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

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.

Mean: 97.3285 F

Median: 97.7 F

Mode: 97.7 F

Variance: 1.64057 F

Standard Deviation 1.28084 F

When comparing both Kathleen’s data and my own data there were some interesting points that could be seen.  First, since we are of different sexes there can be two comparisons made, first comparing our numbers with those of the general population and second, comparing our numbers with our own sex.  First I will examine my results and then will move on to Kathleen’s.  My average temperature (both on the calculator and SPSS) was 98.32571 degrees. For simplicity I will round this to 98.3.  The average temperature for males is 98.1 degrees so when I subtract my own temperature from that I get a result of -.2 degrees.  When compared to the general population I get a result of -.05 degrees.  When squared according to the standard deviation formula, my standard deviation fits nicely within the .7 degree average standard deviation of normal humans situated around either average mean (sex and general population), meaning that I have a relatively normal body temperature.  When computing Kathleen’s data I used the average temperature for females which measures at 98.4 degrees and again used the average temperature for the general population which measures 98.25.  Kathleen’s average temperature was 97.1571 which we will round to 97.2.  When subtracted from the average temperature of females her standard deviation was 1.2 degrees, a full .5 degrees below the standard deviation.  When subtracted from the general population her standard deviation was .95 degrees which is still .25 degrees away from the standard deviation.  These results show that she is a naturally cooler person than the majority.   

Shoemaker, A.L.(1996). What’s normal? Temperature, gender, and heart rate. Journal of Statistic Education. 4, (2).

 If this is the case, my average temperature in degrees C is (97.1571- 32) * 5/9= 36.1980

Pat’s is (98.32571 – 32) *5/9 = 36. 8465 degrees C

Extreme values are rare and unusual depending on how much data is present.  For instance, since this was a relatively small amount of data of only 35 points, only one extreme (91.5) was seen in the data.  This extreme was the result of a poor reading made by a faulty instrument.  If the experiment had continued longer and there was mroe data, more extremes would have been present. Extremes will also be seen more often in a larger sample size.  If we look at just the data from Pat and I, we would see very few extremes, but if we looked at the whole class or maybe even the whole school we would see more and more extremes, caused by many different reasons. 

 Outliers are usually not the most reliable data points because since they are so far away from the rest of the data, it generally means that they do not represent the average information and were most likely caused by a random act that does not pertain to the rest of the data.

Mean: 97.1571

Median: 97.7

Mode: 97.7

Standard Deviation: 1.29144

Variance: 1.668