Calculate the correlation coefficient from the following data and interpret it. x 9 7 6 8 9 6 7 y 19 17 16 18 19 16 17 - Mathematics and Statistics

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Sum

Calculate the correlation coefficient from the following data and interpret it.

x 9 7 6 8 9 6 7
y 19 17 16 18 19 16 17
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Solution

  xi yi xi2 yi2 xiyi
  9 19 81 361 171
  7 17 49 289 119
  6 16 36 256 96
  8 18 64 324 144
  9 19 81 361 171
  6 16 36 256 96
  7 17 49 289 119
Total 52 122 396 2136 916

From the table, we have
n = 7, `sum"x"_"i"` = 52, `sum"y"_"i"` = 122, `sum"x"_"i"^2` = 396, `sum"y"_"i"^2` = 2136, `sum"x"_"i""y"_"i"` = 916

∴ `bar"x" = (sum"x"_"i")/"n" = 52/7`

`bar"y" = (sum"y"_"i")/"n" = 122/7`

∴ `bar"x"bar"y"=(52xx122)/49` = `6344/49`

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

= `916/7-6344/49`

= `(6412-6344)/49`

= `68/49`

`sigma_"x"^2 = (sum"x"_"i"^2)/"n" - (bar"x")^2`

= `396/7 - (52/7)^2`

= `(8772-2704)/49`

= `68/49`

`sigma_"y"^2 = (sum"y"_"i"^2)/"n" - (bar"y")^2`

= `2136/7 - (122/7)^2`

= `(14952 - 14884)/49`

= `68/49`

∴ `sigma_"x" sigma_"y" = sqrt(sigma_"x"^2 sigma_"y"^2)`

= `sqrt(68/49 xx 68/49)`

= `68/49`

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

= `((68/49))/(68/49)` = 1
∴ The value of r indicates perfect positive correlation between x and y.

Concept: Interpretation of Value of Correlation Coefficient
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Chapter 5: Correlation - Miscellaneous Exercise 5 [Page 64]

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