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Math Help - statistics - stuck on the final question

  1. #1
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    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 Iíve done so far:

    Question a

    X Y X2 Y2 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 ∑X2= 1722 ∑Y2= 780.25 ∑XY= 1157


    Normal equations for Y on X are
    ∑Y=a∑X+nb & ∑XY=a∑x2+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 x2 y2
    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 ∑ x2=249.34 ∑ y2=86.86
    X=15.67 Y=10.75

    r=∑xy / √((∑ x2)( ∑ y2))


    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.
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  2. #2
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    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.
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  3. #3
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    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?
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  4. #4
    Master Of Puppets
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    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
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  5. #5
    Newbie Neeraj's Avatar
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    Re: statistics - stuck on the final question

    Hi, try this may be it helps.

    Method Of Least Square
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