Let's break down the problem step by step.

We are tasked with creating a frequency distribution and a histogram for the given data. Here's how to proceed:

Step 1: Organize the Test Scores

The test scores are provided as:

84, 80, 85, 84, 70, 68, 
86, 77, 72, 75, 80, 88, 
88, 76, 69, 87, 73, 69

Step 2: Set the Class Intervals

The class intervals are already set with a class width of 6. They are:

• 66 to 71
• 72 to 77
• 78 to 83
• 84 to 89

Step 3: Tally the Frequencies for Each Interval

Now, count how many scores fall into each interval:

• 66 to 71: 70, 68, 69, 69 → 4 values
• 72 to 77: 77, 72, 75, 76, 73 → 5 values
• 78 to 83: 80, 80 → 2 values
• 84 to 89: 84, 85, 84, 86, 88, 88, 87 → 7 values

Step 4: Complete the Frequency Table

Now, we can fill out the frequency distribution:

Step 5: Construct the Histogram

• X-axis: The test score intervals (66-71, 72-77, 78-83, 84-89).
• Y-axis: The frequency (0 to 7).

The histogram will have 4 bars with heights representing the frequencies:

• Bar for 66 to 71 → Height 4
• Bar for 72 to 77 → Height 5
• Bar for 78 to 83 → Height 2
• Bar for 84 to 89 → Height 7

Would you like more details on how to construct the histogram or have any other questions?

Here are 5 related questions to deepen your understanding:

1. What is the relationship between class width and the shape of the histogram?
2. How would changing the class width impact the frequency distribution?
3. Can you calculate the mean and median from the grouped frequency data?
4. How do outliers in the data affect the histogram?
5. What are the advantages of using a histogram over a bar graph in this context?

Tip: When constructing a histogram, ensure the bars touch each other to indicate the continuous nature of the data.