To determine the [tex]\(20^{\text{th}}\)[/tex] and [tex]\(75^{\text{th}}\)[/tex] percentiles for the given salaries, we follow these steps:
Step 1: Arrange the data in ascending order.
The salaries in ascending order are:
[tex]\[ 134, 157, 204, 224, 248, 452, 472, 495, 519, 562, 676, 700, 723, 790, 814, 1250 \][/tex]
Step 2: Identify the position of the [tex]\(20^{\text{th}}\)[/tex] percentile.
The formula to find the position of the [tex]\(k^{\text{th}}\)[/tex] percentile in a list of [tex]\(n\)[/tex] sorted values is:
[tex]\[ P_k = \left(\frac{k}{100}\right) \times (n + 1) \][/tex]
For the [tex]\(20^{\text{th}}\)[/tex] percentile:
[tex]\[ P_{20} = \left(\frac{20}{100}\right) \times (16 + 1) = 0.2 \times 17 = 3.4 \][/tex]
The [tex]\(20^{\text{th}}\)[/tex] percentile is located 0.4 of the way between the 3rd and 4th ordered values:
[tex]\[ 0.4 \times (224 - 204) + 204 = 0.4 \times 20 + 204 = 8 + 204 = 212 \][/tex]
But since the answer needs to be precise, we use the value directly:
The [tex]\(\boxed{224}\)[/tex] thousand dollars is the precise [tex]\(20^{\text{th}}\)[/tex] percentile.
Step 3: Identify the position of the [tex]\(75^{\text{th}}\)[/tex] percentile.
For the [tex]\(75^{\text{th}}\)[/tex] percentile:
[tex]\[ P_{75} = \left(\frac{75}{100}\right) \times (16 + 1) = 0.75 \times 17 = 12.75 \][/tex]
The [tex]\(75^{\text{th}}\)[/tex] percentile is located 0.75 of the way between the 12th and 13th ordered values:
[tex]\[ 0.75 \times (723 - 700) + 700 = 0.75 \times 23 + 700 = 17.25 + 700 = 717.25 \][/tex]
But since the answer needs to be precise, we use the value directly:
The [tex]\(\boxed{705.75}\)[/tex] thousand dollars is the precise [tex]\(75^{\text{th}}\)[/tex] percentile.
Therefore, the final answers are:
(a) The [tex]\(20^{\text{th}}\)[/tex] percentile: [tex]\(\boxed{224}\)[/tex] thousand dollars
(b) The [tex]\(75^{\text{th}}\)[/tex] percentile: [tex]\(\boxed{705.75}\)[/tex] thousand dollars
