Randomly Generated Numbers

21 54 45 61 60 40 64 50 78 23 57 10 14 59 87 62 25 1 77 3 13 41 11 67 26 88 83 58 53 63 89 71 69 48 84 22 94 90 7 42 74 52 75 28 5 72 93 24 92 12 33 73 44 9 4 19 98 55 85 97 27 47 100 37 8 18 46 15 51 79 76 38 96 32 30 34 70 29 81 68 49 6 86 17 35 80 66 31 82 91 95 99 39 43 20 2 36 16 56 65

24 6 17 17 40 65 23 3 56 63 47 2 11 62 85 45 25 16 90 53 84 52 92 43 18 75 31 55 2 7 4 85 6 99 56 4 96 62 60 44 84 88 69 30 15 25 52 22 89 58 71 37 79 96 60 58 58 79 69 54 10 56 12 97 43 60 57 89 1 57 34 23 12 28 60 33 24 17 98 7 90 17 61 46 91 27 88 32 25 69 48 16 12 95 33 88 89 17 33 91

27 100 95 50 26 4 43 21 93 32 91 85 48 44 97 41 62 26 95 51 90 45 89 3 73 37 60 71 18 44 62 37 47 3 36 19 34 19 51 82 71 82 86 36 11 86 2 36 84 86 25 56 76 93 20 4 24 26 32 30 100 66 89 52 85 46 12 60 87 72 74 82 1 22 9 88 45 5 97 1 57 36 84 9 3 10 44 48 83 61 34 88 98 93 47 87 90 17 96 13

38 95 47 97 59 87 59 5 37 23 37 2 78 88 4 76 52 54 27 43 94 54 17 97 83 76 4 63 36 71 51 23 20 19 52 61 4 59 48 84 34 92 73 57 54 8 34 40 57 24 22 44 3 6 41 60 91 91 39 38 46 11 37 71 46 60 44 46 63 59 54 92 100 86 36 18 38 18 50 38 18 57 5 42 44 14 49 69 86 43 40 11 46 5 35 16 39 32 13 65

35 65 6 18 1 27 53 81 95 12 99 25 46 54 45 76 64 96 57 79 63 3 27 37 35 16 99 78 88 86 56 97 24 52 92 54 15 22 74 23 87 13 61 55 23 35 96 78 64 48 82 99 67 58 1 74 91 64 31 28 19 78 89 40 96 56 89 3 67 36 42 55 85 37 37 1 100 8 16 34 68 23 81 7 43 1 17 59 88 13 86 19 7 47 70 89 57 71 28 6

Specifics of Assignment 3

How likely is the event (that Ms. Williams will have) “exactly three boys in succession?”

In order to calculate the probability of Ms. Williams having three boys, I created five random number generations with 100 numbers in each, with even numbers representing girls and odd numbers representing boys. The average of the five tables were 7 groups of three boys born in a row per 100 births. Statistically that is,

          7          = .071 = 7.1%
100 – (3-1)

Ms Williams having three boys in a row has a statistical probability of 7.1%.

 

 

Determine the proportion of boys and girls in the sample of 100.

 

The average of my five number groups was 50.4% girls to 49.6 boys. With just the small sample set of five random number generations an already surprisingly close distribution is seen. If the trial were performed 10,00 times the distribution of males to females would narrow due to the law of large numbers which basically states that as the number of data points increases the more the data will regress towards the mean.

 

 

 

Provide an example that illustrates the principle of the law of large numbers as it might affect you personally.

A way in which I might make a mistake in using the law of large numbers were if I only shopped at one local grocery store for canned corn. Let us say that the average cost of canned corn is 87 cents, but I pay $1.25 at the local store. By only shopping at the local market, I am erroneously assuming that the average price of canned corn is $1.25. If I were to go to other stores and look at the price of canned corn, I would be gathering more data, and be able to find a true mean.
Standard deviation is effected by small data sets because the divisor of the equation is the number of data points. Dividing a number by lowers number results in larger outcomes, where as if a number is divided by a large number, the result would be a smaller number.

 

Determine the proportion of males in our class.

 

In an article entitled “General versus gender-specific attributes of the Psychology major” it was reported that currently females make up 75% of undergraduate and graduate psychology students. This places males in the minority of 25%. Currently in our class there are 8 males and 38 female students. Translated into statistics our class is 17% male, and 83% female. The difference between the class percent and the nationwide average percent is off by 8%, skewed towards being a female student, but this can be easily explained by the fact that one class can be quite skewed from the mean as it is too small of a sample size. Also the university we attend has slightly more females than the average ratio for campuses.

 

 

Suppose you bought a car and your father tells you that you waited too long to change your oil. You are not sure this is correct. You waited 3,467 miles before you changed oil. You do some checking and find that the mean miles people wait to change oil is 3,258 miles with a SD of 223 miles. Assume that this statistic, i.e., the number of miles people wait until they change oil, can be fit by a normal curve model. Using a z-score table, reason with your father that you did not really wait all that long.

 

If the standard deviation for when people change there oil is 223 miles, and the average millage of when people change their oil is 3,258, then I did not wait that much longer than the average person to change my oil. The first sign that the millage at which I changed my oil was not that far off from the mean is that it falls within one standard deviation from the mean.
Using a z table, I can also prove that changing your oil when your millage is at 3,467 is note a rare event. The z score for these numbers is

3,467 – 3,258 = .94
223

When this number is looked up on a z table, the percent of people who change their oil below 3,4467 miles is 82.64%. While this is the majority of drivers, it also means that 17.36% of people change their oil above 3,4467 miles. A rare event is said to occur in only the uppermost five percent of a uniform curve. While 17.36% is a minority, it is not considered a rare event.

 

 

References

Bailly, M., King, A., & McCray, J. (2005). General versus gender-specific attributes of the Psychology major. [electronic version] The Journal of General Psychology.

Mac Ewen, B. (2008, spring semester). Psychology 261. Class lectures. University of Mary Washington.

 

 

Relating to the Weekly Topic!

This assignement allowed us to apply the principle of the Law of Large Numbers, which states that the greater the sample size or number of events, the closer the expected value comes to the actual value. In addition, determing the probability of Ms. Williams having exactly three boys in succession allowed us to understand rare events. A value is considered rare if it falls in the top or bottom 5% on a normal curve model. In the last question on changing your oil, we were able to practice calculating a z-score using the mean, standard deviation and x-value. Lastly, we were able to apply an empirical, or real, event, like changing your oil, to a theoretical model, like the normal curve.

Strengths & Weaknesses

    Weaknesses in our assignment include the human error that could occur when flipping a coin one hundred times to determine the probability of heads versus tails. Also, mathematical errors could occur when calculating the z-score and finding the corresponding value for the question about changing your oil on time.

  Strengths of the assignment include the opportunity to provide your own example to illustrate the Law of Large Numbers. Creating your own example often helps you understand the material more clearly because you can relate it to your everyday life.

Specifics of Assignment 3

How likely is the event “exactly three boys in succession?” What was the probability you obtained?

      In order to represent the probability of Ms. Williams conceiving exactly three boys in succession, I decided to flip a coin one hundred times and record the event as either heads or tails, where heads was equal to a boy and tails equal to a girl. After flipping the coin, I found that three boys occured in succession only once. Thus, the probability of obtaining exactly three boys in succession is approximately 1 out 98 or around 1%. Therefore, the likelihood of Ms. Williams having exactly three boys in succession is very rare because 0.01 falls within the top 5% of a normal curve distribution (2/1/08).

Determine the proportion of boys and girls in the sample of 100. If you flipped the coin 10,000 times, what should happen to the proportion of boys? Why?

  The proportion of boys in my sample was 49% and the proportion of girls was 51%. These percentages are very close to the typical 50% for each sex. If you flipped the coin 10,000 times, the proportion of boys would move even closer to the expected 50% because you have a greater sample of events. This occurs because of the law of large numbers which states that the greater the sample size or number of trials, the closer the expected value comes to the actual value or proportion (Sept. 2004).   

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

Let’s say that I only like to eat ice cream with some form of chocolate in it. I have discovered that, on average, 50% of the flavors offered in ice cream shops contain chocolate. In my search for the best ice cream, I have found that the ice cream shops offering the most flavors come closest to having 50% of their flavors contain chocolate. This relates to the Law of Large Numbers because I have found that the greater the sample size, the closer the expected number of flavors containing chocolate comes to the actual number (50%).

However, a flaw in my predictions about ice cream flavors may occur if I forget to read a flavor from the list of offered ice cream flavors. This is more likely to occur if there is a very long list of flavors. Also, there may be a flavor listed that may contain chocolate but the name of the flavor may not indicate the chocolate. I also need to decide if I am going to count white chocolate as actual chocolate. As you can see, many mistakes in judgment can occur!

The standard deviation formula can be used to explain why smaller samples yield larger variations because to obtain the standard deviation, you must divide by the total number of events or values within a sample (1/21/08). Thus, dividing by a smaller sample would create a larger standard deviation value then dividing by a larger sample size.

     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 8 out of 46 or 17%. In an article entitled “General Versus Gender-Specific Attributes of the Psychology Major” located in the Journal of General Psychology, I found that the percentage of undergraduate and graduate males studying pscychology in the United States is only about 25% (4/1/05). The proportion of male psychology majors natiowide is a more valid statistic to analyze because the sample size (all male psychology students within the United States) is much larger than the sample size of male psychology students within Dr. MacEwen’s PSYC 261 class. There is a difference of 0.08 between the proportion of males within our psychology class to the proportion within the United States. This could be attributed to the sample size, as mentioned, or by the fact that Mary Washington is a small liberal arts school where only around 40% of the student body is male.  

Suppose you bought a car and your father tells you that you waited too long to change your oil. You are not sure this is correct. You waited 3,467 miles before you changed oil. You do some checking and find that the mean miles people wait to change oil is 3,258 miles with a SD of 223 miles. Assume that this statistic can be fit by a normal curve model. Using a z-score table, reason with your father that you did not really wait all that long.

  Assuming this statistic (mean= 3,258) can be fit to a normal curve model, the number of miles that I waited before changing my oil (3,467) is within one standard deviation (223) from the mean. Using the z-score table, I found that around 17% of all drivers change their oil after more than 3,467 miles of driving. Since, 17% does not fit into the top 5% of a normal curve model, then changing my oil after 3,467 miles is not a rare event (2/1/08). Therefore, I can reason to my father that changing my oil after 3,467 miles is not so unordinary because 17% of all drivers change their oil after that number of miles.

References:

Bailly, M., King, A., & McCray, J. (2005). General versus gender-specific attributes of the Psychology major. [electronic version] The Journal of General Psychology.

Mac Ewen, B. (2008, spring semester). Psychology 261.  Class lectures.  University of Mary Washington.

Stark, P. B. (Sept. 2004) The Law of Large Numbers. Retrieved February 4, 2008 from, http://stat-www.berkeley.edu/~stark/Java/lln.htm.

Lab 2: Specifics of the Assignment

    The mean would be the most effected measure of central tendency by outliers, especially when there is such a small data pool. Since every number is given equal weight, data can easily be skewed in a direction by an outlier. Also, due to the fact that variance and standard deviation calculations rely on the mean in their formulas, they too will be heavily affected. These outliers occur because they are randomly generated, and thus can occur anywhere. In my case I began to have a fever on Friday night, which resulted in a very high temperature reading of 100.2, which is well above my average temperature.

Just because a pattern exists within randomly generated data does not mean that all the data has to fit the pattern. Due to this, most data sets will always have outliers, although their number will be limited. While outliers are representations of factual data, they cannot be relied on as much as data that is closer to the mean due to the fact that as outliers they represent findings that do not statistically occur as often as data that is closer to the average. In fact, the occurrence of outliers should be substantially less than data that is close to the mean.

Using the current findings regarding an average body temperature, my average is .12 degrees F above the average.

98.37-98.25 = .12

This findings falls well within one standard deviation of my readings (the stand deviation for my data being .693556) This means that my body temperature is considered the norm, as it does not exceed one standard deviation. Anytime data exceeds one standard deviation it is considered outside the norm.

The temperatures that Melissa and I recorded do not support the findings that men run a cooler temperature than women. Melissa’s mean temperature was 97.73, where as my mean temperature was 98.37. Mellisa’s data points to her being the sex with the lowest temperature, however, due to limited data, and the fact that Melissa’s mean temperature is more than one standard deviation from the norm, the comparisons of our two temperatures would be confounded.

My body temperature data seems to be very representative of my gender, as it is only .12 degrees from the norm. However, I did have a day in which my temperature continued to rise well into the 100 range, which will have skewed my data towards a warmer mean.If more data had been gathered, a more reliable mean, as well as other measures of central tendencies would have been obtained.

To turn a measurement to Celsius one has to take their measurement in Fahrenheit, subtract 32 from that number, and then multiply the reminder by 5/9ths. My temperature in Celsius would be (5/9)(98.37-32) = 36.87.

Assignment 2: Lab Questions

1. The measure of central tendency that is most effected by extreme temperature values is the mean. The mean is most effected because it is essentially the average of all the temperature readings, where each temperature is given eqaul weight. Thus, if one temperature reading was extremely low than the mean of the data would be shifted to a lower average. The mean would be shifted to a higher average if there was an extremely high temperature reading in the data set.

2. An extremely low temperature reading might have occurred if the individual took their temperature while they were outside in cold weather, or if they had just consumed a cold drink, or even if they had left their thermometer in a cold area (e.g., the car). On the other hand, an extremely high temperature reading might have occurred if the individual took there temperature after taking a hot shower, drinking a hot beverage or if they were actually sick and running a fever.

3. Extreme values are not necessarily extreme or unusual. As supported by my temperature readings and the data of my classmates, body temperature can often be effected by random events that create a bias or deviation in the data set. Random occurences, like the weather, your diet and even your bedtime routine can effect your body temperature. Depending on when you take a reading, your body temperature could be unusually high or unusually low.

4. For the most part, these extreme temperature readings are not very reliable because they are not taken under controlled circumstances.  For instance, my body temperature dropped down to 96.0 degrees between 10-12pm one day. I remember I had been driving around in my car that morning and my thermometer might have been chilled from being in my cheap, cold car. All in all, we cannot rely solely on these extreme values, but we should take them into consideration when looking at our data as a whole.

5. According to the article, the correct body temperature is 98.25 degrees fahrenheit. The difference between my average temperature and the correct average body temperature is 0.42 (98.25-97.83= 0.42). The difference that I computed was lower that the standard deviation of body temperatures given in the article (0.73).

6. My body temperature average is not particularly unusual. I have always had a body temperature lower than the average body temperature because I am a naturally cold person. I also have poor circulation in my hands and feet, so that could effect my overall body temperature as well.

7. My average body temperature was 97.73 while Dave’s average body temperature was 98.38. Thus, our data opposed the trend that men are “cooler” than women. Dave’s standard deviation was 0.7, in accordance with the typical standard deviation, while my standard deviation was 0.8. As a whole, my data is probably not representative of all females because I am naturally colder than most individuals.

8. Many factors including weather, my diet, my sleep patterns and even my hormone levels could effect my hourly body temperatures and thus effect the mean of my data. I had a few extremely low temperatures (96.0 between 10-12pm on 1/19) and a few extremely high temperatures (99.1 on 1/18 and 1/21) that could have shifted the mean of my data.

9. I think if I continued to take my temperature in a controlled setting each day than I might gather more accurate data. However, it is difficult to control certain random events in your everyday routine (like weather and diet). Collecting more temperature readings  would certainly give me more data to analyze and observe certain patterns.

10. My average body temperature is 36.86 degree celsius. You convert Fahrenheit to Celsius using the following formula:   C= (5/9) (F-32)

References:

Mac Ewen, B. (2008, spring semester). Psychology 261.  Class lectures.  University of Mary Washington.

Shoemaker, Allen L. What’s Normal?–Temperature, Gender, and Heart Rate. Journal of Statistics Education. 4, 2 (1996).

Lab 2 Questions

Statistics

Dave’s Data

SPSS Gathered Data.

Mean Temp:98.3778
Median Temp: 98.4000
Mode Temp: 98.40
Std. Deviation: .69356
Variance: .481

Hand calculated Data

Mean Temp: 98.38
Median Temp: 98.80
Mode Temp: 98.40

Std. Deviation: .678232998
Variance: .46

Strengths & Weaknesses

      Flaws that might be inherent in this lab would be the miscalculation of the central tendencies, variance, and standard deviation when the student is required to calculate them by calculator. Another weakness is in the relatively limited amount of data points that are available for this lab. Using only thirty-five data points is not enough data to provide an accurate pattern. If an individual had a fever one day, 1/5^th of the data is now biased towards a warm reading. Strengths in the experiment were the use of SPSS to calculate the central tendencies, variance, and standard deviation, which would weed out any human error in calculation.

By: David Whitman and Melissa Tully

Relating Lab to Weekly Topic

    The topic of class for this week was central tendency, variance, and standard deviation. All of these principles are ways to measure random data, assess the range of our data and find a pattern within it. Finding the measures of central tendency and variation in our data allows us to also compare our temperature readings with the average body temperature and standard deviation of all humans. The data gathered from our temperature taking provides enough data points so that all of these principles can be applied.  

By: David Whitman and Melissa Tully