In the following data, one of the values of Y is missing. Arithmetic means of X and Y series are 6 and 8

X |
6 |
2 |
10 |
4 |
8 |

Y |
9 |
11 |
? |
8 |
7 |

(a) Estimate the missing observation.

(b) Calculate correlation coefficient.

#### Solution

(a)First, we find the missing value of Y and let us denote it by a.

`barY=(SigmaY)/N=(9+11+a+8+7)/5=(35+a)/5`

`=>8=(35+a)/5=>40=35+a=>a=5`

Thus the completed series is

X | 6 | 2 | 10 | 4 | 8 |

Y | 9 | 11 | 5 | 8 | 7 |

(b)Now we Find coefficient correlation

xi | yi | xiyi | `x_i^2` | `y_i^2` |

6 | 9 | 54 | 36 | 81 |

2 | 11 | 22 | 4 | 121 |

10 | 5 | 50 | 100 | 25 |

4 | 8 | 32 | 16 | 64 |

8 | 7 | 56 | 64 | 49 |

Σxi = 30 | Σyi = 40 | Σxiyi = 214 | Σ`x_i^2`=220 | Σ`y_i^2` = 340 |

Here, n= 5, ΣX = 30, Σx^{2} = 40, ΣY=40, ΣY^{2} = 20 and Σxy = -26

Now,

`barX = (SigmaX)/n =30/5=6 " and "barY =(SigmaY)/n=40/5=8`

`:. r = (1/n Σ x iyi - barx bary)/(sqrt[[(Σxi^2)/n - barx^2] [(Σyi^2)/n - bary^2]`

`r = (1/5 xx 214 - 8 xx 6)/(sqrt(220/5 - 6^2) xx sqrt(340/5 - 8^2)`

`= (42.8 - 48)/((sqrt (44 - 36) xx sqrt(68 - 64))`

= `(-5.2)/(sqrt8 xx sqrt4)`

= `(-5.2)/(4sqrt2)`

r = -0.92