Independent solution

How to solve this Normal Distribution question

Setup

Setup

Form the prediction error as the difference of the two normal variables.

D=W−TD=W-T

Model

Model

Independence makes the variances add even though the variables are subtracted.

E⁡[D]=0\operatorname{E}[D]=0
Var⁡(D)=4+12=16,D∼N(0,16)\operatorname{Var}(D)=4+12=16,\qquad D\sim N(0,16)

Compute

Compute

Standardize both endpoints of the symmetric interval.

Pr⁡(∣W−T∣<1)=Pr⁡(∣Z∣<14)\Pr(|W-T|<1)=\Pr\left(|Z|<\frac14\right)
=2Φ(0.25)−1=0.1974126514…=2\Phi(0.25)-1=0.1974126514\ldots

Answer

Answer

The probability rounds to 0.20.

0.20(A)\boxed{0.20\quad\text{(A)}}