To find the 64th percentile ($P_{64}$) from the given data, we will use the following steps:

Step 1: Arrange the data in ascending order

The data is already provided in ascending order:

$11, 12.1, 12.8, 17, 17.6, 20, 20.4, 22.5, 24.9, 26.6, 30.8, 33.9, 35.7, 37.5, 42.3, 42.9, 44.2, 47.3, 47.4, 49$

There are $n = 20$ data points.

Step 2: Determine the position of the 64th percentile

Using the formula for the position of the $k$th percentile:

$\text{Position} = \frac{k}{100} \times (n + 1)$

Here, $k = 64$ and $n = 20$:

$\text{Position} = \frac{64}{100} \times (20 + 1) = 0.64 \times 21 = 13.44$

Step 3: Identify the values at the calculated position

The position 13.44 means that the 64th percentile lies between the 13th and 14th values in the ordered data.

• The 13th value is 35.7.
• The 14th value is 37.5.

Step 4: Interpolate between these two values

Since the position is 13.44, we are 0.44 of the way between the 13th and 14th values:

$P_{64} = 35.7 + 0.44 \times (37.5 - 35.7)$

Calculate the difference:

$P_{64} = 35.7 + 0.44 \times 1.8 = 35.7 + 0.792 = 36.492$

So, the 64th percentile $P_{64}$ is approximately 36.49.

Would you like more details or have any questions?

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Here are five related questions:

1. How would you find the 50th percentile (median) from the given data set?
2. Can you calculate the 90th percentile for this data set?
3. How does the method change if the data set has a much larger number of entries?
4. What is the difference between percentiles and percentages?
5. Can you explain how to use percentiles to interpret test scores?

Tip: Percentile positions are useful for understanding where a particular value stands in relation to the rest of the data. Remember that the interpolation step is crucial when the position is not an integer.