Independent solution

How to solve this Linear Combinations of Independent Random Variables question

Setup

Setup

Let S_n be the sum for n independent employees. Add the means and variances before taking the standard deviation.

Sn∼N ⁣(2500n, 7002n)S_n\sim N\!\left(2500n,\,700^2n\right)
E[Sn]=2500n,SD⁡(Sn)=700n\mathbb{E}[S_n]=2500n,\qquad \operatorname{SD}(S_n)=700\sqrt{n}

Model

Model

A normal 90th percentile is the mean plus z_{0.90} standard deviations.

167168=2500n+z0.90(700n),z0.90=1.2815516…167168=2500n+z_{0.90}(700\sqrt{n}),\qquad z_{0.90}=1.2815516\ldots

Compute

Compute

Set x equal to the square root of n and solve the resulting quadratic. The tiny deviation from 64 comes from using the full normal quantile while the target value reflects table rounding.

2500x2+897.0861…x−167168=02500x^2+897.0861\ldots x-167168=0
x=7.9997875…,n=x2=63.9966007…x=7.9997875\ldots,\qquad n=x^2=63.9966007\ldots
Using z0.90=1.28:2500(64)+1.28(700)(8)=167168\text{Using }z_{0.90}=1.28:\quad 2500(64)+1.28(700)(8)=167168

Answer

Answer

The population count must be an integer, and the matching listed value is 64.

n=64(D)\boxed{n=64\quad\text{(D)}}