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Question
The two regression lines were found to be 4X – 5Y + 33 = 0 and 20X – 9Y – 107 = 0. Find the mean values and coefficient of correlation between X and Y.
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Solution
To get mean values we must solve the given lines.
4X – 5Y = – 33 ……(1)
20X – 9Y = 107 …….(2)
Equation (1) × 5
20X – 25Y = – 165
20X – 9Y = 107
Subtracting (1) and (2),
– 16Y = – 272
Y = `272/16` = 17
i.e., `bar"Y"` = 17
Using Y = 17 in (1) we get,
4X – 85 = – 33
4X = 85 – 33
4X = 52
X = 13
i.e., `bar"X"` = 13
Mean values are `bar"X"` = 13, `bar"Y"` = 17,
Let regression line of Y on X be
4X – 5Y + 33 = 0
5Y = 4X + 33
Y = `1/5` (4X + 33)
Y = `4/5"X" + 33/5`
Y = 0.8X + 6.6
∴ byx = 0.8
Let regression line of X on Y be
20X – 9Y – 107 = 0
20X = 9Y + 107
X = `1/20` (9Y + 107)
X = `9/20"Y" + 107/20`
X = 0.45Y + 5.35
∴ bxy = 0.45
Coefficient of correlation between X and Y is
r = `± sqrt("b"_"yx" xx "b"_"xy")`
r = `± (0.8 xx 0.45)`
= ± 0.6
= 0.6
Both byx and bxy is positive take positive sign.
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