Questions: Find the data value that represents the 24th percentile, P24, from the following data 1200 1400 1600 1900 2600 2700 3000 3100 4600 4700 4900 5100 5300 5400 5500 5900 6000 6100 6200 6500 6700 6800 6900 7300 7400 7800 8100 8400 8500 8600 8700 P24=

Find the data value that represents the 24th percentile, P24, from the following data 1200 1400 1600 1900 2600 2700 3000 3100 4600 4700 4900 5100 5300 5400 5500 5900 6000 6100 6200 6500 6700 6800 6900 7300 7400 7800 8100 8400 8500 8600 8700 P24=
Transcript text: Find the data value that represents the 24th percentile, $\mathrm{P}_{24}$, from the following data \begin{tabular}{|l|l|l|l|l|} \hline 1200 & 1400 & 1600 & 1900 & 2600 \\ \hline 2700 & 3000 & 3100 & 4600 & 4700 \\ \hline 4900 & 5100 & 5300 & 5400 & 5500 \\ \hline 5900 & 6000 & 6100 & 6200 & 6500 \\ \hline 6700 & 6800 & 6900 & 7300 & 7400 \\ \hline 7800 & 8100 & 8400 & 8500 & 8600 \\ \hline 8700 & & & & \\ \hline \end{tabular} \[ P_{24}= \]
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Solution

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Solution Steps

To find the 24th percentile (\(P_{24}\)) of a given dataset, we need to follow these steps:

  1. Combine all the data values into a single sorted list.
  2. Calculate the index for the 24th percentile using the formula \( \text{index} = \frac{24}{100} \times (N + 1) \), where \(N\) is the total number of data points.
  3. If the index is an integer, the value at that index is the 24th percentile. If the index is not an integer, interpolate between the two closest data points.
Step 1: Combine and Sort the Data

First, we combine all the data values into a single list and sort them in ascending order: \[ \text{data} = [1200, 1400, 1600, 1900, 2600, 2700, 3000, 3100, 4600, 4700, 4900, 5100, 5300, 5400, 5500, 5900, 6000, 6100, 6200, 6500, 6700, 6800, 6900, 7300, 7400, 7800, 8100, 8400, 8500, 8600, 8700] \]

Step 2: Calculate the Index for the 24th Percentile

The formula to calculate the index for the 24th percentile is: \[ \text{index} = \frac{24}{100} \times (N + 1) \] where \(N\) is the total number of data points. Given \(N = 31\): \[ \text{index} = 0.24 \times (31 + 1) = 0.24 \times 32 = 7.68 \]

Step 3: Interpolate Between the Closest Data Points

Since the index \(7.68\) is not an integer, we need to interpolate between the 7th and 8th data points. The 7th data point is \(3000\) and the 8th data point is \(3100\). The fractional part of the index is \(0.68\): \[ P_{24} = 3000 + 0.68 \times (3100 - 3000) = 3000 + 0.68 \times 100 = 3000 + 68 = 3068 \]

Final Answer

\(\boxed{3068}\)

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