Comparing weighted means in two sets of data

[bottles]

Homework Statement

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.

Homework 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 https://controls.engin.umich.edu/wiki/index.php/Comparisons_of_two_means. 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|)/S_pooled]*√[Nn/(N+n)]
S_pooled = √{[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|/(s_d/√N)
|d| = the mean of the differences for a sample of the two measurements
s_d = the standard deviation of the sampled differences
N = the number of measurements in the sample​

The Attempt at a Solution

I've done this so far:

weighted averages:
μ_N = Ʃ(x_i/σ²_i)/Ʃ(1/σ²_i)
μ_n = Ʃ(x_i/σ²_i)/Ʃ(1/σ²_i)

standard deviations in the weighted means:
σ_N = √[1/Ʃ(1/σ²_i)]
σ_n = √[1/Ʃ(1/σ²_i)]

difference in the means:
Δτ = |μ_N-μ_n| ± √[σ²_N + σ²_n]

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.

 

[sankalpmittal]

Better articles to be considered are :

http://en.wikipedia.org/wiki/Weighted_mean
http://rulesofreason.wordpress.com/2012/02/13/weighted-variance-and-weighted-coefficient-of-variation/
https://controls.engin.umich.edu/wiki/index.php/Basic_statistics:_mean,_median,_average,_standard_deviation,_z-scores,_and_p-value
http://hiv.cochrane.org/sites/hiv.cochrane.org/files/uploads/Ch09_Analysing.pdf