To find the regression equation and the best predicted diastolic blood pressure given the systolic data, we will go through these steps:

Step 1: Compute the correlation coefficient $r$

We will calculate the correlation coefficient $r$ between systolic and diastolic blood pressure. Using Pearson's correlation formula:

[ 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:

• $n$ is the number of data points,
• $x$ is the systolic blood pressure values, and
• $y$ is the diastolic blood pressure values.

The systolic and diastolic values are:

• $x$: 122, 134, 143, 140, 135, 113, 110, 145
• $y$: 62, 71, 80, 80, 100, 59, 82, 105

Step 2: Calculate the slope and intercept of the regression line

The regression line is represented as: $y = a + bx$ Where:

• $b = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2}$ is the slope, and
• $a = \bar{y} - b\bar{x}$ is the intercept.

Step 3: Predict the diastolic pressure for a systolic pressure of 137 mm Hg

Once the regression equation is determined, substitute $x = 137$ into the equation to predict the diastolic pressure.

Let's proceed by calculating each of these values.### Step 1: Pearson's Correlation Coefficient The calculated correlation coefficient $r = 0.579$, indicating a moderate positive correlation between systolic and diastolic blood pressure.

Step 2: Regression Equation

The regression equation, rounded to two decimal places, is: $y = -11.20 + 0.70x$ Where $x$ is the systolic blood pressure, and $y$ is the predicted diastolic blood pressure.

Step 3: Predicted Diastolic Pressure for a Systolic Pressure of 137 mm Hg

Using the equation $y = -11.20 + 0.70(137)$, the best predicted diastolic blood pressure is approximately: $y = 84.6 \, \text{mm Hg}$

Would you like further details on any of these calculations, or do you have additional questions?

Here are some related questions to consider:

1. How does changing the systolic value affect the diastolic prediction?
2. What are the implications of a correlation coefficient close to 1 or -1?
3. What other methods can be used to predict diastolic pressure based on systolic pressure?
4. How can the accuracy of this regression model be evaluated?
5. What are the limitations of using linear regression for this dataset?

Tip: In regression analysis, checking the residuals (differences between actual and predicted values) helps to evaluate the model's accuracy.