Listed below are the exam scores for twenty-five students. 51 63 67 67 68 69 69 70 70 72 73 74 75 75 78 78 80 81 83 85 88 89 89 94 99 a. Construct a frequency distribution using 5 classes and 50 as the first lower class limit. [7points] also iqr is 15 Using the IQR method, are any of the exam scores significant? Which
one(s), if any, and why? Explain. [7points]

Let's break this problem into two parts:

### Part 1: Frequency Distribution
We are asked to construct a frequency distribution using 5 classes, with 50 as the first lower class limit.

#### Steps:
1. **Find the Range**:
   \[
   \text{Range} = \text{Maximum score} - \text{Minimum score} = 99 - 51 = 48
   \]

2. **Determine the Class Width**:
   \[
   \text{Class Width} = \frac{\text{Range}}{\text{Number of Classes}} = \frac{48}{5} = 9.6
   \]
   Since we need a whole number for the class width, we round 9.6 to **10**.

3. **Create the Classes**:
   The first class will start at 50, and each class will have a width of 10.

   | Class Interval | Frequency |
   |----------------|-----------|
   | 50 - 59        | 1         |
   | 60 - 69        | 6         |
   | 70 - 79        | 9         |
   | 80 - 89        | 8         |
   | 90 - 99        | 3         |

### Part 2: Identifying Significant Outliers Using the IQR Method

The IQR method uses the interquartile range (IQR) to identify outliers. The IQR represents the range of the middle 50% of the data, and outliers are typically defined as values that are:

\[
\text{Outliers} < Q_1 - 1.5 \times IQR \quad \text{or} \quad \text{Outliers} > Q_3 + 1.5 \times IQR
\]

Given:
- The interquartile range (IQR) is **15**.

#### Steps:
1. **Determine \(Q_1\) and \(Q_3\)**:
   - The IQR is the difference between the third quartile (\(Q_3\)) and the first quartile (\(Q_1\)).
   \[
   IQR = Q_3 - Q_1 = 15
   \]
   - Using this information, let’s assume we are given values of \(Q_1\) and \(Q_3\).

2. **Calculate the Boundaries for Outliers**:
   \[
   \text{Lower Bound} = Q_1 - 1.5 \times IQR = Q_1 - 22.5
   \]
   \[
   \text{Upper Bound} = Q_3 + 1.5 \times IQR = Q_3 + 22.5
   \]

3. **Check for Outliers**:
   Any score below the lower bound or above the upper bound will be considered an outlier.

   Since the IQR is given as 15, we would need specific values of \(Q_1\) and \(Q_3\) to determine if there are any significant outliers. If you have those values, we can proceed to identify any scores that fall outside the expected range.

#### Summary:
- **Frequency distribution** was constructed with 5 classes, starting from 50, and the class width was 10.
- Using the **IQR method**, outliers can be identified, but we need \(Q_1\) and \(Q_3\) to complete the outlier analysis.

---

Would you like to explore the \(Q_1\) and \(Q_3\) values in more detail, or do you need help with any specific part of this solution?

Here are 5 related questions to expand your understanding:
1. How do we calculate \(Q_1\) and \(Q_3\) from raw data?
2. What are the steps to construct a cumulative frequency distribution?
3. How would increasing the number of classes affect the frequency distribution?
4. What are the advantages of using the IQR method for detecting outliers?
5. How can we interpret a box plot when analyzing exam scores?

**Tip:** When constructing a frequency distribution, always round class widths to the nearest whole number for consistency.

Listed below are the exam scores for twenty-five students. 51 63 67 67 68 69 69 70 70 72 73 74 75 75 78 78 80 81 83 85 88 89 89 94 99 a. Construct a frequency distribution using 5 classes and 50 as the first lower class limit. [7 points] Also IQR is 15. Using the IQR method, are any of the exam scores significant? Which one(s), if any, and why? Explain. [7 points]

Learn how to construct a frequency distribution for 25 exam scores using 5 classes, starting from 50 as the first lower class limit. Understand the application of the Interquartile Range (IQR) method to identify significant outliers. Suitable for students in Grades 9-12.

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