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To find the correlation coefficient between serum cholesterol levels and diastolic blood pressure from the data provided, we can use the formula for the Pearson correlation coefficient:

\[
r = \frac{n\sum{XY} - \sum{X}\sum{Y}}{\sqrt{(n\sum{X^2} - (\sum{X})^2)(n\sum{Y^2} - (\sum{Y})^2)}}
\]

Where:
- \( X \) represents the cholesterol levels,
- \( Y \) represents the diastolic blood pressure,
- \( n \) is the number of data points.

Given dataTo find the correlation coefficient between serum cholesterol levels and diastolic blood pressure from the data provided, we can use the formula for the Pearson correlation coefficient:

\[
r = \frac{n\sum{XY} - \sum{X}\sum{Y}}{\sqrt{(n\sum{X^2} - (\sum{X})^2)(n\sum{Y^2} - (\sum{Y})^2)}}
\]

Where:
- \( X \) represents the cholesterol levels,
- \( Y \) represents the diastolic blood pressure,
- \( n \) is the number of data points.

Given 
- Cholesterol levels (\( X \)): \( 307, 259, 341, 317, 274, 416, 267, 320, 274, 336 \)
- Diastolic blood pressure (\( Y \)): \( 80, 75, 90, 74, 75, 110, 70, 85, 88, 78 \)

Let's calculate each of the required components step by step.

### Step 1: Calculate \(\sum X\), \(\sum Y\), \(\sum{X^2}\), \(\sum{Y^2}\), and \(\sum{XY}\)
1. \(\sum{X}\) = 307 + 259 + 341 + 317 + 274 + 416 + 267 + 320 + 274 + 336
2. \(\sum{Y}\) = 80 + 75 + 90 + 74 + 75 + 110 + 70 + 85 + 88 + 78
3. \(\sum{X^2}\) = \(307^2 + 259^2 + 341^2 + \ldots + 336^2\)
4. \(\sum{Y^2}\) = \(80^2 + 75^2 + 90^2 + \ldots + 78^2\)
5. \(\sum{XY}\) = \(307 \times 80 + 259 \times 75 + \ldots + 336 \times 78\)

### Step 2: Substitute these sums into the formula for \( r \).

Let me calculate these values.The Pearson correlation coefficient \( r \) between serum cholesterol levels and diastolic blood pressure is approximately **0.809**.

This value suggests a strong positive correlation between the two variables, meaning that as the cholesterol level increases, the diastolic blood pressure tends to increase as well.

Would you like to know more details about this calculation or have any further questions?

Here are some related questions that might interest you:
1. How do we interpret the strength of a correlation coefficient?
2. What are some factors that could affect the correlation between cholesterol levels and blood pressure?
3. How can we test the significance of the correlation coefficient?
4. What are the potential limitations of using the Pearson correlation coefficient?
5. How would you calculate the correlation coefficient if the data were not linear?

**Tip:** Always visualize the data when calculating correlation to check if the relationship is linear, as the Pearson correlation coefficient is most appropriate for linear relationships.

Find the correlation coefficient between serum cholesterol levels and diastolic blood pressure for 10 randomly selected persons.

Learn how to calculate the Pearson correlation coefficient between serum cholesterol levels and diastolic blood pressure using data from 10 randomly selected individuals. This example provides a step-by-step approach with key statistical formulas.

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