The age in years of 7 young couples is given below. Calculate husband’s age when wife’s age is 38 years. Husband (x) 21 25 26 24 22 30 20 Wife (y) 19 20 24 20 22 24 18 - Mathematics and Statistics

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The age in years of 7 young couples is given below. Calculate husband’s age when wife’s age is 38 years.

Husband (x) 21 25 26 24 22 30 20
Wife (y) 19 20 24 20 22 24 18
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Solution

Given, X = Age of husband,

Y = Age of wife

  X = xi Y = yi xi2 yi2 xiyi
  21 19 441 361 399
  25 20 625 400 500
  26 24 676 576 624
  24 20 576 400 480
  22 22 484 484 484
  30 24 900 576 720
  20 18 400 324 360
Total 168 147 4102 3121 3567

From the table, we have

`n = 7, sumx_"i" = 168,  sumy_"i" = 147, sumx_"i"^2 = 4102`

`sumx_"i"y_"i" = 3567, sumy"i"^2 = 3121`

∴ `bar(x) = (sumx_"i")/"n" = 168/7 = 24`

`bar(y) = (sumy_"i")/"n" = 147/7 = 21`

byx = `(sumx_"i"y_"i" - "n"bar(x) bar(y))/(sumx_"i"^2 - "n"bar(x)^2)`

= `(3567 - 7 xx 24 xx 21)/(4102 - 7 xx (24)^2`

= `(3567 - 3528)/(4102 - 4032)`

= `39/70`

= `0.557`

Now, a = `bar(y) - "b"_(yx)  bar(x)`

= `21 – 0.557 × 24`

= `21 – 13.368`

= `7.632`

bxy = `(sumx_"i"y_"i" - "n"bar(x) bar(y))/(sumy_"i"^2 - "n"bar(y)^2)`

= `(3567 - 7 xx 24 xx 21)/(4102 - 7 xx (21)^2`

= `(3567 - 3528)/(3121 - 3087)`

= `39/34`

= `1.147`

Now, a' = `bar(x) - "b"_(xy)  bar(y)`

= `24 – 1.147 × 21`

= `24 – 24.087`

= `– 0.087`

The regression equation of age of husband (X) on age of wife (Y) is

X = a' + bxy Y

`∴ X = – 0.087 + 1.147 Y`

when wife’s age is 38 years, Y = 38

`∴ X = – 0.087 + 1.147 × 38 = 43.5`

∴ Husband’s age is 43.5 years, when wife’s age is 38 years.

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Chapter 2.3: Linear Regression - Q.4

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