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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
116
133
112
107
110
108
105
157
154
134
Diastolic
70
91
75
71
74
69
66
103
94
87

The regression line equation is 2.9987+0.623x

If the systolic pressures of two patients differ by 13 millimeters, by how much would you predict their diastolic pressures to differ? Round the answer to three decimal places.

3 answers

To find out how much the diastolic pressures are predicted to differ based on the difference in systolic pressures, we can use the slope of the regression equation.

The regression line equation you've provided is:

\[ \text{Diastolic Pressure} = 2.9987 + 0.623 \times \text{Systolic Pressure} \]

In this context, the slope \(0.623\) indicates that for every 1 mmHg increase in systolic pressure, the diastolic pressure is predicted to increase by 0.623 mmHg.

If the systolic pressures of two patients differ by \(13\) mmHg, we can calculate the expected difference in diastolic pressures as follows:

  1. Calculate the predicted difference in diastolic pressures: \[ \text{Predicted difference in diastolic pressure} = 0.623 \times \text{Difference in systolic pressure} \] \[ \text{Predicted difference in diastolic pressure} = 0.623 \times 13 \] \[ = 8.099 \]

  2. Round the result to three decimal places: \[ \text{Predicted difference in diastolic pressure} \approx 8.099 \]

Thus, if the systolic pressures of two patients differ by 13 mmHg, we would predict their diastolic pressures to differ by 8.099 mmHg.

Predict the diastolic pressure for a patient whose systolic pressure is 137 millimeters. Round the answer to one decimal place.

To predict the diastolic pressure for a patient with a systolic pressure of 137 mmHg using the provided regression equation:

\[ \text{Diastolic Pressure} = 2.9987 + 0.623 \times \text{Systolic Pressure} \]

Substituting the systolic pressure of 137 mmHg into the equation:

\[ \text{Diastolic Pressure} = 2.9987 + 0.623 \times 137 \]

Now, calculate the value:

  1. Calculate \(0.623 \times 137\): \[ 0.623 \times 137 = 85.611 \]

  2. Add this value to 2.9987: \[ \text{Diastolic Pressure} = 2.9987 + 85.611 = 88.6097 \]

  3. Round to one decimal place: \[ \text{Diastolic Pressure} \approx 88.6 \]

Therefore, the predicted diastolic pressure for a patient with a systolic pressure of 137 mmHg is 88.6 mmHg.