Independent solution

How to solve this Normal Distribution question

Setup

Setup

Standardize the reported percentile to estimate the standard deviation of the raw-score distribution.

z0.80=0.841621…z_{0.80}=0.841621\ldots
σX=24.20−20z0.80=4.99037…\sigma_X=\frac{24.20-20}{z_{0.80}}=4.99037\ldots

Model

Model

Use the integer-variance condition and the moment rules for a positive affine transformation.

Var⁡(X)=25\operatorname{Var}(X)=25
Var⁡(Y)=a2Var⁡(X)=25\operatorname{Var}(Y)=a^2\operatorname{Var}(X)=25
a>0⟹a=1a>0\quad\Longrightarrow\quad a=1

Compute

Compute

Translate the stated relationship between the input variance and output expectation, then solve the affine mean equation.

Var⁡(X)=12E[Y]⟹E[Y]=50\operatorname{Var}(X)=\frac12\mathbb{E}[Y]\quad\Longrightarrow\quad\mathbb{E}[Y]=50
50=E[aX+b]=aE[X]+b=20+b50=\mathbb{E}[aX+b]=a\mathbb{E}[X]+b=20+b
b=30b=30

Answer

Answer

The intercept is the positive value listed under choice D.

b=30(D)\boxed{b=30\quad\text{(D)}}