To compute the least-squares regression line for predicting diastolic pressure (\(y\)) from systolic pressure (\(x\)), we will use the following formulas:
-
Slope (\(b\)): \[ b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \]
-
Y-intercept (\(a\)): \[ a = \frac{\sum y - b(\sum x)}{n} \]
Where:
- \(n\) is the number of data points.
- \(\sum x\) is the sum of the systolic pressures.
- \(\sum y\) is the sum of the diastolic pressures.
- \(\sum xy\) is the sum of the product of systolic and diastolic pressures.
- \(\sum x^2\) is the sum of the squares of the systolic pressures.
Step 1: Gather the Data
First, we will extract our data into convenient variables:
| Systolic (\(x\)) | Diastolic (\(y\)) | |------------------|-------------------| | 133 | 91 | | 112 | 75 | | 107 | 71 | | 110 | 74 | | 108 | 69 | | 105 | 66 | | 157 | 103 | | 154 | 94 | | 134 | 87 | | 115 | 83 |
Step 2: Calculate the Necessary Sums
Let's calculate the required summations:
- \(n = 10\)
- \(\sum x = 133 + 112 + 107 + 110 + 108 + 105 + 157 + 154 + 134 + 115 = 1285\)
- \(\sum y = 91 + 75 + 71 + 74 + 69 + 66 + 103 + 94 + 87 + 83 = 823\)
- \(\sum xy = 133 \times 91 + 112 \times 75 + 107 \times 71 + 110 \times 74 + 108 \times 69 + 105 \times 66 + 157 \times 103 + 154 \times 94 + 134 \times 87 + 115 \times 83 = 12103 + 8400 + 7597 + 8140 + 7452 + 6930 + 16171 + 14476 + 11658 + 9545 = 123329\)
- \(\sum x^2 = 133^2 + 112^2 + 107^2 + 110^2 + 108^2 + 105^2 + 157^2 + 154^2 + 134^2 + 115^2 = 17689 + 12544 + 11449 + 12100 + 11664 + 11025 + 24649 + 23716 + 17956 + 13225 = 134953\)
Step 3: Compute the Slope (\(b\))
Now, we will plug in those sums into the slope formula:
\[ b = \frac{10(123329) - (1285)(823)}{10(134953) - (1285)^2} \]
Calculating the numerator:
\[ 10 \times 123329 = 1233290 \] \[ 1285 \times 823 = 1057465 \] \[ 1233290 - 1057465 = 175825 \]
Calculating the denominator:
\[ 10 \times 134953 = 1349530 \] \[ (1285)^2 = 1656225 \] \[ 1349530 - 1656225 = -306695 \]
Thus:
\[ b = \frac{175825}{-306695} \approx -0.5730 \]
Step 4: Compute the Y-intercept (\(a\))
Now we compute the y-intercept:
\[ a = \frac{\sum y - b(\sum x)}{n} \] \[ a = \frac{823 - (-0.5730)(1285)}{10} \] Calculating \(b \sum x\):
\[ -0.5730 \times 1285 \approx -735.205 \] So:
\[ a = \frac{823 + 735.205}{10} \approx \frac{1558.205}{10} \approx 155.8205 \]
Final Model
The least-squares regression line for predicting the diastolic pressure from the systolic pressure is:
\[ y = -0.5730x + 155.8205 \]
Thus, rounding the slope and y-intercept to four decimal places gives us:
- Slope (\(b\)) = -0.5730
- Y-intercept (\(a\)) = 155.8205
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