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.

Summary

In the spirit of keeping things concise I will keep this as short as I can.  When we took our mean temperatures and compared them to both the general population and our respective sexes, Kathleen and I determined that she was naturally cooler comparitively than I was in terms of sex and population, which is contrary to the statistics found by scientists.  Again the limited amount of time that each of us recorded our temperatures is a potential reason along with  illness, faulty equiptment, events preceding the recording etc.

My data done by hand

Here is the data that I calculated using my hand calculator, it’s pretty much the same as what SPSS calculated.

Mean: 97.3285 F

Median: 97.7 F

Mode: 97.7 F

Variance: 1.64057 F

Standard Deviation 1.28084 F

Allen L. Shoemaker’s Article

In Allen L. Shoemaker’s article entitled What’s normal? Temperature, Gender, and Heart Rate  he challenges the age old assumption that 98.6 degrees is the mean human body temperature.  Due to the advance in thermometer technology amongst other things he determined that 98.6 degrees is actually a few tenths of a degree off of what the true human body temperature is.  Using modern thermometers and updated methods it was determined that the mean male temperature was 98.1 degrees and the average female temperature was 98.4.  When combined the average temperature of humans turned out to be 98.25 degrees.  Shoemaker cited numerous reasons for this including the above mentioned 100 year old thermometers, diurnal fluctuations, and possible rounding errors when converting Centagrade into Fahrenheit. 

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).

Celsius vs Fahrenheit

The formula for converting degrees fahrenheit to degrees celsius is (Degrees F- 32) * 5/9.

 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

Men vs. Women; Battle of the Temperatures

While it has been found that men are generally “cooler” than women, our data appears to be the opposite.  My average body temperatue was 97.1571 F, while Pat’s 98.3257 F. Most men are 0.1 to 0.2 degrees cooler than women, but I am on average 1.1686 degrees F cooler than Pat.  Men and women usually have about the same standard deviation of about 0.7 degrees.  Pat’s standard deviation (.45719) was a bit lower than this, while mine was quite a bit higher (1.29144). This proves that outliers have quite an influence on standard deviation as well, since mine was so high as a result of my one extreme data point. Based on the variance between our data and the average data collected about the temperature of men and women, we are not very representative of most of the population. 

There are many things that will affect the mean of the data, most of them being random.  This could include being sick, being ina freezing or very hot place, or having a broken thermometer. Things that might affect the mean of the entire population could be age, location, and health. 

If we continuted to take measurements throughout the semester, our central tendencies would be much more accurate.  With more data, we would be able to see more patterns and outliers would not have such an affect.  It is certain that we would be able to draw a more accurate mean, median, mode, and standard deviation.  Of course, it is always possible that something completely random might occur, such as one of us getting very sick, and our data would be less accurate, since our average temperature will be much more influenced by extremes.

Which Central Tendency is Most Influenced by Extremes?

I found that the central tendency that is most influenced by extremes is definitely the median.  The reason for this is that the median is found by placing the data points in numerical order, so when the numbers are all lined up and an outlier is added to the data, the median can shift from side to side.  For instance, if I had the temperatures 97.0, 97.2, 98.2, 98.5, and 98.6, the median is 98.2.  But,  if for some reason an exteme outlier is added, such as 91.5, it would have to go on the lowest end, forcing the median to shift from 98.2 to 97.7, which is quite a difference.  Of course, if an outlier was added on the higher end, such as 100.0 F, the two extremes would balance each other out at the median would remain the same. Extremes can occur for any number of randomly attributed instances, such as catching a cold or fever, or they can even be caused by a faulty intsrument. 

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.

My Data from SPSS

Here is the data that SPSS calculated regarding my body temperature.

Mean: 97.1571

Median: 97.7

Mode: 97.7

Standard Deviation: 1.29144

Variance: 1.668