1. Present Data
The following bar graph displays the result of 100 completely random coin flips; these coin flips represent the random variation affecting one’s probability of having a girl or a boy child. In this experiment, Drew and I were testing the idea that Ms. Williams could be wired for boys after having three sons in a row. When the quarter was flipped, tails represented girls and heads represented boys.
As you can see in the graph, the number of times that tails (girls) came up when the quarter was flipped greatly exceeds the number of times heads (boys) came up, by a ratio of 64-36. Therefore, according to the results of this particular experiment, Ms. Williams is actually not wired for boys at all simply because she was found to have more girls than boys due to random variation.
2. Provide Answers To Questions Related to Weekly Topic
a. Making Inferences From Small Samples
# times coin came up tails: 64/100 = 64 girls
# times coin came up heads: 36/100 = 36 boys
# times 3 heads (boys) came up in a row: 3 = S
# times possible that 3 heads (boys) could come up in a row: 98 = 100-(3-1)
Based on this experiment, the probability of having three heads come up in a row is only 3%. This percentage was found by dividing the value of S by 98, as stated above, and multiplying by 100. Therefore, since Ms. Williams did indeed have three boys in a row, she fits into this small percentage. However, it is not safe to say that she is “wired” for boys because this particular experiment also reflects that random variation can cause an abundance of one side (girls) over another (boys) of a normally 50/50 chance event (girls or boys). If we flipped the quarter 10,000 times instead of only 100 times, the occurrence of heads (boys) would rise. Likewise, the occurrence of tails (girls) would steadily decrease, thus creating a proportion of boys v. girls closer to 50%.
b. The Principle of the Law of Large Numbers
One example of where the law of large numbers affected me was when I developed a brief devotion to Outback Steakhouse. I had not been to the restaurant for years. I went to eat there, enjoyed the food, and went back three more times in the next year. However, when I went the fourth time, I acquired a bad case of food poisoning. I even got a phone call from the restaurant a week later as an apology (apparently many people who ate there that night had been poisoned by tainted seafood). Nonetheless, I have never been back to the restaurant due to this bad memory.
My example illustrates one condition in which I would make a mistake in judgment. I may have gotten poisoned the fourth time I went to the restaurant, but this certainly does not mean that people who go to Outback have a 25% chance of being served tainted food. Despite the fact that I could acknowledge this reasoning, I still remained aversive to going back, as if the probability of 1:4 was possible. I made an error about the situation, similar to that of the mother who had only boys and that of the orange inspector from our class assignment paper.
Going to Outback again was probably not going to result in another bad experience, yet I had a fixed impression of the event just like the orange inspector thought all the oranges were bad and the mother thought that she could not have girls.I also could have experienced other rare conditions at Outback that would have affected my judgment of the restaurant. For example, the oven could have broke and we would have not received a meal, the server could have brought us the wrong dishes and we may have received the original meals for free as a result, or we may have driven to the restaurant to discover that they were closing down and we would have to leave. All of these potential influences could affect my judgment of the restaurant, because they are rare events and are extreme cases. If any of these events occurred in the four times I went to Outback, then I would have a rare event in a small sample size. With less experience going to the restaurant, I would be prone to thinking that such an unusual event was the norm (even if I knew otherwise that this was not the case).
Standard deviation formula: SD=square root (sum of a given variable X minus the mean squared) /n or the number of times the event occurred
Assuming that n is the only variable that changes when comparing two SD equations, the equation where n equals a smaller number will equate to a larger end answer. This is due to the fact that a greater bottom value (a larger sample) results in a smaller answer (less variation), than does a tiny bottom value (a smaller sample) with the same top value sum which results in a larger answer (more variation).
c. Proportion of Males in Class v. Male Psychology Majors (nationwide)
Professor MacEwen mentioned that “there are 8 males in our class out of 46 total people”. However, I found from a web article that “73% of bachelor’s degrees awarded in psychology were given to women”. The article titled “Major Conclusions” proceeded to conclude that “within psychology graduate programs, women constitute the majority” (Major Conclusions).
8/46 = .1739 73/100 = .7300
Despite the fact that the article is out of date (the figure of 73% was from 1991), the article mentions the trend of females roles in psychology increasing and surmises that their numbers among psych majors will continue to rise as the number of men declines. Therefore, the lower frequency of males in our class could be due to the fact that, over a period of about two decades, men’s numbers in psych programs have indeed experienced a dramatic reduction. However, the results may not have been related to nationwide trends, but instead to state college trends or even differences in various psych classes. For instance, do other Virginia colleges have more or less psych majors who are males? Does our statistics class happen to have a disproportionately high or low amount of males relative to all the males who are currently majoring in psych at UMW?
As demonstrated in class with the law of large numbers, we may be obtaining an inaccurate sample. For instance, if our class ratio of males to total students is 8:46 and there are only 10 male psych majors currently at UMW than our sample would be disproportionately inaccurate. However, if there were 200 pysch majors who were male, our sample may be unrepresentative as well, since it would be abnormally low instead of abnormally high. Also, the proportion of psych majors at UMW compared to other VA schools might be higher or lower.
d. Using Z-Scores
# miles I waited to change oil: 3467 miles
Mean # miles people wait to change oil: 3258 miles, SD = 223 miles
When first addressing this debate with my father, given the statistics listed above, I would initially argue that I chose to change my oil well within one standard deviation of the population mean. When looking at the structure of the normal distribution curve, I would show my dad that 68% of people will change their oil between 3035-3481 miles:
Population mean (3258) +/- standard deviation (223) = 1 SD (3035-3481)
Therefore, choosing to change my oil at 3467 miles falls within one positive standard deviation of the population mean. When using the Z-table, we first had to convert our mileage into z-scores, aka standard scores. We plugged in our values as follows:
Z = (x-value – population mean) / SDZ = (3467 mi. – 3258 mi.) / 223 mi. = 0.9372
After obtaining 0.9372 as the z-score, I rounded it off to 0.94 and then moved to the z-table provided on blackboard to find the area between the mean and our z-score. When looking at the table, a z-score of 0.94 correlated with an area of 0.3264. So, speaking in terms of probability, it can be said that 0.3264 of scores fall between the mean (3258 mi.) and a z-score of 0.94 (3467 mi.). To convert this percentage, I used the following formula:
% = (probability)(100) % = (0.3264)(100) = 32.64%
Therefore, in this particular argument with my father, I would tell him that 17.36% of people wait longer than I did to change my oil. To find this value, I simply subtracted 32.64% from 50% (the area above the mean).
3. Relation to Statistical Topic From Class
This experiment exemplifies the fact that despite randomness, most events occurring in the world today will develop a normal or bell-shaped curve. This frequency distribution is theoretical in the sense that it only occurs in our mind, although we can make educated inferences based upon it. A normal frequency distribution can be converted into a probability distribution by simply converting percentage into probability, as outlined earlier. This procedure was put into practice in the earlier examples. By finding the probability distribution, as was done in part (d) of section 2, the area under the distribution curve can also be found, making for easy comparison. Because we are usually given the standard deviation scores, which are not convertible in probability distributions, we must convert them into standard scores, aka z-scores. By using the equation listed above to calculate z-scores, we can then look at the z-table to find the corresponding area. Again, part (d) of section 2 put this concept into practice.
4. Weaknesses/Limitations and Strengths
Flaws/Weaknesses: Girls came up significantly more frequently than boys in the sample of 100 coin tosses (64 vs. 36), which is not close to the 50-50 chance that it should be around. Also, boys appeared 3 times in a row only on 3 occasions.
Strengths: From the data we collected, it can still be concluded that Mrs. Williams can in fact have girls. Our data also explain how Mrs. Williams could have figured that she could only have boys (she fits into the scenario of those 3 cases where boys appeared 3 times in a row).
5. References
1. Unknown Author. “Major Conclusions” http://www.apa.org/pi/taskforce/majorconcl.html
2. MacEwen, Professor. PowerPoint presentations (lectures from 1/30, 1/31, and 2/4)
