To find the regression equation and use it to predict the diastolic blood pressure for a systolic reading of 137 mmHg, follow these steps:

Step 1: Calculate the Regression Equation

The regression equation is in the form: $\hat{y} = a + b x$ where:

• $b = \frac{\text{Covariance of } x \text{ and } y}{\text{Variance of } x}$
• $a = \bar{y} - b \bar{x}$

From the table:

• $x$: Systolic values: 129, 119, 126, 145, 126, 125, 142, 110
• $y$: Diastolic values: 94, 64, 68, 84, 93, 91, 104, 79

Step 2: Compute Means ($\bar{x}$ and $\bar{y}$)

$\bar{x} = \frac{\sum x}{n}, \quad \bar{y} = \frac{\sum y}{n}$

Step 3: Compute Slope ($b$) and Intercept ($a$)

Use the formulas: $b = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (x - \bar{x})^2}$ $a = \bar{y} - b \bar{x}$

Step 4: Predict Diastolic Blood Pressure

Substitute $x = 137$ into the regression equation.

I'll calculate as np

Data

systolic = np.array([129, 119, 126, 145, 126, 125, 142, 110]) diastolic = np.array([94, 64, 68, 84, 93, 91, 104, 79])

Calculate means

mean_systolic = np.mean(systolic) mean_diastolic = np.mean(diastolic)

Calculate slope (b)

numerator = np.sum((systolic - mean_systolic) * (diastolic - mean_diastolic)) denominator = np.sum((systolic - mean_systolic) ** 2) b = numerator / denominator

Calculate intercept (a)

a = mean_diastolic - b * mean_systolic

Regression equation: y = a + b * x

Predict diastolic blood pressure for systolic = 137

predicted_diastolic = a + b * 137

a, b, predicted_diastThe regression equation is:

$\hat{y} = 6.60 + 0.61x$

Using this equation, the predicted diastolic blood pressure for a systolic reading of 137 mmHg is approximately:

$\hat{y} = 90.27 \, \text{mmHg}$

Would you like a detailed breakdown of these calculations or further assistance? Here are 5 questions to expand on this:

1. How are the slope ($b$) and intercept ($a$) derived in regression analysis?
2. What does the Pearson correlation coefficient signify for this data?
3. Can we assess the accuracy of this regression equation using residual analysis?
4. How can we calculate the coefficient of determination ($R^2$) for this dataset?
5. What are some potential sources of error in this prediction?

Tip: Always plot the data to visually inspect the relationship before using regression for predictions