A New Approach for Comparing Means of Two Populations

1
A New Approach for Comparing Means of Two
Populations

• By Brad Moss
• And Sponsored by Dr. Chand Chauhan

2
The Problem

• The research that is being presented is about a
  current problem in statistics.
• The problem is how do you compare two means of
  different populations when the variance of each
  population is unknown and unequal.
• There are many ways to deal with this problem.
  We will compare a new method to one of the other
  methods during this presentation.

3
The Idea!

• After consulting with Dr. Chauhan and reading A
  Note on the Ratio of Two Normally Distributed
  Variables
• It was decided that we could try to get around
  the unknown/unequal variance problem by taking a
  ratio of the estimated means which has not been
  done before.
• We would approximate this ratio as a normal
  distribution and from that, decide whether or not
  the means are different.

4
A Context for the Idea

• Lets say we are employees of Get Better Drug
  Company and that we are in the process of
  developing a drug for weight loss.
• Our scientists have developed two drugs, A and B.
• The company can only mass produce one of these
  drugs.

5
Things We Will Want to Know

• Overall, what is the mean weight loss for people
  taking each drug?
• Is one drug more effective than the other?

6
How To Answer

•  

7
How this Works

• A ratio (otherwise know as a fraction) of the two
  means should be one or close to one.
• How close to 1 is close enough?
• For this, we create what is know as a Confidence
  Interval using the estimated ratio and an
  estimate of the variance.
• If 1 falls into this interval, we will say that
  the means are statistically the same.

8
The Ratio Can Be Approximated by a Normal
Distribution.

• According to the paper mentioned on slide 3, a
  ratio of two normally distributed random
  variables can be approximated as a normal
  distribution.
• This happens when the standard deviation of one
  of the random variables is significantly smaller
  than the other.
• The ratio of the standard deviations should
  exceed 19 to 9 for this to work assuming the
  means are equal.

9
How Can We Use This

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10
How Can We Use This

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11
How Can We Use This

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How Does that Change Things

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Now Substituting

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The Confidence Interval!

• Now we need to determine how close to 1 do we
  have to be in order to say that the means are the
  same.
• So we will create an interval around our
  approximate mean.
• In statistics, we convert our normal values back
  to standard normal by subtracting the mean and
  then dividing by the standard deviation.

15
The Confidence Interval!

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16
The Confidence Interval!

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17
Doing Some Algebra

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18
Results

• So, given those two endpoints, if 1 is between
  them, then the means are statistically the same.
• Otherwise the means are different.

19
Results

• Using a program called Minitab, I ran 5000
  simulations for several cases and the results are
  on the following slides.

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Case 1 Equal Sample Sizes

• YN(100,10), XN(100,0.5), Sample Size 30
• Confidence Interval of 94.04
• Mean Length 0.0711821
•  
• YN(100,10), XN(100,2), Sample Size 30
• Confidence Interval of94.42
• Mean Length 0.0726839
• YN(100,10), XN(100,5), Sample Size 30
• Confidence Interval of 94.54
• Mean Length 0.0795576

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Case 2 Y has smaller sample size

• YN(100,10) Sample Size 15, XN(100,0.5) Sample
  Size 20
• Confidence Interval of 93.34
• Mean Length 0.100127
• YN(100,10) Sample size 15, XN(100,2) Sample
  Size 20
• Confidence Interval of 92.88
• Mean Length 0.101451
• YN(100,10) Sample Size 15, XN(100,5) Sample
  Size 20
• Confidence Interval of 93.68
• Mean Length 0.108929

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Case 3 X has smaller sample size

• YN(100,10) Sample Size 20, XN(100,0.5) Sample
  Size 15
• Confidence Interval of 93.8
• Mean Length 0.0869802
• YN(100,10) Sample Size 20, XN(100,2) Sample
  Size 15
• Confidence Interval of 93.56
• Mean Length .0891369
• YN(100,10) Sample Size 20, XN(100,5) Sample
  Size 15
• Confidence Interval of 93.84
• Mean Length 0.100611

23
Sources

• Jack Hayya, Donald Armstrong, and Nicolas
  Gressis. A Note on the Ratio of Two Normally
  Distributed Variables Management Science 21.11
  (1975). 14 Jan 2011 lt http//www.jstor.org/stabl
  e/2629897 gt
• Roussas, George. An Introduction To Probability
  and Statistical Inference. San Diego Elsevier
  Science, 2003