The following are the P/E ratios (price of stock divided by projected earnings per share) for 19 banks. \[ 29,31,22,18,18,21,19,21,20,14,17,23,22,18,38,20,29,34,22 \] Send data to calculator 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: \( \boxed{18.5} \)

Step 7 ：The \(80^{\text{th}}\) percentile: \( \boxed{29.0} \)