Let's work through the problem step-by-step.

Given Data Set:

36.3, 41.3, 22.9, 41.4, 43.2, 26.2, 24.2, 30.3, 28.4, 35.7.

Part (a) - Find the range of the data set.

To find the range, subtract the smallest number in the set from the largest number.

1. Identify the maximum and minimum values in the data set:

   • Maximum: 43.2
   • Minimum: 22.9

2. Calculate the range: $\text{Range} = \text{Maximum} - \text{Minimum} = 43.2 - 22.9 = 20.3$

So, the range is 20.3 (as given in the image).

Part (b) - Change 43.2 to 63.7 and find the range of the new data set.

Now, we need to replace 43.2 with 63.7 and calculate the new range:

1. Updated data set: 36.3, 41.3, 22.9, 41.4, 63.7, 26.2, 24.2, 30.3, 28.4, 35.7.

2. Identify the new maximum and minimum values:

   • Maximum: 63.7
   • Minimum: 22.9

3. Calculate the new range: $\text{Range} = 63.7 - 22.9 = 40.8$

So, the new range is 40.8.

Let me know if you'd like further clarification or have any questions!

Here are 5 related questions to explore:

1. What would happen to the range if you added more numbers to the data set?
2. How does the range compare to the standard deviation?
3. What other measures of spread can be used in statistics?
4. How can the range be affected by outliers?
5. Why is the range considered a simple measure of variability?

Tip: The range is quick to calculate but can be sensitive to outliers, so it's often useful to combine it with other measures like the interquartile range (IQR) for a fuller picture of the data's spread.