228 = 0.9104

Convert to percentage:

• 0.9104 × 100% = 91.04%

Verification:

• Option D: 91.04% ✓
• This represents the area under the normal curve between 2 standard deviations below the mean and 1.5 standard deviations above the mean
• Approximately 95.44% of data falls within ±2σ, and 86.64% falls within ±1.5σ, so 91.04% is reasonable for the asymmetric interval [-2σ, +1.5σ]