QuestionFind the 39th percentile of the sorted data set with $n=117$ . Calculate it as $P_{39}=$ .

Studdy Solution

STEP 1

Assumptions1. The data set is sorted in ascending order. . The total number of data points (n) is117.
3. We are asked to find the39th percentile, denoted as $_{39}$ .

STEP 2

First, we need to calculate the rank of the percentile, which is the index of the data point that falls under the desired percentile. The rank (R) can be calculated using the formula $R = \frac{}{100} \times n$ where is the percentile and n is the total number of data points.

STEP 3

Now, plug in the given values for the percentile and the total number of data points to calculate the rank.
$R = \frac{39}{100} \times117$

STEP 4

Calculate the rank.
$R = \frac{39}{100} \times117 =45.63$

STEP 5

Since the rank is not a whole number, we round it up and down to get two ranks, $R_{1}$ and $R_{2}$ , which are the indices of the data points that the39th percentile falls between.
$R_{1} = \lfloor R \rfloor = \lfloor45.63 \rfloor =45$ $R_{2} = \lceil R \rceil = \lceil45.63 \rceil =46$

STEP 6

Now that we have the two ranks, we can find the data points that correspond to these ranks. Let's denote these data points as $_{1}$ and $_{2}$ .
$_{1} = Data\, point\, at\, R_{1} = Data\, point\, at\,45$ $_{2} = Data\, point\, at\, R_{2} = Data\, point\, at\,46$

STEP 7

Looking at the data set, we find the data points that correspond to the ranks.
$_{1} =48.$ $_{2} =49$

STEP 8

Since the rank was not a whole number, the39th percentile falls between two data points. We calculate the39th percentile by taking the average of these two data points.
$_{39} = \frac{_{1} + D_{2}}{2}$

STEP 9

Plug in the values for the data points to calculate the39th percentile.
$_{39} = \frac{48.8 +49}{2}$

STEP 10

Calculate the39th percentile.
$_{39} = \frac{48.8 +49}{2} =48.9$ The39th percentile of the data set is48.9.

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