Independent solution

How to solve this Normal Distribution question

Setup

Setup

A normal distribution is symmetric and unimodal, so its mode identifies its mean. Let sigma denote its unknown standard deviation.

μ=56\mu=56
Φ−1(0.40)=−0.2533471031…\Phi^{-1}(0.40)=-0.2533471031\ldots

Model

Model

Standardize the known lower quantile to determine the scale.

52.20−56σ=−0.2533471031…\frac{52.20-56}{\sigma}=-0.2533471031\ldots
σ=14.9991847271…\sigma=14.9991847271\ldots

Compute

Compute

Convert the target score to a z-score and evaluate the standard-normal cumulative distribution.

z=65.50−5614.9991847271=0.6333677578…z=\frac{65.50-56}{14.9991847271}=0.6333677578\ldots
Φ(z)=0.7367532425…\Phi(z)=0.7367532425\ldots

Answer

Answer

A cumulative probability of about 0.737 corresponds to the 74th percentile.

74th percentile(B)\boxed{\text{74th percentile}\quad\text{(B)}}