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 Let’s 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

12. How Does that Change Things

13. Now Substituting

14. 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!

17. Doing Some Algebra

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.

20. Case 1: Equal Sample Sizes Y~N(100,10), X~N(100,0.5), Sample Size 30 Confidence Interval of 94.04% Mean Length: 0.0711821 Y~N(100,10), X~N(100,2), Sample Size 30 Confidence Interval of94.42% Mean Length: 0.0726839 Y~N(100,10), X~N(100,5), Sample Size 30 Confidence Interval of 94.54% Mean Length: 0.0795576

21. Case 2: Y has smaller sample size Y~N(100,10) Sample Size 15, X~N(100,0.5) Sample Size 20 Confidence Interval of 93.34% Mean Length: 0.100127 Y~N(100,10) Sample size 15, X~N(100,2) Sample Size 20 Confidence Interval of 92.88% Mean Length: 0.101451 Y~N(100,10) Sample Size 15, X~N(100,5) Sample Size 20 Confidence Interval of 93.68% Mean Length: 0.108929

22. Case 3: X has smaller sample size Y~N(100,10) Sample Size 20, X~N(100,0.5) Sample Size 15 Confidence Interval of 93.8% Mean Length: 0.0869802 Y~N(100,10) Sample Size 20, X~N(100,2) Sample Size 15 Confidence Interval of 93.56% Mean Length:.0891369 Y~N(100,10) Sample Size 20, X~N(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 http://www.jstor.org/stable/2629897 Roussas, George. An Introduction To Probability and Statistical Inference. San Diego: Elsevier Science, 2003