Question A company is offering $401 \mathrm{k}$ matching for its employees who stay with the company for more than 10 years. The company's CFO finds that the average retirement account holds $\$ 490,000$, with a standard deviation of $\$ 55,000$, distributed normally. What is the average amount of money that separates the lowest $20 \%$ of the means of retirement accounts from the highest $80 \%$ in a sampling of 80 employees? Use the $z$-table below: \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|} \hline$z$ & 0.00 & 0.01 & 0.02 & 0.03 & 0.04 & 0.05 & 0.06 & 0.07 & 0.08 & 0.09 \\ \hline-0.8 & 0.212 & 0.209 & 0.206 & 0.203 & 0.201 & 0.198 & 0.195 & 0.192 & 0.189 & 0.187 \\ \hline-0.7 & 0.242 & 0.239 & 0.236 & 0.233 & 0.230 & 0.227 & 0.224 & 0.221 & 0.218 & 0.215 \\ \hline-0.6 & 0.274 & 0.271 & 0.268 & 0.264 & 0.261 & 0.258 & 0.255 & 0.251 & 0.248 & 0.245 \\ \hline-0.5 & 0.309 & 0.305 & 0.302 & 0.298 & 0.295 & 0.291 & 0.288 & 0.284 & 0.281 & 0.278 \\ \hline-0.4 & 0.345 & 0.341 & 0.337 & 0.334 & 0.330 & 0.326 & 0.323 & 0.319 & 0.316 & 0.312 \\ \hline-0.3 & 0.382 & 0.378 & 0.374 & 0.371 & 0.367 & 0.363 & 0.359 & 0.356 & 0.352 & 0.348 \\ \hline \end{tabular} Round the $z$-score and $\bar{x}$ to two decimal places. Provide your answer below:
Solution
Step 1 ：Find the z-score that corresponds to the lowest 20% of the distribution. From the z-table, this z-score is approximately -0.84.
Step 2 ：Use the z-score to find the average amount of money that separates the lowest 20% of the means of retirement accounts from the highest 80%. The formula for the z-score is \(z = \frac{X - \mu}{\sigma}\), where \(z\) is the z-score, \(X\) is the value from the dataset, \(\mu\) is the mean of the dataset, and \(\sigma\) is the standard deviation of the dataset.
Step 3 ：Rearrange the formula to solve for \(X\): \(X = z\sigma + \mu\).
Step 4 ：Substitute the values into the formula: \(X = -0.84 * 55000 + 490000\).
Step 5 ：Calculate \(X\): \(X = -46200 + 490000\).
Step 6 ：\(\boxed{X = 443800}\). So, the average amount of money that separates the lowest 20% of the means of retirement accounts from the highest 80% is approximately $443,800.
