https://file.aiquickdraw.com/mathbot/file/1727113833022ekj6uiis.jpg

Give a step by step solving of the correlation in the question

Let's go step by step in calculating the correlation coefficient using the data provided in the image. The data includes age (\(X\)) and systolic blood pressure (\(Y\)) for 18 adult females.

### Step 1: Organize the Data
The data given for age (\(X\)) and systolic blood pressure (\(Y\)) are:

\[
\begin{array}{|c|c|}
\hline
\text{Age (X)} & \text{Blood Pressure (Y)} \\
\hline
8 & 122 \\
10 & 172 \\
13 & 132 \\
15 & 133 \\
16 & 145 \\
18 & 137 \\
\hline
10 & 116 \\
12 & 128 \\
13 & 130 \\
14 & 128 \\
15 & 145 \\
16 & 137 \\
\hline
13 & 122 \\
14 & 172 \\
16 & 135 \\
18 & 155 \\
\end{array}
\]

### Step 2: Calculate Summation Values
To calculate the Pearson correlation coefficient (\(r\)), we need to compute the following:

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

We will calculate the following:

1. \( \sum X \) (sum of ages)
2. \( \sum Y \) (sum of systolic blood pressures)
3. \( \sum X^2 \) (sum of squared ages)
4. \( \sum Y^2 \) (sum of squared systolic blood pressures)
5. \( \sum XY \) (sum of the products of age and blood pressure)

### Step 3: Use Formula for Correlation Coefficient
Once we have these values, we will plug them into the correlation formula.

Let me compute the necessary sums and values for you now.The Pearson correlation coefficient between age and systolic blood pressure for the given data is approximately \( r = 0.244 \).

### Interpretation:
This positive correlation of \( r = 0.244 \) suggests a weak positive linear relationship between age and systolic blood pressure. As age increases, systolic blood pressure tends to increase slightly, but the relationship is not very strong.

Do you need further details or clarifications?

Here are five related questions for further exploration:
1. How can we interpret the strength of correlation values?
2. What does a correlation coefficient close to 1 or -1 imply about the data?
3. What are the assumptions behind Pearson correlation?
4. Can correlation imply causation in this dataset?
5. How would the correlation change if outliers were removed?

**Tip:** Always plot your data first (e.g., scatter plot) to visually assess the relationship before calculating the correlation coefficient.

Learn how to calculate the Pearson correlation coefficient between age and systolic blood pressure using step-by-step instructions. This problem involves statistics, correlation, and the Pearson correlation formula. Suitable for high school students in Grades 10-12.

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