Independent solution

How to solve this Normal Distribution question

Setup

Setup

Translate the two given cumulative probabilities into standard-normal quantiles.

Φ(0.72)=0.7642\Phi(0.72)=0.7642
Φ(1.32)=0.9066\Phi(1.32)=0.9066

Model

Model

Standardization makes the observed cutoffs affine functions of their z-scores.

2000−μσ=0.72\frac{2000-\mu}{\sigma}=0.72
3000−μσ=1.32\frac{3000-\mu}{\sigma}=1.32

Compute

Compute

Subtract the equations, solve the parameters, and standardize the target.

σ=10001.32−0.72=1666.6667\sigma=\frac{1000}{1.32-0.72}=1666.6667
μ=2000−0.72σ=800\mu=2000-0.72\sigma=800
z1000=1000−8001666.6667=0.12z_{1000}=\frac{1000-800}{1666.6667}=0.12

Answer

Answer

The standard-normal cdf at 0.12 is 0.5478.

Φ(0.12)=0.5478(C)\boxed{\Phi(0.12)=0.5478\quad\text{(C)}}