Question

the data set below represents the ages of 30 executives. which ages are above the 75th percentile?

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$$\begin{matrix} 44 & 58 & 65 & 46 & 58 & 42 & 56 & 54 & 61 & 56 \\\\ 56 & 50 & 67 & 56 & 50 & 60 & 46 & 39 & 49 & 42 \\\\ 54 & 40 & 48 & 46 & 29 & 34 & 39 & 42 & 41 & 43 \\end{matrix}$$

determine the values in the data set above the 75th percentile. if a data value above the 75th percentile appears more than once in the data set, make sure to include that value in your answer the appropriate number of times.

the ages above the 75th percentile are
(use a comma to separate answers as needed.)

Explanation:

Sort the dataset in ascending order

Using the Percentiles and Quantiles knowledge point
The original dataset of \(N = 30\) ages is:
Row 1: 44, 58, 65, 46, 58, 42, 56, 54, 61, 56
Row 2: 56, 50, 67, 56, 50, 60, 46, 39, 49, 42
Row 3: 54, 40, 48, 46, 29, 34, 39, 42, 41, 43

Let's count the frequency of each value:

• 29: 1
• 34: 1
• 39: 2
• 40: 1
• 41: 1
• 42: 3
• 43: 1
• 44: 1
• 46: 3
• 48: 1
• 49: 1
• 50: 2
• 54: 2
• 56: 4
• 58: 2
• 60: 1
• 61: 1
• 65: 1
• 67: 1

Total count check: \(1+1+2+1+1+3+1+1+3+1+1+2+2+4+2+1+1+1+1 = 30\).

Sorted dataset:
29, 34, 39, 39, 40, 41, 42, 42, 42, 43, 44, 46, 46, 46, 48, 49, 50, 50, 54, 54, 56, 56, 56, 56, 58, 58, 60, 61, 65, 67

Calculate the index of the 75th percentile

Using the Percentiles and Quantiles knowledge point

$$ i = \frac{P}{100} \times N = \frac{75}{100} \times 30 = 22.5 $$

Since the index \(i = 22.5\) is not an integer, we round up to the next integer, which is 23.
The 75th percentile is the 23rd value in the sorted list.

Identify the 75th percentile value

Using the Percentiles and Quantiles knowledge point
Let's find the 23rd value in the sorted list:

• 1st to 10th: 29, 34, 39, 39, 40, 41, 42, 42, 42, 43
• 11th to 20th: 44, 46, 46, 46, 48, 49, 50, 50, 54, 54
• 21st: 56
• 22nd: 56
• 23rd: 56

Thus, the 75th percentile value is 56.

Determine the values above the 75th percentile

Using the Percentiles and Quantiles knowledge point
We find all values in the sorted list that are strictly greater than 56:

• 24th: 56 (not strictly greater)
• 25th: 58
• 26th: 58
• 27th: 60
• 28th: 61
• 29th: 65
• 30th: 67

The values above the 75th percentile are: 58, 58, 60, 61, 65, 67.

Answer:

The ages above the 75th percentile are <blank>58, 58, 60, 61, 65, 67</blank>
(Use a comma to separate answers as needed.)