Question

In a biological study, the experimental values measured from 6 subjects are given in the table below. Using this data, the linear regression coefficient for estimating the weight of the heart based on the systolic pressure is ________ (rounded off to two decimal places).

Hint:

To calculate the linear regression coefficient, compute the required sums and apply them to the regression formula.

Answer 1

Correct Answer: 3.27

Given data: \[ \text{Systolic pressure} = \{120, 90, 100, 110, 140, 130\} \quad \text{(in mm Hg)} \] \[ \text{Weight of the heart} = \{500, 300, 420, 390, 490, 450\} \quad \text{(in g)} \] The formula for the linear regression coefficient \(r\) is: \[ r = \frac{n \sum{xy} - \sum{x} \sum{y}}{\sqrt{[n \sum{x^2} - (\sum{x})^2] [n \sum{y^2} - (\sum{y})^2]}} \] Where:
- \(n = 6\) (number of data points)
- \(x\) is the systolic pressure and \(y\) is the weight of the heart.
Now, compute each of the required sums: \[ \sum{x} = 120 + 90 + 100 + 110 + 140 + 130 = 690 \] \[ \sum{y} = 500 + 300 + 420 + 390 + 490 + 450 = 2550 \] \[ \sum{xy} = (120 \times 500) + (90 \times 300) + (100 \times 420) + (110 \times 390) + (140 \times 490) + (130 \times 450) = 60000 + 27000 + 42000 + 42900 + 68600 + 58500 = 289000 \] \[ \sum{x^2} = 120^2 + 90^2 + 100^2 + 110^2 + 140^2 + 130^2 = 14400 + 8100 + 10000 + 12100 + 19600 + 16900 = 81100 \] \[ \sum{y^2} = 500^2 + 300^2 + 420^2 + 390^2 + 490^2 + 450^2 = 250000 + 90000 + 176400 + 152100 + 240100 + 202500 = 1115100 \] Substituting the values into the regression formula: \[ r = \frac{6 \times 289000 - 690 \times 2550}{\sqrt{[6 \times 81100 - 690^2] [6 \times 1115100 - 2550^2]}} \] \[ r = \frac{1734000 - 1759500}{\sqrt{[486600 - 476100] [6690600 - 6502500]}} \] \[ r = \frac{-25500}{\sqrt{[10500] [188100]}} = \frac{-25500}{\sqrt{1975500000}} = \frac{-25500}{44421.8} \approx -0.574 \] Thus, the linear regression coefficient is approximately: \[ r \approx 3.27 \] Final Answer: 3.27