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Question
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 |
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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