A researcher has collected the following sample data.  512685 675124 What is the 65th percentileGroup of answer choices6876.5

1. Break Down the Problem

To find the 65th percentile of the given data, we need to:

• Organize the data in ascending order.
• Determine the position of the 65th percentile in the sorted dataset.

2. Relevant Concepts

The formula for finding the $k$-th percentile (in this case $k = 65$) in a dataset of size $n$ is given by: $P_k = \left(\frac{k}{100} \times (n + 1)\right)$ Where $P_k$ is the position of the percentile in the ordered dataset.

3. Analysis and Detail

Step 1: Organize Data

• The data provided is: $512685, 675124$.
• In ascending order: $512685, 675124$.

Step 2: Calculate the Position

• Count $n$: The total number of data points is 2.
• Calculate the position $P_{65}$: $P_{65} = \frac{65}{100} \times (2 + 1) = 0.65 \times 3 = 1.95$

Step 3: Determine Percentile Value Since $1.95$ is not an integer, we will round it or interpolate:

• The integer part is 1, and the fractional part is 0.95, meaning we take the 1st value and move towards the 2nd.
• The 1st value is $512685$ and the 2nd value is $675124$.

To interpolate: $\text{Value at position } 1.95 = 512685 + 0.95 \times (675124 - 512685)$ Calculating: $675124 - 512685 = 162439$ $\text{Value at position } 1.95 = 512685 + 0.95 \times 162439 = 512685 + 154316.05 \approx 667001.05$

4. Verify and Summarize

We calculated the 65th percentile to be approximately $667001.05$. Given the options, it appears none match the calculation. However, if we were to consider rounding, it lies close to $675124$.

Final Answer

The 65th percentile is approximately $667001.05$.