Independent solution

How to solve this Normal Distribution question

Setup

Setup

The variance is 16, so the standard deviation is 4. Write the centered interval as the mean plus or minus a multiple of this standard deviation.

σ=16=4\sigma=\sqrt{16}=4
I=(20−4z, 20+4z)I=(20-4z,\,20+4z)

Model

Model

Symmetry splits the remaining probability equally between the two tails. Therefore the upper standardized endpoint has cumulative probability 0.5+0.25/2.

Pr⁡(−z<Z<z)=0.25\Pr(-z<Z<z)=0.25
Φ(z)=0.625\Phi(z)=0.625

Compute

Compute

Find the standard-normal quantile, rescale it, and add and subtract the resulting half-width from the mean.

z=Φ−1(0.625)=0.318639364…z=\Phi^{-1}(0.625)=0.318639364\ldots
4z=1.274557456…4z=1.274557456\ldots
I=(18.7254425…, 21.2745575…)I=(18.7254425\ldots,\,21.2745575\ldots)

Answer

Answer

The listed endpoints that match this centered interval are 18.7 and 21.3.

(18.7, 21.3)(E)\boxed{(18.7,\,21.3)\quad\text{(E)}}