To determine the number of classes, class intervals, and create a frequency table based on the data provided, we will follow these steps:

1. Organize the data: List all the values.
2. Determine the range: The range is the difference between the highest and lowest value.
3. Choose the number of classes: This can be done using Sturges' Rule $k = 1 + 3.322 \log N$, where $N$ is the number of data points (34 in this case).
4. Calculate the class width: The class width is determined by dividing the range by the number of classes.
5. Create the frequency distribution: Tally the data within each class.

Let's begin by listing and analyzing the data from the image:

Data: $\{71, 75, 57, 88, 64, 80, 75, 82, 90, 68, 90, 88, 71, 75, 71, 81, 81, 48, 82, 72, 62, 68, 74, 79, 84, 75, 57, 75, 75, 68, 68, 68, 75, 80\}$

Step 1: Determine the Range

• Minimum value: 48
• Maximum value: 90
• Range = $90 - 48 = 42$

Step 2: Determine the Number of Classes

Using Sturges' Rule: $k = 1 + 3.322 \log N = 1 + 3.322 \log 34 \approx 1 + 3.322 \times 1.531 = 6.09 \approx 6$ Thus, the number of classes is approximately 6.

Step 3: Determine the Class Width

$\text{Class width} = \frac{\text{Range}}{\text{Number of classes}} = \frac{42}{6} = 7$ So, each class will have a width of 7 points.

Step 4: Create the Class Intervals

The first class will start at the minimum value (48), and each subsequent class will increase by the class width (7).

1. $48 - 54$
2. $55 - 61$
3. $62 - 68$
4. $69 - 75$
5. $76 - 82$
6. $83 - 90$

Step 5: Frequency Distribution Table

Thus, the class intervals and frequencies have been determined.

Would you like more details on any of these steps or calculations? Feel free to ask!

Related Questions:

1. How is Sturges' Rule used in determining the number of classes?
2. Why is it important to choose an appropriate class width?
3. Can we use a different method to determine the number of classes?
4. What are the other methods for constructing frequency tables?
5. How does the range affect the number of classes?

Tip: Always round up the number of classes when applying Sturges' Rule to avoid under-representation of data in classes.