Definition 

Law of Large Numbers – the rule or theorem that the average of a large number of independent measures of a random quantity tends toward the theoretical average of that quantity (2006).

Relate to class topic

This assignment relates to our class topic, because it deals with using the theoretical model of the normal curve to find the probability of something occuring. The law of large numbers can also be seen in this assignment, especially when flipping the coin. When one side of the coin had repeated a few times, after a few more trials the other side would then repeat, to make it equal again.

Data

Shown throughout the questions

Questions

Probability of Three Boys 

a)Ms. Williams : I used the TI-83 calculator’s random application and used heads to be boys and tails to be girls, and I had the occurrance of exactly three boys only three times. I counted how many groups of three there could be with a girl in between and found that there could be twenty five times that exactly three boys would be born in a row. When I did the calculation I found that there was a twleve percent chance of having exactly three boys born in a row. (I divided 3(the times it happened when I flipped the coin) by 25 (the possible times that three boys could have occurred in a row with in the sample of 100)). The total amount of boys was 49, which made the total amount of girls 51. This is close to 50%, but not exactly. If I were to continue flipping for a total of 10,000 times the proportion of boys to girls would stay about the same, but hopefully it would be 50/50. The proportion of boys to girls should be equal with more trials, because of the law of large numbers (Brittany).

Only one time did exactly three boys in succession occur during Matt’s trial using the random application of the TI-83 calculator. (Matt)

The Law of Large Numbers in Real Life

b) If I were to not study for a test, or wait until the night before to cram, I could still get a good grade, the first few times that I use this studying technique. After a few times though, I may not get a good grade on a test. The good grades that I got, the first few times that I waited to study or didn’t study at all, random factors could have influenced the fact that I still managed to get a good grade. As I continued to use this strategy, there would be sometimes that it worked and other times that it didn’t. If I based my decision of whether this technique would continue to work or not on one or two tests, I would make a bad decision. (Brittany)

The law of large numbers is dependent upon a large number of trials. A good example is the flip of a coin. If you only flip the coin a few times the outcome could be slanted toward heads or tails. If the coin is flipped 1000 times then according the law of large numbers the outcomes will be almost even between heads and tails. An example of the law of large numbers that affects me personally takes place at the beginning of our weekly backyard football games. It is a tradition that my friends and I have managed to carry on for many years now. After teams have been chosen, we flip a coin to see who will receive that ball first just like in professional football. I am usually designated as the person who decides for my team. I decided when we first started this tradition that I would always pick tails. Over the long term this has almost guaranteed that I will win at least half of the coin flips. If I for lack of a better term flipped back and forth between the two choices, the likelihood of me being at about 50% correct would be slim. By choosing to pick the same side week in and week out, it is safe to say that I will win the toss at least one of those weeks. I chose to pick tails every week, because the first few times that was the outcome. That was only by chance, since there is always a 50/50 chance of either heads or tails.

Smaller samples yield larger variation, because there is less data and random numbers can skew it. The more data that is acquired the more accurate the mean and other statistical measures will be. The denominator in the standard deviation formula is the number of trials minus 1. The larger this number becomes the more telling and accurate the standard deviation will become. This is because it will be a true representation of the overall data and not just a few numbers that may include an extremely unlikely number that skews it. (Matt)

The Proportion of Males 

c) Of the 28 people in our lab section there are 6 males and 22 females. The proportion of boys in our class is 21.4%. In 1991, the proportion of male psychology majors in the United States was 27% for Baccalaureate degrees (1993). Since this statistic is fifteen years old, it could be off, but it is the most relevant data we could find. We now know that the percentage of women who go to college is higher than the percentage of males. In 1991, 42% of college students were females (1993).  In October 2005, 57% of college students were said to be female (2005). Psychology is a field dominated by women and since more women are going to college than before, we think that the proportion of female psychology majors has increased, therefore decreasing the proportion of males.

Did you wait too long to change your oil?

d) The average amount of miles that people wait to get their oil changed was 3258 miles, with a standard deviation of 223 miles. The person in question waited 3467 miles. If you subtract the mean from the value, you get 209 miles which is within one standard deviation from the mean. When looking at the normal curve, the percent of people who wait until they have driven 3467  miles can be calculated. By using the z-score formula, it can be seen that 31.33% of people wait longer than 3258 miles before they change their oil. You can show this to his father to prove to him that he did not wait too long. Some people even wait longer than he did, because he was not one whole standard deviation away from the mean.

Strengths and Weaknesses

We had difficulty finding the proportion of males psychology majors in the United States, so our percent could be outdated and irrelevant now, which is a weakness in our data. We used a calculator to flip the coin to limit random factors, like how we flipped the coin or if it had landed and then fallen, which is a strength.

Sources

(2006). The American Heritage Dictionary of the English Language Fourth Edition, Retrieved January 31, 2008, from Dictionary.com Website: http://dictionary.reference.com/browse/law%20of%20large%20numbers

American Psychological Association. (1995). Changing Gender Composition of Psychology: October 1995, Retrieved January 31, 2008, from APA Online Website: http://www.apa.org/pi/taskforce/homepage.html

American Psychological Association. (2003). Apa Style: Electronic references. Retrieved February 3, 2008, from http://www.apastyle.org/elecref.html

MacEwen, Professor. (2008, January). Lecture presented in Chandler Hall. University of Mary Washington, Fredericksburg, VA.

Marklein, M.B. (2005, October). College gender gap widens: 57% are women. Retrieved January 31, 2008, from USA Today Website: http://www.usatoday.com/news/education/2005-10-19-male-college-cover_x.htm 

The mean is the measure of central tendency most influenced by outliers. The mean will be influenced by outliers because they are added to the other data pieces to make the total and they can skew the data to be either lower than it should be or higher than it would be if all the data points were with in a certain range and closer to the mean. (Brittany)

 Why might these extreme values have occured?

In the example of the body temperatures, low temperatures could have occurred because the person or thermometer had been outside. High temperatures could have been caused by a fever, or taking a hot shower. Any of the high or low temperature readings could also have been due to a faulty thermometer. These are all weaknesses of the data collection.(Brittany)

 Are these extreme values reliable?

These outlier values are not reliable because random events have influenced the body temperature reading. Extreme values occur less often than data that are closer to the mean which can be seen by using the theoretical model of a normal curve. When a frequency distribution is created for the set of data, it can be seen that the values at the extremes occur the least often and most of the values fall around the middle or mean. The more frequent values are more reliable than the outliers. (Brittany)

Comparing the article values with my data

The standard deviation in the article was 0.73 and the difference between the average body temperature and my average body temperature was 0.98714. My average body temperature was between one and two standard deviations away from the mean in the article. The difference between the two means could be because of unreliable data that I collected and outlier temperatures. I do not think that my temperature is unusual because it is within two standard deviations from the mean. (Brittany)

Men vs. Women

Brittany’s average temperature was 97.26286 degrees Fahrenheit while Matt’s was 97.436 degrees Fahrenheit. Typically men’s temperatures are cooler than women’s, which was not the case for our team. (Matt and Brittany

 Is your data representative?

Our data does not seem representative of the typical pattern. This could be because of inconsistencies in data collection. Both of us were in different places when we took our temperatures and the environment that we took our temperatures in could affect our readings. (Matt and Brittany)

 Do you think that continuing to collect data would produce more or less accurate data?

If we continued to collect data and controlled the enviornments to minimize the effects of randomness, our data would be more accurate than the little data that we have now. (Brittany and Matt) 

3. Mode: 97.3 degree F

Median: (35+1)/2=18. The median was also 97.3 degrees F

Mean: (96.8+96.8+96.9+96.9+96.9+97+97.1+97.1+97.1+97.2+97.2+97.2+97.2+97.3+

97.3+97.3+97.3+97.3+97.4+97.4+97.4+97.4+97.5+97.5+97.6+97.7+97.8+97.8+97.9+

97.9+97.9+98+98.1+98.2+98.3)/35 = 97.436

The mean is most influenced by extreme outliers. If there would have been an unusually high temperature recorded, then the mean would have increased. Extremely low or high temperatures can be the result of many things, including your environment.

Variance: (96.8-97.436)2+(96.8-97.436)2+(96.9-97.436)2+(96.9-97.436)2+(96.9-97.436)2+(97-97.436)2+(97.1-97.436)2+(97.1-97.436)2+(97.1-97.436)2+(97.2-97.436)2+(97.2-97.436)2+(97.2-97.436)2+(97.2-97.436)2+(97.3-97.436)2+(97.3-97.436)2+(97.3-97.436)2+(97.3-97.436)2+(97.3-97.436)2+(97.4-97.436)2+(97.4-97.436)2+(97.4-97.436)2+(97.4-97.436)2+(97.5-97.436)2+(97.5-97.436)2+(97.6-97.436)2+(97.7-97.436)2+(97.8-97.436)2+(97.8-97.436)2+(97.9-97.436)2+(97.9-97.436)2+(97.9-97.436)2+(98-97.436)2+(98.1-97.436)2+(98.2-97.436)2+(98.3-97.436)2 /35 = .1651

Standard Deviation: square root of variance = .4063

Calculations from SPSS:

Mode: 97.30

Median: 97.3000

Mean: 97.4314

Variance: .158

Standard Deviation: .39688

 Extreme values are usually rare and unusual when talking about something like body temperature. The body does a good job of maintaining an extremely consistent temperature. When there are increases or decreases it is a sign that something is wrong, for example a fever. Many times an extreme value concerning body temperature can be the result of faulty equipment. This can be corrected by taking one’s temperature again or purchasing a new thermometer.

4. Sources or references used

MacEwen, Bruce. Lecture. Chandler Hall 202. 24 Jan. 2008.

Shoemaker, Allen L. “What’s Normal? Temperature, Gender, and Heart Rate.” Journal of Statistics Education 4 (1996). 27 Jan. 2008.

5. Flaws, weaknesses, or limitations related to the data collection.

There was only 35 temperature readings recorded over the course of 5 days. This is not a long enough period to get an accurate average of one’s body temperature. It was also extremely cold during the 5 days, which most likely accounted for the drops in temperature throughout the day. It was difficult to consistently take the temperature 7 times a day at the exact same time over the 5 days. A strength was that the other 20 or so students were doing the same thing, so that was a good means of comparison.

Other calculations:

The mean body temperature is really 98.2 degrees F. The article said that 98.6 was incorrect because of problems with Wunderlich’s original methodology, diurnal fluctuations, and unreliable thermometers.

98.25-97.436=0.814

The articles standard deviation was .73

My body temperature is between 1 and 2 standard deviations away from the mean that was in the article. My body temperature is not unusual, because it is not that far from the mean that was calculated in the article.

degrees C = (degrees F – 32) * 5/9

(97.436-32)5/9 = 36.353 degrees C

Mode: 97.2 degrees F

Median: (35+1)/2 = 18 The 18th body temperature was 97.2 degrees F.

Mean: (94.2 + 95.8 + 96.2 + 96.3 + 96.3 + 96.5 + 96.6 +96.6 + 96.8 + 96.9 + 97.0 + 97.0 + 97.2 + 97.2 + 97.2 +97.2 + 97.2 + 97.2 + 97.2 + 97.4 + 97.6 + 97.6 + 97.7 + 97.7 + 97.7 + 97.9 + 98.1 + 98.1 + 98.1 + 98.1 + 98.1 + 98.2 + 98.3 + 98.4 + 98.6)/ 35 = 97.26286

Variance: (94.2 – 97.26286)2 + (95.8 - 97.26286)2 + (96.2 – 97.26286)2 + (96.3 – 97.26286)2 + (96.3  – 97.26286)2 + (96.5  – 97.26286)2 + (96.6 – 97.26286)2 + (96.6 – 97.26286)2 + (96.8 – 97.26286)2 + (96.9 – 97.26286)2 + (97.0 – 97.26286)2 + (97.0 – 97.26286)2 + (97.2 – 97.26286)2 + (97.2 – 97.26286)2 + (97.2 – 97.26286) 2+ (97.2 – 97.26286)2 + (97.2 – 97.26286)2 + (97.2 – 97.26286)2 + (97.2 – 97.26286)2 + (97.4 – 97.26286)2 + (97.6 – 97.26286)2 + (97.6 – 97.26286)2 + (97.7 – 97.26286)2 + (97.7 – 97.26286)2 + (97.7 – 97.26286)2 + (97.9 – 97.26286)2 + (98.1 – 97.26286)2 + (98.1 – 97.26286)2 + (98.1 – 97.26286)2 + (98.1 – 97.26286)2 + (98.1 – 97.26286)2 + (98.2 – 97.26286)2 + (98.3 – 97.26286)2 + (98.4 – 97.26286)2 + (98.6 – 97.26286)2 )/ 35 = .75236

Standard Deviation: Square root of variance = .86739

Calulations using SPSS

Mode: 97.20

Median: 97.2000

Mean: 97.2629

Variance: .772

Standard Deviation: .87853

Other Calculations

98.25-97.26286 = .98714

Degrees Celsius = (Degrees Fahrenheit – 32) * (5/9) = (97.26286 – 32) * (5/9) =36.25714

2) Traffic is another example of random variation. You can travel on the same roads every day to get to school or work, but the exact time that you are traveling could change, the amount of people on the road at a certain time can change depending on their schedules and random events that affect them.

3) Your mood can also vary from day to day or during the day. Seeing someone on campus could positively or negatively affect your mood, or not affect it at all. Hearing a song on the radio or seeing a show on tv could also affect your mood. The amount of sleep that you got the night before, which can be affected by where you live, if you live in a dorm, what the other people in your building were doing could keep you up at night, all these things can happen or not and have an affect on your mood or not.

2. Being unprepared for something can cause an individual’s performance to decrease significantly. If there was a person engaging in late night exercise of a sexual nature in the next dorm room, it could lead to a bad test grade due to the lack of sleep.

3. Bad luck. Having someone rear-end your car while you are driving can lead to rage prior to an important business meeting. This distraction could cause the accident victim to snap, throw something at their boss, and lose their job.

4. No matter the level of effort put forth towards one’s job, in some lines of work there is the possibility of being layer off. This can have a drastic impact on a person’s life leading to numerous other problems ranging from family to mental stability.

For five days I measured my temperature every two hours, seven times a day, using a digital thermometer. The highest temperature that I measured in the five days was 98.6 degrees F which is the normal body temperature that is used to decide if someone has a fever or not. I only measured that temperature once and the rest of the time my temperature was below the normal level and some times significantly. The lowest point was 94.2 degrees F which I took after walking back from the gym in shorts while it was snowing. (Probably not a smart idea, but I lived). The thermometer was also in my pocket while I walked back so it was fairly cold when I returned to my room. This change in temperature seems to have a clear cause, but the other fluctuations and the fact that I only had the normal body temperature once, are due to random factors. What I ate or drank before taking my temperature, or what I was doing could have influenced my measurement and all to different extents. The mean temperature was 97.26286 degrees F. I could have used this number to try to estimate my next temperature, but as I discussed before, sometimes the measurement would not have been close to the mean, due to random events.

Brittany Cooney

Data must be recorded over the long run to reduce systematic biases and give a more accurate representation of how things actually are. It is impossible to predict exactly what my temperature is at this very moment. There are too many things that come into play and affect the outcome. Had this class been in the summer at this project assigned when the heat index is 110 degrees, my average temperature would have most likely been higher. It is not just the weather or the temperature of the room that changes a human’s body temperature. It fluctuates and that is normal. 98.6 is an average, not an exact result every time. The point of this was to demonstrate that there are many things that account for randomness.