Summary Post

This assignment allowed us to understand proportions, probabilty, and the Law of Large Numbers using physical representations.  In experimenting with 100 coin tosses, we were able to see that the probability of flipping heads (or tails) is about 50/50, even given the effects of randomness.  We were also introduced to the Law of Large Numbers, which in turn helped us understand how smaller samples yield higher variation.  This assignment also gave us a chance to work with the z-score table, and how to interpret data in relation to the normal bell curve.  A strengths of ours this week was applying class material to the assignment, since we needed to understand all of that information to complete the questions.  Again our weakness would be understanding wordpress, since we had trouble inserting gif files and formatting some of our text.

Law of Large Numbers

Provide an example that illustrates the principle of the law of large numbers as it might affect you personally.  In your example explain under which conditions you will, on average, make more mistakes in judgment and why? 

A good example of the interaction of the Law of Large Numbers and our everyday lives as students is personal health.  If you were to take the average number of days a regular person feels ill over the course of a year it would average around 10 days (according to a Warwickshire survey in 2004-2005).  Given that there are 365 days in a year and around 10 of them are days when people are feeling ill, the average person gets sick about once every five weeks, give or take a few days.  Now the Law of Large Numbers makes this seem ordinary because it is looked at through the lenses of a 365 day period and not say, a month during flu season.  If we were to only take the amount of sickness over the 2 month or so time period in January and February when flu is rampant (I got it a week ago!) and translated that to over a year long time span it would appear that people get sick a ton more than they actually do.  This large number changes dramatically when the study is conducted over a year’s time because more values get added into the equation which limits the variation a few rare cases can have on the overall outcome of the statistic.

Explain, using the standard deviation formula, why smaller samples yield larger variation.

Smaller sample sizes yield larger variation in large part because they allow rare values to disturb the integrity of the statistic.  Like I said above, the rare 6 or 7 week span of flu season will see a lot of sick days, but that segment of the year isn’t applicable, to say, the third week of June.  So in formulaic terms, a small number of x’s could result in drastic differentiations from the mean, and let randomness of rare numbers distort the final conclusion of the statistic.

Determine the proportion of males in our class.  Now, do some research and find the proportion of male psychology majors, nationwide.  How might you explain the difference in the two statistics you found?  Relate this difference to class lecture. The proportion of males in our class is 6/25 which works out to be exactly 24%.  Ironically the national average according to the website http://www.encyclopedia.com/doc/1G1-132241867.html is very close, at 25% male in the psychology major.  I would have expected that Mary Washington to have a smaller population of males compared to the national average given that this is a predominantly female institution and that our business program attracts a disproportionate amount of males, but this information suggests otherwise.

Did we take too long to change our oil?

We were given a number of statistics to determine this answer.  They were:
Mean amount of miles driven before an oil change: 3258
Our Actual Driven miles: 3467
Standard Deviation: 223In order to see if we fit within a normal driver’s range of appropriate oil change we determined the Z score of our numbers.
This was done by subtracting the mean from our actual number and dividing it by the standard deviation.
3457-3258/ 223= .9327
The answer we ended up with was a z score of .9327 which fits within the normal standard deviation of oil changes.  Also when analyzed using a Z-Chart the score turned out to be .1736 which means over 17% of the population were treating their cars worse than I was.

References
Bailly, M.D., King, A.R., McCray, J.A. (2005). General versus gender-specific attributes of the Psychology major. Journal of General Psychology, Retrieved February 4, 2008, from http://www.encyclopedia.com/doc/1G1-132241867.html
(2005, July). Retrieved February 4, 2008, from National Average Sickness Absence Web site: http://www.warwickshire.gov.uk/CORPORATE/WDC%20ADMIN%20R5.NSF/0/01416c39e9aac1dd802571e300468c91/$FILE/Sickness%20F-2006%20(15KB).pdf Z-distribution. Retrieved February 4, 2008 fro http://math2.org/math/stat/distributions/z-dist.htm

50 Tosses

How likely is the event “exactly three boys in sucession?”  

I decided to test Ms. Williams’ theory of her being “wired” for boys by tossing a coin 100 times, assigning heads to represent boys and tails to represent girls.  Of the one hundred tosses, I found that exactly three “boys” occured in a succession four times.  There were also four “boys” in a row and another time there were six, but we are only focusing on sets fo three.  So, the probablity of having three boys in a row is S/ 100-(3-1) , or 4/98, which is .04. This means that out of 100 tosses, three boys will occur in succession 4 percent of the time.

 What is the proportion of “boys” to “girls?”

After tossing a coin 100 times, I found that proportion was about exactly what one would expect it to be, 50/50.  Out of 100 tosses, I tossed 49 heads and 51 tails. The proportion of heads to tails is then 49:51.   If I was to flip the coin 10,000 times, this proportion should remain the same, since the probability of tossing heads is just as probable as tossing tails, meaning that their proportion should always remain about 50:50.