Question

Find the number of pairs of observations from the following data,
r = 0.15, `sigma_"y"` = 4, `sum("x"_"i" - bar"x")("y"_"i" - bar"y")` = 12, `sum("x"_"i" - bar"x")^2` = 40.

Answer 1

Given, r = 0.15, `sigma_"y"` = 4, `sum("x"_"i" - bar"x")("y"_"i" - bar"y") `= 12, `sum("x"_"i" - bar"x")^2` = 40

Since, `sigma_"x" = sqrt(1/"n" sum("x"_"i" - bar"x")^2) = sqrt(40/"n"`

Cov (x, y) = `1/"n" sum("x"_"i" - bar"x")("y"_"i" - bar"y")`

= `1/"n" xx 12`

∴ Cov (x, y) = `12/"n"`

Since, r = `("Cov (x, y)")/(sigma_"x" sigma_"y")`

∴ 0.15 = `(12/"n")/(sqrt(40/"n") xx 4)`

∴ 0.15 = `3/("n" xx sqrt(40/"n")`

∴ 0.15 = `1/(sqrt("n") xx sqrt(40)`
Squaring on both the sides, we get

0.0025 = `1/("n" xx 40)`

∴ n = `1/(0.0025 xx 40)`

= `10000/(25 xx 40)`

= `10000/1000`

∴ n = 10