https://file.aiquickdraw.com/mathbot/file/17271005655396na4xz7q.jpg

Let's break down the problem step by step based on the provided image, which is a frequency distribution table with the following columns:

- \( X \): Class intervals (grouped data)
- \( f \): Frequency (number of occurrences in each interval)
- \( cf \): Cumulative frequency
- \( c\% \): Cumulative percentage

The given questions are:

**a. Find the 50th percentile for this distribution.**

The 50th percentile, or median, is the value below which 50% of the observations fall. To find this:

1. Look at the cumulative percentage column (\(c\%\)).
2. The 50th percentile lies between the intervals where the cumulative percentage crosses 50%.

**Step-by-step:**
- The cumulative percentage for the interval 30-34 is 56%, meaning the 50th percentile is within this group (30-34).
- The formula for finding the exact 50th percentile within a class interval is:
  
  \[
  P_{50} = L + \left( \frac{N/2 - cf_{previous}}{f} \right) \times h
  \]
  where:
  - \(L\) is the lower bound of the interval containing the 50th percentile (in this case, 30),
  - \(N\) is the total number of observations (here \(N = 25\)),
  - \(cf_{previous}\) is the cumulative frequency before the 50th percentile class (here \(cf_{previous} = 14\)),
  - \(f\) is the frequency of the interval containing the 50th percentile (here \(f = 5\)),
  - \(h\) is the width of the interval (5, since the intervals are equally spaced).

Now, let's compute it:

\[
P_{50} = 30 + \left( \frac{25/2 - 14}{5} \right) \times 5 = 30 + \left

Consider the data in the following table. Find the 50th percentile for this distribution. Find the 60th percentile. What is the percentile rank for X = 42? What is the percentile rank for X = 44.5?

Learn how to calculate the 50th and 60th percentiles and percentile ranks for X = 42 and X = 44.5 using a cumulative frequency table. This detailed step-by-step guide covers key statistical concepts and formulas suitable for high school students.

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