The following are the annual salaries of 19 chief executive officers of major companies. (The salaries are written in thousands of dollars.) \( 633,224,204,157,134,542,405,75,767,1250,814,676,84,315,586,362,495,381,472 \) Send data to calculator Find \( 20^{\text {th }} \) and \( 75^{\text {th }} \) percentiles for these salaries. (a) The \( 20^{\text {th }} \) percentile: \( \square \) thousand dollars (b) The \( 75^{\text {th }} \) percentile: \( \square \) thousand dollars

To find the \(20^{\text{th}}\) and \(75^{\text{th}}\) percentiles of the given salaries, we will follow these steps: 1. **Sort the Data**: First, we need to sort the salaries in ascending order. 2. **Calculate the Percentiles**: Use the formula for percentiles to find the values. ### Step 1: Sort the Data The given salaries are: \[ 633, 224, 204, 157, 134, 542, 405, 75, 767, 1250, 814, 676, 84, 315, 586, 362, 495, 381, 472 \] Sorting these values in ascending order gives: \[ 75, 84, 134, 157, 204, 224, 315, 362, 381, 405, 472, 495, 542, 586, 633, 676, 767, 814, 1250 \] ### Step 2: Calculate the Percentiles The formula for the \(k^{\text{th}}\) percentile is: \[ P_k = \frac{k}{100} \times (n + 1) \] where \(n\) is the number of data points. #### (a) The \(20^{\text{th}}\) Percentile - \(k = 20\) - \(n = 19\) Calculating the position: \[ P_{20} = \frac{20}{100} \times (19 + 1) = 0.2 \times 20 = 4 \] The \(20^{\text{th}}\) percentile is the value at the 4th position in the sorted list. #### (b) The \(75^{\text{th}}\) Percentile - \(k = 75\) Calculating the position: \[ P_{75} = \frac{75}{100} \times (19 + 1) = 0.75 \times 20 = 15 \] The \(75^{\text{th}}\) percentile is the value at the 15th position in the sorted list. ### Final Values - The \(20^{\text{th}}\) percentile corresponds to the 4th value in the sorted list, which is \(157\) thousand dollars. - The \(75^{\text{th}}\) percentile corresponds to the 15th value in the sorted list, which is \(633\) thousand dollars. Thus, the answers are: (a) The \(20^{\text{th}}\) percentile: \(157\) thousand dollars (b) The \(75^{\text{th}}\) percentile: \(633\) thousand dollars