The following table give information about the blood pressure level of different age at different weight

Blood Pressure 147 125 160 118 149 128
Age 56 42 72 36 63 47
Wt 65 90 60 85 75 80

What will be the blood pressure of a person at age 55 and having weight 85 kg. Also compute the standard error.

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Solution:

Let Y = Blood Pressure

X1 = Age

X2 = Weight

Y X1 X2 X12 X22 Y2 X1Y X2Y X1X2
147 56 65 3136 4225 21609 8232 9555 3640
125 42 90 1764 8100 15625 5250 11250 3780
160 72 60 5184 3600 25600 11520 9600 4320
118 36 85 1296 7225 13924 4248 8850 3060
149 63 75 3969 5625 22201 9387 11175 4725
128 47 80 2209 6400 16384 6016 10240 3760
ΣY = 827 ΣX1 = 316 ΣX2 = 395 ΣX12 = 17558 ΣX22 = 35175 ΣY2 = 115343 ΣX1Y = 44653 ΣX2Y = 60670 ΣX1X2 = 23285

Y = a + b1X1 + b2X2    …… (i)

By using least square method, normal equations

ΣY = na + b1ΣX1 + b2ΣX2     …… (ii)

ΣX1Y = aΣX1 + b1ΣX12 + b2ΣX1X2   ….. (iii)

ΣX2Y = aΣX2 + b1ΣX1X2 + b2ΣX22   …..(iv)

Now, substituting the values min equation in (ii), (iii) and (iv)

827 = 6a + 316b1 + 395b2   …… (a)

44653 = 316a + 17558b1 + 23285b2   …… (b)

60670 = 395a+ 23285b1 + 35175b2   ……. (c)

Solving equation (a), (b) and (c), we get

a = 177.06

b1 = -2.4

b2 = 1.33

Putting the value of a, b1, b2 in equation (1), We get

Y = 177.06 – 2.4X1 + 1.33X2    …….. (d)

If person’s age(X1) is 55 and weight(X2) is 85 then the blood pressure will be

Y = 177.06 – 2.4 × 55 + 1.33 × 85

Y = 158.11

When the person age is 55 and weight is 85 at that time, blood pressure will be 158.11

Standard Error:

The standard error is given by

\(σ_{yX_{1}X_{2}} = \sqrt{\frac{\sum{Y^2} – a.\sum{Y} – b_1\sum{X_1Y} – b_2\sum{YX_2} }{n-3}}\)

\( = \sqrt{\frac{115343 – 177.06 \times 827 – (-2.4) \times 44653 – 1.33 \times 60670 }{6-3}}\)

= 228.6

Therefore, The standard deviation is 228.6

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