• How likely is “exactly three boys in succession?”
  • The occurrence of three boys in succession, boys represented by “1s,” appears in my set of 100 random numbers about 5 times. To determine the probability of exactly three boys in succession, I divide actual number of occurrences by number of possible occurrences. Number of occurrences is 98 and as I mentioned, number of actual occurrences is 5. I plug these numbers into the following equation:
  • # actual occurrences/# possible occurrences = 5/98 = .051
  • Therefore, there is a 5.1% likelihood of having exactly three boys in one succession according to my data.

• What is the proportion of boys and girls in the sample of 100?
  • In my sample there are 38 boys and 62 girls.
  • Since n=100 in this sample, to determine proportions of boys and girls, I can take the value for boys and the value for girls and divide them each by 100.
  • I end up with these percentages for each:
    • Boys: 38%
    • Girls: 62%

• What would happen to the proportion of boys if 10,000 numbers were generated?
  • The law of large numbers states that in a sample of independently and equally distributed random variables, the values of the variables will more closely approach the expected values as number of trials increases (Law of large numbers, 2008).
  • Since there are two variables in this sample, each with equal chance to occur randomly, both boys and girls will each have a 50% chance of occurring.
  • According to the law of large numbers, if 10,000 more numbers were generated, then the proportion of boys should rise from 38% and become extremely close to 50%.

• An example illustrating the law of large numbers that personally affects me
  • An example of the law of large numbers that personally affects me is how often I have to sell gold. My fiance (currently in Alaska) has me watch over his gold while he is gone. Every once and a while I will get a call and hear that I have to sell right away. Because the price of gold can change by the minute, I usually have to drop what I am doing and leave right away when this happens. The particular coin shop I have to go to is in Gaithersburg, MD, so I have to drive about an hour and a half to get there. The shop is a male dominated environment and sometimes there’s some sketchy men in there. Then I have to wait out I-95 traffic and don’t leave for home until after 7pm. On these days I lose a lot of time traveling and waiting, and the experience overall is not very pleasant.
  • On a typical day, I will not have to go sell gold. In the past few weeks however, the price of gold had risen and lingered around the same price, so I have had to go sell on a number of occasions during this time. If I were to record how often I had gone to sell gold over this period of time, the rate would be unusually high. This would not be representative of how often I have to go on my gold runs overall because this only represents a small period of time.
    
  • Overall, I only have to go sell gold about 20% of the time. If I were to take data for a few months to year, the rate that had sold gold would probably more closely approach and resemble 20%.
  • I usually will not expect to have to sell gold because I do not have to do it very often. I do know however that if the price of gold significantly rises, then the probability that I will have to sell also rises. I tend to make more mistakes in judgement when I do not go to kitco.com and monitor the price of gold. I don’t really look at this site much at all because I am doing other things and don’t have a huge interest in the price of gold. So when I get a gold call, I am usually caught off guard and surprised that I have to leave. If I were to monitor the price of gold more often, I would have a better idea of whether or not I will have to go sell.

• Why do smaller samples yield larger variation?
  • As a rule, small samples (under 30) uses the Standard Deviation equation with “(n-1).” Larger samples use the Standard Deviation equation with “n.”
  • In using n-1 in small samples, any bias is adjusted for. To account for this bias, the idea is that more variation is allowed for.
  • Using the equation for small sample sizes over the one for large sample sizes, mathematically this makes sense. If you divide a number by a smaller denominator than the result will be larger.
    • For example, dividing 10 by 5 will be 2, but dividing 10 by 2 will be 5 – a larger number.
      
  • When dealing with large samples, if we were to use n-1 in the Standard Deviation equation, there wouldn’t be much of a difference in the end result. For this reason, n is used instead of n-1 (MacEwen, 2008).
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  • Portion of males in our class
    • There are 8 males out of our class of 46 (MacEwen, 2008).
    • By dividing the number of males by number of students in the class, I can determine the proportion of males in the class.
      • 8/46 = .174
    • Therefore, males make up 17.4% of the class

• Proportion of male Psychology majors nationwide
  • According to the US Census Bureau, nationwide there are currently 741,000 students who are psychology majors. 306,000 of these are men and 434,000 of them are women (2004).
  • Dividing number of males by total number of students, I can find the proportion of males as Psychology majors.
  • Proportion of Males: 41.3%
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  • Graph: This represents the number of students as Psychology Majors in 2004 according to the US Census Bureau. Note that values represent thousands. A visual of this data and not of the class data is shown because this more closely resembles the true proportions.
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• Explanation of the statistical differences
  • This relates to the law of large numbers. While Our class had a sample size of 46, the US Census Bureau had a much larger sample size of 741,000. For whatever reason (some of which are discussed in section 5), our class proportion my not represent the true proportion of males as Psychology majors nationwide. To review, the law of large numbers states that as a sample size increases, proportion of males in this case should approach and more closely resemble the true proportion. Because the US Census Bureau’s sample size is larger than that of the class, according to the law of large numbers, this should be a more accurate value and more closely resembles the true proportion of male psychology majors.

• Why I did not wait too long to change my oil
  • Given
    • mean = 3,258 miles
    • SD = 223 miles
    • x = 3,467 miles
  • Calculations
    • z = (x-mean)/SD
    • z = (3,467-3,258)/223=.937
  • Translation with z-table
    • Area = 0.3264
  • After calculating the area (adjacent to the right of the mean) based on the information given, I can see that my area of 32.64% does not fall in the top 5 %of the normal curve. If this were the case, than my area would be 45% or greater. Therefore, the mileage at which I changed my oil does not fall into the rare area. Thus, I did not wait too long to change my oil (MacEwen, 2008).

This assignment has covered several of the topics discussed in lecture for the past few weeks. We had to generate random numbers for this assignment, relating to the topic of randomness. We know that these numbers do not have any meaning or reason behind them. We learned how to give order to randomness with statistical equations like central tendencies and standard deviation. Likewise, we used standard deviation in this assignment as we discussed the reasoning behind the different formulas used for small and large samples and used a standard deviation to solve a problem.  In this problem that we solved we applied more of what we learned, the z-score,  z-table, and rare region. We also frequently calculated probability to determine several percentages. Finally, we predominantly applied the law of large numbers to this assignment. This was used to explain the differences in proportion of males in our class vs. the nationwide statistic for males in Psychology. Using this theorem, we also hypothesized what would happen had we generated more random numbers. Finally we related the law of large numbers to a personal example, bringing what we had learned in class into real life.

Using StatTrek.com, I was able to generate a series of 100 random numbers. I only allowed numbers 1 and 2 to appear and did allow for duplicate entries to occur. I will let the “1s” represent boys and the “2s” represent girls (2007).

Law of large numbers. (2008, January 21). Wikipedia. Retrieved January 27, 2008, from http://en.wikipedia.org/Law_of_large_numbers

MacEwen, B. (2008, Spring Semester). Psychology 261. Class Lectures. University of Mary Washington.

Random Number Generator. (2007). Stat Trek. Retrieved January 31, 2008, from http://stattrek.com/Tables/Random.aspx

Table of the Standard Normal (z) distribution. (2008). SixSigma. Retrieved Febuary 4, 2008, from http://www.isixsigma.com/library/content/zdistribution.asp

What it’s Worth: Field of Training and Economic Status in 2004 Detailed Tables. (2004). US Census Bureau. Retrieved January 31, 2008, from http://www.census.gov/population/www/socdemo/education/sipp2004w2.html

Weaknesses

• One weakness deals with the source I used to collect my set of random numbers. I used an online number generator. As far as I know, this is not a certified website and I do not know how these numbers were determined. I have no way of knowing if the numbers were generated truly randomly.
• One thing I did notice that was wrong with the US Census Bureau source is the date of publication. This information was released in 2004 – 4 years ago. There could have been changes in the psychology field in that time and the true statistic today may well be different.
• We obtained the proportion of males in this class for this assignment. A few things threaten the reliability of this score. The data I received from the US Census Bureau referred to advanced education, which could have included graduate school. Our class only represents undergraduate students. Since this is a beginning level course, many may choose to change their majors in the future. It is also difficult to get into this class and had they been able to, a lot more males may have been in the class. Finally we do not know for sure if everyone in the class is actually a psychology major.

Strengths

• The nice thing about using the random number generator online is that it took less than a minute. I could imagine how long it would take to manually flip a coin 100 times and record as I go.
• Another strength of using a quicker method over flipping a coin is that human error does not enter in. I could have miscounted flips, incorrectly recorded flips, not flipped with enough force so that the coin would not have flipped at all when airborne, to name a few. Using this online source, I could not make any of these mistakes.
• I had e-mailed the US Census Bureau to obtain my statistics for the proportion of males nationwide in psychology majors. I was fortunate to hear back from them and I know that the US Census Bureau is a reliable source. I know that data was carefully collected to obtain these numbers and that I can trust these numbers to be true.

• Which of the three measures of central tendency is influenced most by outlier temperature values? Why?
  • To determine which calculated measure of central tendency based off of my data was most influenced by my outlier temperature value (100) I threw out my outlier and recalculated the mean, median, and mode. Both the median (98.1) and mode (98.2) calculations not including the score of 100 F were exactly the same as those from my entire data set. The mean value without my outlier included was slightly lower than the mean of my entire data set (98.0967). Therefore, the mean is the most affected by the outlier temperature value. In general, the mode would not be affected because removal of one extreme point would not change what the most frequently occurring score was. The median may only change slightly because if data points are similar, then the difference of the closest relative scores should not be large. The mean however has the most potential to be changed by the outlier because the mean will adjust itself too high or too low in order to accommodate for an extreme score.
• Why might these extremes have occurred?
  • My most extreme value was 100.0 degrees F. For all of my other data points, temperatures were only 0.3 degrees apart from their one another at most. This outlier however was separate from its closest data point by 1.1 degrees. I vaguely recall taking this particular measurement and have a good idea as to why this extreme occurred. I hypothesised last week that there was one particular systematic affect present in my data, continually drinking hot tea. As I recall, I had been drinking hot tea prior to this measurement. I did not feel sick and had not been doing any extremely strenuous activity prior to measurement. The most likely contributor to this unusually high reading was the hot tea. While most of my other temperature readings tended to remain within a fairly small range, only this one reading was abnormal. From this, I can assume that this outlier was not my true body temperature at that time. This must have occurred due to the systematic error of hot tea’s presence.
• Are extreme values rare and unusual or not?
  • From analysis of my own data, I can assume that extreme values in general tend to occur due to error and do not represent true values of any measure. Usually, if an experiment is well planned and controlled, data can be gathered accurately. While nothing is perfect however, every once and a while, a participant or the experimenter may make a significant mistake which can dramatically alter some of the data points. Since these mistakes will happen only on occasion, these outliers should only make up a very small percentage of the data. These extreme values are rare and unusual in comparison to the vast number of similar values in one set of data. Looking at data sets in general however, extreme values seem to appear in almost all experiments. Humans do tend to make occasional mistakes so in this sense, it is not unusual for extreme values to be present in data sets, even if in only small percentages.
• How reliable are extreme data values? Can we count on them?
  • As discussed above, extreme data values do not likely represent true values and are more likely distortions of true values. Because of this, we should not count on them when evaluating data, especially if evaluating central tendencies in a set of data. These incorrect measurements do not represent true values and calculated tendencies that represent data as a whole should not be based on these false values. These extreme values if included in calculation of central tendency for example can drastically alter calculated central tendency. With this, the representation of the entire set of data would be distorted just because of one or two distorted data points. We cannot count on these extreme data points and should not include them when computing central tendencies.
• Correct body temperature, Why 98.6 is incorrect, and Where the incorrect value came from
  • According to Shoemaker’s article, the correct body temperature is 98.25 F. The commonly assumed correct body temperature of 98.6 is in fact incorrect for multiple reasons. This score was originally obtained at least 100 years ago so something that changed in the environment may have affected body temperature over that vast time period. Thermometers used to measure these temperatures were unreliable. The original methodology to collect this data was flawed. One thing that played into flukes in the methodology was the fact that body temperature fluctuates daily. The incorrect and outdated value was determined by a man named Wunderlich.
• Difference in deviation from Shoemaker’s mean and how unusual my average body temperature is
  • Subtracting my average computed temperature (98.1581) from the mean body temperature mentioned in Shoemaker’s article (98.25), I get a difference of 0.0919. Given Shoemaker’s standard deviation of body temperature (0.73), my deviation from the mean is not even close to being one standard deviation away from the mean. Therefore, my average body temperature is not at all unusual – it is quite normal.
• Representativeness of my body temperature data in terms of standard deviation and sex differences
  • According to Shoemaker’s data, the mean temperature for men is 98.1 and the mean temperature for women is 98.4. As a woman, my body temperature should really be compared to the mean score of 98.4. Since standard deviation for both sexes’ body temperature is .7 and my average temperature’s deviance from the true female mean .2409, my temperature is considerably less than one standard deviation away from the mean. Therefore, my temperature representative of body temperature for the whole (female) population.
• What might affect the mean of my data?
  • As discussed above, the mean can be greatly altered by outlier data points, and my mean was in fact different that what it would have been had my outlier point not been present. The other factor affecting the mean of my data was the initial collection of data points themselves. As discussed last week, several factors related to randomness and systematic bias affected my data collection and resulted in not completely accurate data. Inaccurate data logically speaking will yield an inaccurate mean.
• Degree of accuracy upon taking more temperature measurements
  • Only a bit more than 30 data points is a fairly small sample. According to the law of large numbers, collection of more data points would in fact make the mean of my data more accurate. The law of large numbers asserts that in cases of random variation (such as body temperature), as sample size increases, the mean of the data will more closely approach the true population mean.
• How to convert Fahrenheit to Celsius and my mean body temperature in degrees Celsius
  • One can convert degrees Fahrenheit to degrees Celsius with the following equation:
    • degrees C = (degrees F – 32) * 5/9)
  • Plugging in my mean body temperature:
    • (5/9)*(98.15806-32)=36.75448
    • So my mean body temperature in Celsius degrees is 36.8 C.

• This week’s theme in lecture was making sense out of randomness. Among some of the topics are the means of central tendency, normal distributions, standard deviation/variance, the law of large numbers, and how to determine when a data point falls within a normal range. Following this week’s theme, in this assignment we took last week’s data that was random and made generalizations of the data as a whole by determining means of central tendency. This made the data easy to look at as an organized understandable whole. We worked very closely with some of the calculations discussed in class (mean, median, mode , and standard deviation). After performing basic calculations, we were able to get meaning out of these generalized values by determining if our mean body temperatures were representative of the true average body temperature and how accurate our mean body temperatures were in relation to the number of data points initially collected. A milestone of this week was learning how to operate the statistical package, SPSS. This was mentioned in class and discussed in detail in lab. We used this package to determine most of our statistical calculations needed for the assignment (in addition to manual calculations).

• Last week’s data of body temperature:
  • Graph Information: This graph represents consecutive measurements of my body temperature using a digital thermometer seven times a day roughly every two hours for five days. Degrees are shown in Fahrenheit. Note that the four missing data points include the time-slots: 10-12pm, 2-4pm, 8-10pm, and 10-12am on the fourth day of measurement (Sunday, January 20, 2008).

- Finding the measures of central tendency helps to find my typical body temperature. While my data points were random, they did occur only in a range of about three degrees. The measures of central tendency for my data are good representatives for my body temperature overall.

• Manual calculation
  • Mean – 98.1581
    • After finding the sum of all temperature values from my data, I divided by my number of values (n=31) to find the arithmetic average.
  • Median – 98.1
    • By dividing n(31) in half, I was able to locate the middlemost value after sorting the values in ascending order.
  • Mode – 98.2
    • This temperature reading occurred four times, the highest frequency among the other temperature values.
  • Varience – 0.2992
    • This was determined by adding each data point’s difference from the mean squared, and then dividing by the number of data points.
  • Standard Deviation – 0.547
    • This is the square root of varience.
• SPSS
  • Mean – 98.1581
  • Median – 98.1
  • Mode – 98.2
  • Varience – 0.31
  • Standard Deviation – 0.56

  * I found calculations on SPSS to be much easier as I didn’t feel the need to double check my work incase of manual errors in calculation. I also did not have to do any sorting or counting of my data values – SPSS amazingly did this automatically. My manual calculations of central tendency matched those of SPSS. Variance and standard deviation values were similar between the two calculating methods.

MacEwen, B. (2008, Spring Semester). Psychology 261. Class Lectures. University of Mary Washington.

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

Law of large numbers. (2008, January 21). Wikipedia. Retrieved January 27, 2008, from http://en.wikipedia.org/Law_of_large_numbers

• last week’s information
  • As no new data was collected, calculations this week were based off of last week’s data. Strengths and weaknesses of last week’s data collection, discussed in detail, can be found in last week’s Assignment 1 post under section 5. To briefly revisit the strengths and weaknesses of the data collection, weaknesses included temperature of food/beverages altering mouth temperature, degree of activity altering body temperature, environmental conditions affecting the temperature readings, times when data was not to be recorded, leaving useful potential temperatures out of the data set, and conflicting schedules of the experimenter leading to inconsistent time intervals for measurements. Strengths included more accurate measurements in using a digital over a mercury thermometer, more representative data from including a fairly large number of data points, and more representative data from taking measurements over several days.
• Consistencies and inconsistencies between hand calculations and SPSS
  • While all three of my measures of central tendency calculated both manually and on SPSS matched, variance and standard deviation varied between the two methods. Differences could have been due to how many decimal places were included in calculations. When dealing with numbers less than zero especially, number of decimal places can make a huge difference in calculations. I was not sure how many decimal places SPSS accounted for so my calculations may not have matched those of SPSS perfectly. Differences here make it difficult to determine how accurate either of the values are if they do not confirm one another. In the case of the central tendencies however, we can be confident that these values are accurate as both calculation methods generated the same values.
• Alternate formulas for Variance/Standard Deviation
  • The equation used to determine variance/standard deviation may make a difference in the results of calculations. From my initial manual calculations, for variance I got 0.2992 and for standard deviation I got 0.547. While the n-1 is used typically in sample sizes less than 30, and my sample size was 31, I divided by n when computing variance and standard deviation. To see how much of a difference this would make, I made the same calculations but used n-1 in the computations. From this equation I got a variance of 0.3092 and a standard deviation of 0.556. Interestingly, these values are the same as those computed in SPSS. A better explanation as to why my manual calculations and SPSS’s calculations differed is that SPSS always uses n-1 in equations for standard deviation and variance. It is difficult to say which equation would be better to use with my data because my sample size of 31 is close to borderline of being considered a small sample (less than thirty) or a large sample. In turn, this makes it difficult to decide which calculations are more accurate, my manual calculations or those of SPSS.
• Effects from last week’s weaknesses
  • One major factor that threatens the validity of both the central tendency and variance/standard deviation calculations is the initial collection of data points. As discussed, several factors had interfered with data collection, resulting in much of the data not being representative of true temperature values. Using laws of deduction, any calculation or generalization based off of this inaccurate data will also be inaccurate.