statistics - stuck on the final question

The torque, *T *Nm, required to rotate shafts of different diameters, *D *mm, on a machine has been tested and recorded below.

*D *(mm) | 6 | 10 | 14 | 18 | 21 | 25 |

*T *(Nm) | 5.5 | 7.0 | 9.5 | 12.5 | 13.5 | 16.5 |

(a) Use the method of least squares to determine the linear regression equation that relates the torque to the diameter.

(b) Estimate the value of torque required when the diameter is 16 mm.

(c) Calculate the coefficient of correlation between these variables and comment on your prediction obtained in (b).

What I’ve done so far:

Question a

X | Y | X^{2} | Y^{2} | XY |

6 | 5.5 | 36 | 30.25 | 33 |

10 | 7 | 100 | 49 | 70 |

14 | 9.5 | 196 | 90.25 | 133 |

18 | 12.5 | 324 | 156.25 | 225 |

21 | 13.5 | 441 | 182.25 | 283.5 |

25 | 16.5 | 625 | 272.25 | 412.5 |

∑X= | ∑Y= 64.5 | ∑X^{2}= 1722 | ∑Y^{2}= 780.25 | ∑XY= 1157 |

Normal equations for Y on X are

∑Y=a∑X+nb & ∑XY=a∑x^{2}+b∑X

Substitute values from tables gives

64.5=94a=6b & 1157=1722a+94b

Equation 1 gives b= (64.5-94a)/6

Equation 2 gives 1157=1722a+94((64.5-94a)/6)

Solving gives a=0.59 & b=1.507

**The regression line equation is Y=0.59X+1.507**

Question b

**11Nm**

Question c

X | Y | x=X-__X__ | y=Y-__Y__ | xy | x^{2} | y^{2} |

6 | 5.5 | -9.67 | -5.25 | 50.77 | 93.51 | 27.56 |

10 | 7 | -5.67 | -3.75 | 21.26 | 32.15 | 14.06 |

14 | 9.5 | -1.67 | -1.25 | 2.09 | 2.79 | 1.56 |

18 | 12.5 | 2.33 | 1.75 | 4.08 | 5.43 | 3.06 |

21 | 13.5 | 5.33 | 2.75 | 14.66 | 28.41 | 7.56 |

25 | 16.5 | 9.33 | 5.75 | 53.65 | 87.05 | 33.06 |

∑X= 94 | ∑Y= 64.5 | | | ∑XY= 146.51 | ∑ x^{2}=249.34 | ∑ y^{2}=86.86 |

__X__=15.67 | __Y__=10.75 | | | | | |

r=∑xy / √((∑ x^{2})( ∑ y^{2}))

r= 146.51 / √ (249.34 x 86.86)

r= 1

**not sure if this is the final answer for c need some help here and guidance if Im completely wrong here on all of the above. **

Re: statistics - stuck on the final question

I get

a) T = 0.5876D+1.5448

b) T = 10.95

c) r = 0.9954

So you're on the right track. Rounding error is most likely the differnce between your answers and mine.

Re: statistics - stuck on the final question

Thanks for that but how would you comment on your prediction from answer b in relation to what you got for answer c?

Re: statistics - stuck on the final question

D = 16 is an interpolation of the data set and as r is close to one you can say the prediction is very reliable

Re: statistics - stuck on the final question

Hi, try this may be it helps.

Method Of Least Square