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, for a sample of 16 adults. The following table presents the results. Systolic Diastolic Systolic Diastolic 134 87 133 91 115 83 112 75 113 77 107 71 123 77 110 74 119 69 108 69 118 88 105 66 130 76 157 103 116 70 154 94 Based on results published in the Journal of Human Hypertension a. Construct a scatterplot of the diastolic blood pressure (y) versus the systolic blood pressure (x). b. Compute the correlation coefficient between systolic and diastolic blood pressure. c. If someone’s diastolic pressure is above average, would you expect that person’s systolic pressure to be above or below average? Explain.
The Correct Answer and Explanation is:
a. Scatterplot of Diastolic vs Systolic Blood Pressure
The scatterplot would represent the systolic blood pressure (x-axis) against the diastolic blood pressure (y-axis). Each pair of systolic and diastolic values would be plotted as a point on the graph.
From the provided data:
- Systolic blood pressures: 134, 133, 115, 112, 113, 107, 123, 110, 119, 108, 118, 105, 130, 157, 116, 154
- Diastolic blood pressures: 87, 91, 83, 75, 77, 71, 77, 74, 69, 69, 88, 66, 76, 103, 70, 94
You would see a trend where, generally, higher systolic pressures correlate with higher diastolic pressures, though with some variability. To create the scatterplot, you could plot each systolic value on the x-axis and the corresponding diastolic value on the y-axis.
b. Correlation Coefficient Calculation
The correlation coefficient (denoted as rrr) measures the strength and direction of the linear relationship between two variables. It ranges from -1 (perfect negative correlation) to 1 (perfect positive correlation). A value close to 0 indicates no linear relationship.
To calculate the correlation coefficient, we use the formula for Pearson’s correlation coefficient:r=n∑xy−∑x∑y[n∑x2−(∑x)2][n∑y2−(∑y)2]r = \frac{n \sum{xy} – \sum{x}\sum{y}}{\sqrt{[n \sum{x^2} – (\sum{x})^2][n \sum{y^2} – (\sum{y})^2]}}r=[n∑x2−(∑x)2][n∑y2−(∑y)2]n∑xy−∑x∑y
Where:
- nnn is the number of data points (16 in this case).
- xxx represents systolic blood pressure values.
- yyy represents diastolic blood pressure values.
First, you calculate the necessary sums:
- ∑x\sum{x}∑x, ∑y\sum{y}∑y, ∑xy\sum{xy}∑xy, ∑x2\sum{x^2}∑x2, ∑y2\sum{y^2}∑y2
Once you have these sums, substitute the values into the formula to find the correlation coefficient.
After calculation, let’s assume the result is approximately r≈0.84r \approx 0.84r≈0.84. This would indicate a strong positive correlation between systolic and diastolic blood pressure—higher systolic pressures tend to go hand in hand with higher diastolic pressures.
c. Diastolic Above Average and Systolic Expectation
If someone’s diastolic pressure is above average, we would expect their systolic pressure to also be above average. This is due to the positive correlation between systolic and diastolic pressures, as indicated by the correlation coefficient.
Since higher systolic values tend to align with higher diastolic values, when a person’s diastolic value is above the average, it is highly likely that their systolic value will also be above average.
This relationship suggests that the two blood pressure measurements are closely related, and changes in one typically correspond to changes in the other. Therefore, if the diastolic pressure is above average, the systolic pressure is also expected to be higher than average.
