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Solve it using assumed mean formula of Karl Pearson 

To solve the problem using the assumed mean formula of Karl Pearson's correlation coefficient, we will follow these steps:

1. **Identify the Data Sets**: The data from the image includes two series, A and B.

2. **Choose Assumed Means**: Select assumed means for both series to simplify the calculations. It's common to select one of the data points close to the middle of each series.

3. **Calculate Deviations**: For each series, calculate the deviation of each data point from the assumed mean.

4. **Apply the Formula**:
   The formula for Karl Pearson's correlation coefficient using assumed means is:
   \[
   r = \frac{\sum{(x - \bar{x})(y - \bar{y})}}{\sqrt{\sum{(x - \bar{x})^2} \sum{(y - \bar{y})^2}}}
   \]
   Where \( x \) and \( y \) are the values from the two series, and \(\bar{x}\) and \(\bar{y}\) are the respective assumed means.

5. **Calculate the Required Sums**:
   - Sum of the products of deviations \((x - \bar{x})(y - \bar{y})\)
   - Sum of the squares of deviations for each series

6. **Compute the Correlation Coefficient**.

Let's proceed with these steps using the data in the image:

### Extracted Data:

- **Series A**: 15.1, 16.3, 16.8, 17.1, 16.6, 17.3, 16.5, 17.3, 16.3, 16.0, 17.2
- **Series B**: 16.7, 19.2, 20.5, 22.1, 22.6, 22.6, 23.2, 22.0, 22.0, 21.6, 22.5

### Steps:

1. **Choose Assumed Means**:
   - Assumed mean for Series A (\(\bar{x}\)): Let's choose 17.0.
   - Assumed mean for Series B (\(\bar{y}\)): Let's choose 22.0.

2. **Calculate Deviations**:
   - Deviations for Series A: \((15.1 - 17.0), (16.3 - 17.0), \ldots, (17.2 - 17.0)\)
   - Deviations for Series B: \((16.7 - 22.0), (19.2 - 22.0), \ldots, (22.5 - 22.0)\)

3. **Calculate the Required Sums**:
   - Compute \(\sum (x - \bar{x})^2\)
   - Compute \(\sum (y - \bar{y})^2\)
   - Compute \(\sum (x - \bar{x})(y - \bar{y})\)

4. **Compute \( r \)**.

Let's calculate these using Python:The correlation coefficient between Series A and Series B, using Karl Pearson's method with assumed means, is approximately \(0.762\). This value indicates a strong positive correlation between the two sets of data.

### Additional Questions:
1. What is Karl Pearson's correlation coefficient, and how is it used in statistics?
2. Why do we use assumed means in calculating the correlation coefficient?
3. How would changing the assumed means affect the final correlation coefficient?
4. What is the significance of a positive correlation coefficient?
5. How can outliers in the data affect the correlation coefficient?
6. What are some real-world examples where Pearson's correlation coefficient is applied?
7. How does Pearson's correlation coefficient differ from Spearman's rank correlation?
8. Can a correlation coefficient of 1 or -1 occur in practical data, and what does it signify?

### Tip:
When using assumed means in correlation calculations, selecting a mean close to the center of the data distribution can simplify computations and help avoid errors.

Learn how to calculate Karl Pearson's correlation coefficient using the assumed mean formula. Follow step-by-step instructions with specific data points and Python calculations. Suitable for advanced high school or college-level students studying statistics.

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