To solve this problem, we can start with the given equation for \( y \), which represents diastolic pressure in terms of \( X \), which represents systolic pressure:
\[ y = 0.58755X + 7.16117 \]
Given that the difference in systolic pressures of two patients is \( 19 \) mm, we can express their systolic pressures as \( X_1 \) and \( X_2 \):
\[ |X_2 - X_1| = 19 \]
Let's find the difference in diastolic pressures (\( y_2 - y_1 \)) corresponding to the difference in systolic pressures \( X_2 - X_1 \).
The diastolic pressures can be expressed as:
\[ y_1 = 0.58755X_1 + 7.16117 \]
\[ y_2 = 0.58755X_2 + 7.16117 \]
Now, subtract the two equations to find the difference in diastolic pressures:
\[ y_2 - y_1 = (0.58755X_2 + 7.16117) - (0.58755X_1 + 7.16117) \]
This simplifies to:
\[ y_2 - y_1 = 0.58755(X_2 - X_1) \]
We need to calculate \( |y_2 - y_1| \):
\[ |y_2 - y_1| = 0.58755 |X_2 - X_1| \]
With \( |X_2 - X_1| = 19 \):
\[ |y_2 - y_1| = 0.58755 \times 19 \]
Calculating this gives:
\[ |y_2 - y_1| = 11.15645 \]
Finally, rounding this to three decimal places:
\[ |y_2 - y_1| \approx 11.156 \]
Thus, the predicted difference in diastolic pressures is approximately 11.156 mm.
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