Kindly solve the sum properly step-wise & give

Let's go through the steps to solve this problem properly. We'll calculate the percentiles and then determine the range for the middle 30% of the data set.

### 1. **Arrange the Data in Ascending Order**
First, arrange the data set in ascending order:

\[ 5, 7, 10, 12, 15, 17, 19, 21, 24, 26 \]

### 2. **Calculate the Percentiles**
Percentiles are values below which a certain percentage of the data falls. The formula to find the \(k^{th}\) percentile (P\(_k\)) in a sorted data set is:

\[
P_k = \frac{k(N + 1)}{100}
\]

where \(k\) is the desired percentile and \(N\) is the number of observations in the data set.

In this case, \(N = 10\).

### 3. **Find the Relevant Percentiles**
We need to find the percentiles P35, P65, P40, P70, P45, and P75 to understand where the middle 30% lies.

#### **Calculate P35:**
\[
\text{Position} = \frac{35 \times (10 + 1)}{100} = \frac{385}{100} = 3.85
\]

The 3.85th position lies between the 3rd and 4th values. We interpolate between the 3rd value (10) and the 4th value (12):

\[
P35 = 10 + 0.85 \times (12 - 10) = 10 + 1.7 = 11.7
\]

#### **Calculate P65:**
\[
\text{Position} = \frac{65 \times (10 + 1)}{100} = \frac{715}{100} = 7.15
\]

The 7.15th position lies between the 7th value (19) and the 8th value (21):

\[
P65 = 19 + 0.15 \times (21 - 19) = 19 + 0.3 = 19.3
\]

#### **Calculate P40:**
\[
\text{Position} = \frac{40 \times (10 + 1)}{100} = \frac{440}{100} = 4.4
\]

The 4.4th position lies between the 4th value (12) and the 5th value (15):

\[
P40 = 12 + 0.4 \times (15 - 12) = 12 + 1.2 = 13.2
\]

#### **Calculate P70:**
\[
\text{Position} = \frac{70 \times (10 + 1)}{100} = \frac{770}{100} = 7.7
\]

The 7.7th position lies between the 7th value (19) and the 8th value (21):

\[
P70 = 19 + 0.7 \times (21 - 19) = 19 + 1.4 = 20.4
\]

#### **Calculate P45:**
\[
\text{Position} = \frac{45 \times (10 + 1)}{100} = \frac{495}{100} = 4.95
\]

The 4.95th position lies between the 4th value (12) and the 5th value (15):

\[
P45 = 12 + 0.95 \times (15 - 12) = 12 + 2.85 = 14.85
\]

#### **Calculate P75:**
\[
\text{Position} = \frac{75 \times (10 + 1)}{100} = \frac{825}{100} = 8.25
\]

The 8.25th position lies between the 8th value (21) and the 9th value (24):

\[
P75 = 21 + 0.25 \times (24 - 21) = 21 + 0.75 = 21.75
\]

### 4. **Determine the Range for the Middle 30%**
The middle 30% of the observations should be between the percentiles:

- **P35 to P65**: From \(11.7\) to \(19.3\)
- **P40 to P70**: From \(13.2\) to \(20.4\)
- **P45 to P75**: From \(14.85\) to \(21.75\)

Since the middle 30% should be centered around the 50th percentile and span equally across the central part of the data, the correct range is:

**b) P35 to P65**

This range, from \(11.7\) to \(19.3\), best represents the middle 30% of the observations.

Would you like to explore any more details or have other questions? Here are five related questions:

1. How do you calculate the interquartile range (IQR) of a data set?
2. What is the difference between the median and the 50th percentile?
3. How does interpolation work in finding percentiles between data points?
4. What is the significance of different percentile ranges in statistics?
5. How do you use a cumulative frequency table to determine percentiles?

**Tip:** When dealing with percentiles, always make sure your data is sorted, and remember that percentiles divide the data into 100 equal parts, helping you understand the spread and concentration of your data.

Explore a detailed step-by-step solution for calculating percentiles with interpolation. Learn how to determine percentiles using the formula P_k = (k(N + 1)) / 100, illustrated with example data. Suitable for high school students and anyone interested in statistics and data analysis.

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