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Small sample test concerning 2 means with different variances

  1. Nov 22, 2015 #1
    1. The problem statement, all variables and given/known data
    The deterioration of many municipal pipeline networks across the county is a growing concern. An article stated that the fusion process increased the average tensile strength. Data on tensile strength (psi) of linen specimens when a certain fusion process was used and when this process was not given are provided. The data are given below.

    Tensile strength (psi)
    Nofusion 2748 2700 2655 2822 2511 3149 3257 3123 3220 2753
    Fused 3027 3356 3359 3297 3125 2910 2889 2902

    Carry out a test to see whether the data support this conclusion. Use α = 0.05. Assume
    σ1 ≠ σ2

    n1 = 10 x1 = 2902.8 s1 = 277.3 n2 = 8 x2 = 3108.1 s2 = 205.9


    3. The attempt at a solution
    μ1 = nonfusion μ2 = fused
    H0: μ1 - μ2 = 0
    H1: μ1 - μ2 < 0

    for a small sample with unequal standard devations we use t'
    reject H0 when t' < -tα = -t.05 for the estimated degrees of freedom v where

    v = (s1^2/n1 + s2^2/n2)^2/[(s1^2/n1)^2/(n1-1) + (s2^2/n2)^2/(n2-1)]
    = (277.3^2/10 + 205.9^2/8)^2/[(277.3^2/10)^2/9 + (205.9^2/8)^2/7]
    = 15.9 ≅ 16

    so reject H0 when t' < -1.746

    t' = [(x1 - x2) - 0]/sqrt(s1^2/n1 + s2^2/n2) = (2902.8 - 3108.1)/sqrt(277.3^2/10 + 205.9^2/8)
    = (-205.3)/113.97 = - 1.8
    t' = -1.8 < -1.746
    we must reject H0. the data supports the conclusion.

    is this correct? I am slightly confused about if i should use t' or not because the problem never states that the populations are normal
     
  2. jcsd
  3. Nov 22, 2015 #2

    Svein

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    Science Advisor

    Seems OK to me (I am not a statistician, just a mathematician that has read up on basic statistics).
     
  4. Nov 22, 2015 #3
    Thank you. hopefully the assumption that the populations are normal doesn't cause me too much trouble
     
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