Let's break down the problem into steps to solve it.

Step 1: Break the Data into Classes of Width 5

We start with the given data set:

$\text{{Data}} = \{47, 33, 46, 33, 36, 47, 42, 36, 51, 41, 30, 35, 43, 30, 30, 34, 42, 40, 46, 39, 47, 31, 46, 42, 38, 42, 35, 38, 32, 57, 32, 42, 37, 45, 43, 34, 40, 39, 39, 36, 37, 37, 35, 30, 33, 49, 47, 39, 36, 31\}$

Classes are formed with a width of 5, starting at 30:

• Class 1: 30 - 34
• Class 2: 35 - 39
• Class 3: 40 - 44
• Class 4: 45 - 49
• Class 5: 50 - 54
• Class 6: 55 - 59

Step 2: Tally the Frequencies

Now, let's count the frequencies for each class.

• Class 1 (30 - 34): 12 values
• Class 2 (35 - 39): 14 values
• Class 3 (40 - 44): 12 values
• Class 4 (45 - 49): 9 values
• Class 5 (50 - 54): 1 value
• Class 6 (55 - 59): 1 value

Step 3: Calculate Proportion of Values Between 35 and 44

Values that are 35 or higher but less than 45 fall into Classes 2 and 3.

• Total number of values in Classes 2 and 3: $14 + 12 = 26$
• Total number of data points: 49

Proportion = $\frac{26}{49} \approx 0.531$

Step 4: Calculate Probability of a Number Being Less Than 50

We need to count the number of values less than 50.

• Total number of values less than 50: $12 + 14 + 12 + 9 = 47$
• Total number of data points: 49

Probability = $\frac{47}{49} \approx 0.959$

Histogram

Here's a histogram representation:

\text{Class} & \text{Frequency} \\ \hline 30-34 & 12 \\ 35-39 & 14 \\ 40-44 & 12 \\ 45-49 & 9 \\ 50-54 & 1 \\ 55-59 & 1 \\ \end{array}$$ ### Summary of Answers 1. **Proportion of values between 35 and 44 (inclusive):** 0.531 2. **Probability that a randomly selected number is less than 50:** 0.959 Would you like further details or have any other questions? ### Follow-Up Questions: 1. How would the results change if the class width was 10 instead of 5? 2. What is the cumulative frequency for values less than 40? 3. How would you calculate the median value for this data set? 4. What is the mode of this data set? 5. How does the histogram change if the class range was shifted to start from 25? **Tip:** When creating histograms, adjusting the class width can significantly change the shape and interpretation of the distribution.