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From the data given below:

Marks in Economics: |
25 | 28 | 35 | 32 | 31 | 36 | 29 | 38 | 34 | 32 |

Marks in Statistics: |
43 | 46 | 49 | 41 | 36 | 32 | 31 | 30 | 33 | 39 |

Find

- The two regression equations,
- The coefficient of correlation between marks in Economics and Statistics,
- The mostly likely marks in Statistics when the marks in Economics is 30.

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#### Solution

Marks in Economics (X) |
Marks in Statistics (Y) |
x = `"X" - bar"X"` |
y = `"Y" - bar"Y"` |
x^{2} |
y^{2} |
xy |

25 | 43 | − 7 | 5 | 49 | 25 | − 35 |

28 | 46 | − 4 | 8 | 16 | 64 | − 32 |

35 | 49 | 3 | 11 | 9 | 121 | 33 |

32 | 41 | 0 | 3 | 0 | 9 | 0 |

31 | 36 | − 1 | − 2 | 1 | 4 | 2 |

36 | 32 | 4 | − 6 | 16 | 36 | − 24 |

29 | 31 | − 3 | − 7 | 9 | 49 | 21 |

38 | 30 | 6 | − 8 | 36 | 64 | − 48 |

34 | 33 | 2 | − 5 | 4 | 25 | − 10 |

32 | 39 | 0 | 1 | 0 | 1 | 0 |

320 |
380 |
0 |
0 |
140 |
398 |
− 93 |

N = 10, ∑X = 320, ∑Y = 280, ∑x^{2} = 140, ∑y^{2} = 398, ∑xy = − 93, `bar"X" = 320/100` = 32, `bar"Y" = 380/100` = 38

**(a)** Regression equation of X on Y.

b_{xy} = `"r"(sigma_"x")/(sigma_"y") = (sum"xy")/(sum"y"^2) = (-93)/398` = − 0.234

`"X" - bar"X" = "b"_"xy"("Y" - bar"Y")`

X − 32 = − 0.234(Y − 38)

X = − 0.234Y + 8.892 + 32

X = − 0.234Y + 40.892

Regression equation of Y on X.

`"Y" - bar"Y" = "b"_"xy"("X" - bar"X")`

b_{yx} = `"r"(sigma_"x")/(sigma_"y") = (sum"xy")/(sum"y"^2) = (-93)/140` = − 0.664

Y − 38 = − 0.664(X − 32)

Y = − 0.664X + 21.248 + 38

Y = − 0.664X + 59.248

**(b)** Coefficient of correlation (r) = `±sqrt("b"_"xy" xx "b"_"yx")`

= `sqrt((-0.234)(-0.664))`

= − 0.394

**(c)** When X = 30, Y = ?

Y = − 0.664(30) + 59.248

= − 19.92 + 59.248

= 39.328

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