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Let's analyze the histograms and question in detail.

### Problem:
There are two histograms representing two sets of data, each consisting of 23 integers. The task is to determine the smallest possible difference between the mean of data set A and the mean of data set B.

### Breakdown of Histograms:
The histograms show frequency distributions for different intervals of integers:
- **Intervals**: 
  - 10 to 20
  - 20 to 30
  - 30 to 40
  - 40 to 50
  - 50 to 60
- **Frequencies** for each interval are given for both data sets A and B.

### Steps to Calculate the Mean:
1. **Midpoint Calculation**: For each interval (10-20, 20-30, etc.), calculate the midpoint:
   - Midpoint of 10-20 = 15
   - Midpoint of 20-30 = 25
   - Midpoint of 30-40 = 35
   - Midpoint of 40-50 = 45
   - Midpoint of 50-60 = 55

2. **Weighted Mean**: The mean of a data set can be found using the formula:
   \[
   \text{Mean} = \frac{\sum (\text{midpoint} \times \text{frequency})}{\text{total number of data points}}
   \]
   Since each data set consists of 23 integers, the total number of data points for each data set is 23.

3. **Calculate the Mean for Each Data Set**:
   - Multiply the midpoint of each interval by the frequency for that interval in the histogram.
   - Sum these products and divide by 23 to get the mean for each data set.

Let me calculate the means and determine the smallest possible difference for you.The calculated means are as follows:
- **Mean of Data Set A**: 33.26
- **Mean of Data Set B**: 35.43

The **smallest possible difference** between the mean of Data Set A and Data Set B is approximately **2.17**.

Let me know if you need further details or have any questions! Here are some related questions to expand on:

1. How would the means change if the frequencies were altered for one of the intervals?
2. What does the difference in means tell us about the central tendency of the two data sets?
3. How could you verify that this is the smallest possible difference without recalculating everything?
4. How would you compute the median for both data sets?
5. What impact would adding an outlier to one of the data sets have on the mean?

**Tip**: Always double-check the total number of points in a histogram to ensure your mean calculation is accurate.

Two data sets of 23 integers each are summarized in the histograms shown. For each of the histograms, the first interval represents the frequency of integers greater than or equal to 10, but less than 20. The second interval represents the frequency of integers greater than or equal to 20, but less than 30, and so on. What is the smallest possible difference between the mean of data set A and the mean of data set B?

Analyze two histograms representing data sets of 23 integers each, and calculate the smallest possible difference between their means. Learn how to apply weighted mean calculation and histogram analysis to solve this statistical problem. Suitable for high school students in Grades 9-12.

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