# Homework Help: Probability and Hypothesis Testing Problem

1. Oct 21, 2005

### tomcue

I need some help with a problem. Let me first post the results.

289 (10 lottery ball random draws) = 4.66 ave......5.50 mean
67 (9 lottery ball random draws) = 3.30 ave........5.00 mean
49 (8 lottery ball random draws) = 4.08 ave .......4.50 mean
37 (7 lottery ball random draws) = 3.00 ave........4.00 mean
14 (6 lottery ball random draws) = 2.79 ave .......3.50 mean
2 (5 lottery random draws) = 2.50 ave .............3.00 mean

In each of the above samples, the lottery balls were numbered 1 thru 10,9,8,7,6,5. The results have in all cases deviated on the low side. I need help in determining the odd of such a deviation and how to test the hypothesis for each. I also need to know if there is a way to hypothesis test the entire population. You are dealing with a novice.

After I get the answers I will explain why I have presented this question.

Thanks,

Tom

Last edited: Oct 21, 2005
2. Oct 24, 2005

### EnumaElish

In the first set of experiments, you have made 289 random draws from balls numbered 1-10 and the average is 4.66, vs. the expected value 5.5. You can indeed test whether this 4.66 is statistically distinguishable from 5.5. You can also test whether the six averages you have calculated are jointly distinguishable from their respective expectations.

I think what you are asking is, what is the probability that your average is 4.66 or lower given a true mean of 5.50. A statistics theorem says that sample averages tend to be distributed normally. Draw a bell curve with 5.50 corresponding to the peak in the middle (the true mean). Now to the left of the middle, mark your calculated average of 4.66. To calculate the probability you'll need to calculate the standard deviation for the sample average. Just like you know the true mean (5.50), you should be able to come up with the true standard deviation. You need to convert this standard deviation to the standard deviation of the sample average by dividing it into $\sqrt{289}$ ($\sqrt{n}$ for n experiments) -- I guess, better check it. Once you have the mean and the standard deviation, you can calcuate the probability of $\bar x < 4.66$.

3. Oct 24, 2005

### EnumaElish

In the first set of experiments, you have made 289 random draws from balls numbered 1-10 and the average is 4.66, vs. the expected value 5.5. You can indeed test whether this 4.66 is statistically distinguishable from 5.5. You can also test whether the six averages you have calculated are jointly distinguishable from their respective expectations.

I think what you are asking is, what is the probability that your average is 4.66 or lower given a true mean of 5.50. A statistics theorem says that sample averages tend to be distributed normally. Draw a bell curve with 5.50 corresponding to the peak in the middle (the true mean). Now to the left of the middle, mark your calculated average of 4.66. To calculate the probability you'll need to calculate the standard deviation for the sample average. Just like you know the true mean (5.50), you should be able to come up with the true standard deviation. You need to convert this standard deviation to the standard deviation of the sample average by dividing it into $\sqrt{289}$ ($\sqrt{n}$ for n experiments) -- I guess, better check it. Once you have the mean and the standard deviation, you can calcuate the probability of $\bar x < 4.66$.

4. Oct 24, 2005

### tomcue

EE,

Please keep in mind that I am a complete novice with regard to this stuff. You are talking with someone that cannot do much more then figure the averages and see the problem.

What I have done at this point is I tried to duplicate the process using Excel. I first generated random numbers using the descrete method. In each column I generated 289 (1-10) and used all 256 columns in excel. After averaging those 256 results, I used them to produce descriptive statistics. Those results were used to calculate normal distribution. When I did this for the 1-10 group, I get a number like this 0.000027%. Naturally, if I re-write the numbers, that figure will change slightly but not by much and I would think that the odds of this being random are nearly impossible.

Let me now explain why I presented this to the forum in the first place. These are not experiments at all. They are actual results from a horse trainer in my state that trains and races standardbreds. Those 458 experiments are not experiments but are results from what was supposed to be 458 random draws over a 12 month period. I sorted the field sizes for testing. The problem comes about in that the trainers spouse was the person performing the random draw that does not appear to be random. As drawing inside post positions gives a decided advantage, many folks smell a rat here. That is why I need the proper method of testing the results.

I would like to know if my method of testing the results is proper. I also need to know if the results would pass a hypothesis test. If you or anyone else reading this does not mind helping me here, send me your email address and I will email the file for review. My email address is hrbinsurance@starband.net