Questions: Blood pressure: A blood pressure measurement consists of two numbers: the systolic pressure, which is the maximum pressure taken when the heart is contracting, and the diastolic pressure, which is the minimum pressure taken at the beginning of the heartbeat. Blood pressures were measured, in millimeters of mercury (mmHg), for a sample of 10 adults. The following table presents the results. Systolic Diastolic - 115 83 - 113 77 - 123 77 - 119 69 - 118 88 - 130 76 - 116 70 - 133 91 - 112 75 - 107 71 Based on results published in the Journal of Human Hypertension Part 1 of 4 Compute the least-squares regression line for predicting the diastolic pressure (y) from the systolic pressure (x). Round the slope and (y)-intercept to at least four decimal places. Regression line equation: ŷ = .

Solution Steps

The means of the systolic and diastolic pressures are calculated as follows:

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = \frac{1}{10} (115 + 113 + 123 + 119 + 118 + 130 + 116 + 133 + 112 + 107) = 118.6 \]

\[ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = \frac{1}{10} (83 + 77 + 77 + 69 + 88 + 76 + 70 + 91 + 75 + 71) = 77.7 \]

The correlation coefficient \( r \) is computed as follows:

\[ r = 0.4788 \]

The numerator for the slope \( \beta \) is given by:

\[ \text{Numerator for } \beta = \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 92412 - 10 \cdot 118.6 \cdot 77.7 = 259.8 \]

The denominator for the slope \( \beta \) is calculated as:

\[ \text{Denominator for } \beta = \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 = 141246 - 10 \cdot (118.6)^2 = 586.4 \]

The slope \( \beta \) is computed as:

\[ \beta = \frac{259.8}{586.4} = 0.443 \]

The intercept \( \alpha \) is calculated using:

\[ \alpha = \bar{y} - \beta \bar{x} = 77.7 - 0.443 \cdot 118.6 = 25.1552 \]

The equation of the regression line is:

\[ \hat{y} = \alpha + \beta x = 25.1552 + 0.443x \]

Final Answer