The following are the P/E ratios (price of stock divided by projected earnings per share) for 19 banks.
$$29,31,22,18,18,21,19,21,20,14,17,23,22,18,38,20,29,34,22$$

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Find ${25}^{\text{th }}$ and ${80}^{\text{th }}$ percentiles for these ratios.
(a) The ${25}^{\text{th }}$ percentile:
(b) The ${80}^{\text{th }}$ percentile:

Step 1 ：Sort the P/E ratios in ascending order: $[14,17,18,18,18,19,20,20,21,21,22,22,22,23,29,29,31,34,38]$

Step 2 ：Calculate the position of the ${25}^{\text{th}}$ percentile using the formula $P_k=(n+1)\times \frac{k}{100}$ where $n=19$ and $k=25$: $P_{25}=(19+1)\times \frac{25}{100}=5.0$

Step 3 ：Find the value at the ${25}^{\text{th}}$ percentile position: Since the position is a whole number, take the average of the 5th and 6th values in the sorted list: $\frac{18+19}{2}=18.5$

Step 4 ：Calculate the position of the ${80}^{\text{th}}$ percentile using the same formula: $P_{80}=(19+1)\times \frac{80}{100}=16.0$

Step 5 ：Find the value at the ${80}^{\text{th}}$ percentile position: Since the position is a whole number, take the average of the 16th and 17th values in the sorted list: $\frac{29+29}{2}=29.0$

Step 6 ：The ${25}^{\text{th}}$ percentile: $\boxed{{\displaystyle 18.5}}$

Step 7 ：The ${80}^{\text{th}}$ percentile: $\boxed{{\displaystyle 29.0}}$