To solve the given problem, we need to answer two questions based on the dataset provided:

Data Set (in ascending order):

32, 33, 35, 36, 37, 43, 54, 55, 59, 62, 66, 69, 75, 78, 89

This is a set of 15 values. Let's proceed with the two questions:

1. Percentile of the data value 55

The formula to calculate the percentile rank of a given value $x$ is:

$P = \frac{\text{Number of values less than } x}{\text{Total number of values}} \times 100$

For 55:

• There are 7 values less than 55 (i.e., 32, 33, 35, 36, 37, 43, 54).
• Total number of values = 15.

Thus, the percentile rank is:

$P = \frac{7}{15} \times 100 = 46.67$

Rounding to the nearest whole number, the percentile of the data value 55 is 47.

2. Data value representing the 20th percentile

To find the value at the 20th percentile, we use the formula:

$\text{Index} = \frac{P}{100} \times (n + 1)$

Where $P$ is the desired percentile, and $n$ is the number of data points.

For the 20th percentile:

• $P = 20$
• $n = 15$

$\text{Index} = \frac{20}{100} \times (15 + 1) = 0.2 \times 16 = 3.2$

Since the index is between the 3rd and 4th value, we interpolate between the 3rd value (35) and 4th value (36). A value closer to the 3rd value would be approximately 35.2.

Thus, the data value representing the 20th percentile is 35.

Do you have any questions or need further details on these steps?

Here are some related questions you could explore:

1. What is the 90th percentile for this data set?
2. What is the median of the data set?
3. How do you calculate percentiles for a larger data set?
4. How is the interpolation method used to estimate percentiles?
5. What are quartiles, and how do they differ from percentiles?

Tip: Always sort your data in ascending order before calculating percentiles!