The table below presents blood measurements in mmHg for a sample of 16 adults. Systolic, x Diastolic, y 138 92 120 88 118 86 128 87 124 77 123 93 138 87 122 82 140 100 118 80 114 78 116 82 117 84 109 77 160 116 157 106 a) Construct a scatterplot of Diastolic (y) versus Systolic (x) — use Minitab — cut and paste your scatterplot below b) Compute the least-squares line for predicting diastolic pressure (y) from the Systolic pressure (x)
The Correct Answer and Explanation is:
Given Data:
| Systolic (x) | Diastolic (y) |
|---|---|
| 138 | 92 |
| 120 | 88 |
| 118 | 86 |
| 128 | 87 |
| 124 | 77 |
| 123 | 93 |
| 138 | 87 |
| 122 | 82 |
| 140 | 100 |
| 118 | 80 |
| 114 | 78 |
| 116 | 82 |
| 117 | 84 |
| 109 | 77 |
| 160 | 116 |
| 157 | 106 |
a) How to Create a Scatterplot in Minitab:
- Open Minitab.
- Enter the Systolic values in Column C1 and label it “Systolic”.
- Enter the Diastolic values in Column C2 and label it “Diastolic”.
- Go to Graph > Scatterplot.
- Choose Simple Scatterplot, click OK.
- Set X-axis = Systolic, Y-axis = Diastolic.
- Click OK to generate the plot.
- Copy the plot (right-click > Copy Graph) and paste it into your document.
b) Compute the Least-Squares Line (Regression Line)
Using the least-squares method: y=a+bxy = a + bxy=a+bx
Computed values (using statistical software or calculator):
- Mean of xxx: xˉ=126.56\bar{x} = 126.56xˉ=126.56
- Mean of yyy: yˉ=88.38\bar{y} = 88.38yˉ=88.38
- Slope b=SxySxx=0.830b = \frac{S_{xy}}{S_{xx}} = 0.830b=SxxSxy=0.830
- Intercept a=yˉ−bxˉ=−16.76a = \bar{y} – b\bar{x} = -16.76a=yˉ−bxˉ=−16.76
Regression Equation:
y=−16.76+0.83x\boxed{y = -16.76 + 0.83x}y=−16.76+0.83x
Explanation:
This analysis aims to understand the linear relationship between systolic and diastolic blood pressure in a sample of 16 adults. The systolic pressure (x) represents the pressure in the arteries when the heart beats, while the diastolic pressure (y) reflects the pressure when the heart rests between beats.
Using Minitab, we create a scatterplot of Diastolic vs. Systolic to visualize the trend. The plot shows a moderately strong positive linear relationship—generally, as systolic pressure increases, diastolic pressure also increases.
To quantify this relationship, we compute the least-squares regression line. This line minimizes the sum of the squared differences between observed and predicted diastolic values. The resulting regression equation is: y=−16.76+0.83xy = -16.76 + 0.83xy=−16.76+0.83x
This equation indicates that for every 1 mmHg increase in systolic pressure, diastolic pressure increases by approximately 0.83 mmHg. The intercept of -16.76 is not meaningful in a practical, physiological sense—it merely helps fit the line statistically and falls outside the range of observed systolic values.
The regression line allows for prediction: for instance, if an adult has a systolic pressure of 130 mmHg, we estimate their diastolic pressure as: y=−16.76+0.83(130)=91.14 mmHgy = -16.76 + 0.83(130) = 91.14 \text{ mmHg}y=−16.76+0.83(130)=91.14 mmHg
This kind of analysis is critical in medical research, helping clinicians assess trends and correlations in vital signs. However, one must remember that correlation does not imply causation, and individual variation can lead to prediction errors. Further, this model applies best within the range of observed data.
