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Question
Given that r = 0.4, `sigma_"y"` = 3, `sum("x"_"i" - bar"x")("y"_"i" - bar"y")` = 108, `sum("x"_"i" - bar"x")^2` = 900. Find the number of pairs of observations.
Solution 1
Given, r = 0.4, `sigma_y = 3, sum(x_"i" - bar(x))(y_"i" - bar(y)) = 108, sum(x_"i" - bar(x))^2 = 900`
Cov (X, Y) = `1/"n" sum(x_"i" - bar(x))(y_"i" - bar(y))`
= `1/"n" xx 108`
∴ Cov (X, Y) = `108/"n"`
`sigma_x = sqrt(1/"n" xx sum(x_"i" - bar(x))^2`
= `sqrt(1/"n" xx 900)`
= `sqrt(900/"n") = 30/sqrt("n")`
Since, r = `("Cov (X, Y)")/(sigma_x sigma_y)`
∴ 0.4 = `(108/"n")/(30/sqrt("n") xx 3)`
∴ 0.4 = `108/"n" xx sqrt("n")/(30 xx 3)`
∴ 0.4 = `12/(10sqrt("n")`
∴ `sqrt("n") = 12/4` = 3
∴ n = 9.
Solution 2
Given, r = 0.4, `sigma_"y"` = 3, `sum("x"_"i" - bar"x")("y"_"i" - bar"y")` = 108, `sum("x"_"i" - bar"x")^2` = 900.
Cov (x, y) = `1/"n" sum("x"_"i" - bar"x"("y"_"i" - bar"y")`
= `1/"n" xx 108`
∴ Cov (x, y) = `108/"n"`
`sigma_"x" = sqrt(1/"n" xx sum("x"_"i" - bar"x")^2`
= `sqrt(1/"n" xx 900)`
= `sqrt(900/"n") = 30/sqrt("n")`
Since, r = `("Cov (x, y)")/(sigma_"x" sigma_"y")`
∴ 0.4 = `(108/"n")/(30/sqrt("n") xx 3)`
∴ 0.4 = `108/"n" xx sqrt("n")/(30 xx 3)`
∴ 0.4 = `12/(10sqrt("n")`
∴ `sqrt("n") = 12/4` = 3
∴ n = 9
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