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Comparing weighted means in two sets of data

  1. Feb 19, 2013 #1
    1. The problem statement, all variables and given/known data

    For simplicity, I'm leaving out extraneous details (like actual numbers). Also, apologies for my formatting; I don't know how to use Latex, but I tried to make this as readable as possible. I have a set of N measurements for τ which each have their own standard deviations, and I need to determine whether the weighted average of the N measurements is significantly different from the weighted average of the first n (= N-2) of the measurements.

    2. Relevant equations

    I've read a bunch on t-tests, but I'm confused and not sure how to use one. Part of the problem is that I don't know exactly what true means/variances refer to (so I don't know if they're known or unknown in this case). I considered using a paired test somehow, but I don't think that will work since the sample sizes are different. This is for a physics assignment, but I'm trying to answer a question asking me to discuss the difference in the two means, and I don't know how to do that other than with testing for statistical significance (and I don't know how to do that in this case... things were so much more clear back in Intro to Statistics).

    I'm getting most of my information about this from here. These are the four tests that wiki lists:
    • σ = the known standard deviation of the population
    • s = the standard deviation of the data set
    • |a| = average of a

    I. unknown true means; sample standard deviations approx. equal:
    t = [(|x1|-|x2|)/Spooled]*√[Nn/(N+n)]
    Spooled = √{[s21(N-1) + s22(n-1)]/(N+n-2)}

    II. unknown true means; known, true, unequal standard deviations:
    z = (|x1|-|x2|)/√[σ21/N + σ22/n]

    III. unknown true means; unknown true standard deviations:
    t = (|x1|-|x2|)/√[s21/N + s22/n]​

    IV. paired data:
    t = |d|/(sd/√N)
    |d| = the mean of the differences for a sample of the two measurements
    sd = the standard deviation of the sampled differences
    N = the number of measurements in the sample​

    3. The attempt at a solution

    I've done this so far:

    weighted averages:
    μN = Ʃ(xi2i)/Ʃ(1/σ2i)
    μn = Ʃ(xi2i)/Ʃ(1/σ2i)

    standard deviations in the weighted means:
    σN = √[1/Ʃ(1/σ2i)]
    σn = √[1/Ʃ(1/σ2i)]

    difference in the means:
    Δτ = |μNn| ± √[σ2N + σ2n]

    The first four are equations given by the professor for other parts of the assignment. They're supposed to give a 68.3% chance of the true value lying within μ ± σ and a 95.5% of the true value lying within μ ± 2σ. I want to believe that 95.5% means I have a 95.5% confidence interval, but that seems off to me. Also, I don't know whether the σ in these equations is the same as the σ in the equations I found for the t-test.

    Any help would be much appreciated.
  2. jcsd
  3. Feb 20, 2013 #2
    Last edited by a moderator: May 6, 2017
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