Independent solution

How to solve this Normal Distribution question

Setup

Setup

Let Y be the average of the two independent normal measurement errors.

Y=X1+X22Y=\frac{X_1+X_2}{2}

Model

Model

A linear combination of independent normal variables is normal. Divide the sum of their variances by four to obtain the variance of the average.

Var⁡(Y)=(0.0056h)2+(0.0044h)24\operatorname{Var}(Y)=\frac{(0.0056h)^2+(0.0044h)^2}{4}
σY=0.0035609h\sigma_Y=0.0035609h

Compute

Compute

The standard deviation of Y is 0.0035609h. Standardizing the symmetric limits ±0.005h and subtracting the two tails gives a central probability of 0.839723.

Pr⁡(∣Y∣≤0.005h)=2Φ(0.005/0.0035609)−1=0.839723\Pr(|Y|\le0.005h)=2\Phi(0.005/0.0035609)-1=0.839723

Answer

Answer

The probability that the averaged error lies within the stated tolerance rounds to 0.84, which is choice D.

0.84(D)\boxed{0.84\quad\text{(D)}}