To find the regression equation, we first need to calculate the slope (b) and the intercept (a) of the line. The regression equation can be written as:

$\hat{y} = a + bx$

Where:

• $\hat{y}$ is the predicted diastolic blood pressure,
• $x$ is the systolic blood pressure,
• $a$ is the intercept,
• $b$ is the slope of the regression line.

Step 1: Calculate the Mean of X and Y

Let's denote:

• $x$ as systolic blood pressure,
• $y$ as diastolic blood pressure.

Given dataTo find the regression equation, we first need to calculate the slope (b) and the intercept (a) of the line. The regression equation can be written as:

$\hat{y} = a + bx$

Where:

• $\hat{y}$ is the predicted diastolic blood pressure,
• $x$ is the systolic blood pressure,
• $a$ is the intercept,
• $b$ is the slope of the regression line.

Step 1: Calculate the Mean of X and Y

Let's denote:

• $x$ as systolic blood pressure,
• $y$ as diastolic blood pressure.

Given

• Systolic (X): 150, 124, 149, 120, 114, 135, 136, 116
• Diastolic (Y): 109, 91, 110, 85, 82, 74, 72, 87

Calculate the means: $\bar{x} = \frac{\sum x}{n}, \quad \bar{y} = \frac{\sum y}{n}$

Step 2: Calculate the Slope (b)

The slope $b$ is given by:

$b = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (x - \bar{x})^2}$

Step 3: Calculate the Intercept (a)

The intercept $a$ is given by:

$a = \bar{y} - b\bar{x}$

Let's calculate these values.The regression equation, with systolic blood pressure as the independent variable (x), is:

$\hat{y} = 15.44 + 0.56x$

Step 4: Predict the Diastolic Blood Pressure

To predict the diastolic blood pressure when the systolic blood pressure is 118 mm Hg, substitute $x = 118$ into the regression equation:

$\hat{y} = 15.44 + 0.56(118)$

Let's calculate this value.The best predicted diastolic blood pressure for a systolic reading of 118 mm Hg is approximately 81.73 mm Hg.

Would you like any more details or have any questions? Here are a few related questions you might find useful:

1. How is the Pearson correlation coefficient calculated?
2. What does the slope of the regression line represent in this context?
3. How can we assess the accuracy of this regression model?
4. What is the significance of the intercept in the regression equation?
5. How would the prediction change if the systolic value was different?

Tip: Always check the residuals (differences between actual and predicted values) to understand how well your regression model fits the data.