Ms. Williams Problem

Online Random Number Generator

1= Girl

2=Boy

2 2 2 1 1 1 1 1 2 1 2 1 2 1 2 2 2 2 2 1 2 1 2 1 1 1 2 1 2 1 1 1 1 1 1 2 1 2 2 2 2 2 1 2 2 2 2 1 2 2 2 2 2 2 1 2 1 2 1 1 1 2 2 1 2 1 2 2 2 2 1 2 2 1 2 2 1 2 1 1 1 2 2 2 2 2 2 2 2 1 2 1 1 1 1 1 1 2 2 1

Ref: Daniels (2003)

Exactly three boys occured once in our data.

• 1/98= .0102

The probability of randomly selecting a mother who has three boys in a row is 1.02%.

There were 56% boys to 44% girls. If we extended the random number generator to 10,000, the proportion of boys would decrease and the percent of girls would increase because of the Law of Large Numbers. Small samples contain higher variability than bigger samples because larger samples even out randomness.  Therefore, larger samples are more reliable because the values vary less and cluster more around the mean.

Personal Example of the Law of Large Numbers

In a freshman student’s first semester at Mary Washington, they take five classes with five random teachers. Two of the five professors are poor teachers. The student assumes that all teachers at Mary Washington are bad teachers because 40% of the teachers their first semester were bad. If they stay at the University, they will see that the percentage is much lower than 40% (maybe closer to 10%) because the Law of Large Numbers will reduce the impact of randomness on the small sample of five teachers.

Male Psychology Majors

There are 8 out of 46 males in our class

• 8/46=.1739 or 17.39%

According to an article in the Journal of General Psychology, the percentage of male psychology students was 25% nationwide in 2005. Our proportion of males in our class is smaller than the national average because chance has such a pervasive impact on small sample sizes.  The uneven distribution of males to females at Mary Washington also biases our results.

Oil Change

You waited 3,467 miles to change your oil. The mean is 3, 258 miles with a SD of 223 miles.

z-score= (3,467-3,258)/223= .93721

Area below z-score .93721 = .17618

Area between median and z-score = .5-.1761= .3239= 32.39%

Above and below the mean= .5 + .3239= .8239= 82.39

Therefore, 82.39% of drivers wait 3, 467 miles before getting an oil change. You’re mild deviation does not seem unreasonable when phrased this way.

Strengths/Weaknesses

Strengths:

• There are multiple ways to answer the oil change problem and can display your knowledge of statistics in many ways.

Weakness:

• Human error

References:

Baily, M. (April 2005). General versus gender-specific attributes of the Psychology major. Journal of General Psychology. http://209.85.165.104/search?q=cache:K0kDBfts1X0J:www.encyclopedia.com/doc/1G1-132241867.html+statistic+male+psychology+major+undergraduate&hl=en&ct=clnk&cd=11&gl=us

Daniels, Michael (2003). Random Number Generator. Accessed January 31, 2008 at http://www.mdani.demon.co.uk/para/random.htm

MacEwen, B. (2008, spring). Psychology 261. Class lectures. University of Mary Washington

Done by Mike and Mara

Mara’s Results  

Calculator 

Mode: My data was tri-modal, meaning I had three modes; 95.9, 98.4, and 98.9.  These are the most commonly occurring temperatures in my data.

Median:  98.55.  The median is the middle number when the data are listed in ascending order.  This is the middle-most number in the data. 

Arithmetic Mean:  98.226.  The arithmetic mean is the average of all of the temperatures.  If asked to give one number to represent the person’s overall temperature, it would be the arithmetic mean.  

Standard Deviation: 1.57402741.  The standard deviation is the average departure from the mean.

Variance: 1.339581105

SPSS

N	Valid	34
	Missing	0
Mean			98.2294
Median			98.5500
Mode			95.90(a)
Std. Deviation			1.16060
Variance			1.347

 Multiple modes exist. The smallest value is shown.

Outlier temperatures would most affect the arithmetic mean because it is included in the calculation.  The outlier would make the mean more extreme than it truly is.  It wouldn’t disrupt the mode because it is unlikely to repeat many times and therefore would be disregarded.  The median is a number in the middle of the data and outliers lay beyond this boundary.  Therefore, they would not affect the median. 

Outliers may occur because of systematic events such as taking your temperature outside.  The extreme value may also be the result of faulty equipment or a random event.  Either way, these events are rare and are not reliable data values.  The normal curve shows that most data points should be clustered about the mean. 

Randomness plays a part in creating outliers.  Since randomness is not predictable, then outliers that are influenced by randomness are not reliable.   98.6 degrees F is not the correct body temperature.  It is actually 98.25 degrees F.  The confusion is attributed to unreliable thermometers and an insufficient account for random fluctuation.  As the instrument of measure becomes more precise, our mean because more accurate.  Also, the original estimate did not take into account daily fluctuations in temperature; in part, randomness. 

(My arithmetic average temperature) – (Correct Average Temperature) = SD

98.226 degrees F - 98.25 degrees F = -.024 degrees F

My average was .024 degrees F colder than the average correct temperature.  This is significantly closer to the correct temperature than the sample SD of .73.  In other words, I am particularly average.  One reason that my temperature was so close to average could be that I used a Vick’s Comfort-Flex thermometer, a $17 thermometer that is highly accurate.  Overall, my data are representative. The mean of my data is probably lowered by my morning temperature, which was consistently around 95.9 F.  Extending the length of the experiment would make it more accurate because of the Law of Large Numbers, which states that the more data that is used, the closer the answer will be to the mean.  

98.226 degrees F= 36.792 degrees Celsius

Degrees Celsius= (Degrees Fahrenheit-32) x 5/9   

Mike’s Results 

My Mode was 97.3 and appeared 6 times in my data.

My median data is also 97.3F.  This is the value in the direct middle of all my temperatures. Since I had only 34 data points, the median is my 17.5^th point, an average of my 17^th and 18^th values.  Since they were the same value, the median is 97.3.

The mean, my average temperature, was 97.06F

My standard deviation by calculator was .7071, which is the square root of 2 divided by 2. This is the average departure from the mean temperature.

My variance was .5

Statistics from SPSSN    Valid    34
Missing    36
Mean        97.0618
Median        97.3000
Mode        97.30
Std. Deviation        .70711
Variance        .500

Outlier temperatures would most affect the mean, because a slight change in the algebra involved in obtaining the mean results in different values. Take for example the data set of 9, 10 and 11. The median is 10 and the mean is 10. If we throw in a 2 however, the median becomes 9 and the mean becomes 5.5. A larger change occurs in the mean, due to the more complex methods by which we obtain it. 

Outliers occur mostly due to systemic events such as drinking hot tea, or eating ice cream before taking your temperature. I can clearly remember times when I did both, and without thinking directly took my temperature right after. Such times resulted in spikes in both directions, as is seen in my data. These events were rare however, but are easily explainable. Outliers are not very reliable when looking at average temperatures in such a short period, but if we were tracking temperature over a number of months, such data may be acceptable.  

My average body temperature was 97.06 degrees F, while the average body temperature given by the article was 98.25 degrees F. 97.06 F – 98.25F = -1.19F I apparently have a huge deviation from the normal mean temperature. The average SD was .73, and even then, I am off by about .46. This could be due to my growing up in a colder climate in New York, or some medical conditions/medication. My data is very varied, which goes along with my lifestyle. I am never really doing the same thing twice, running around with different schedules each day, spending a lot of time outside, going back and forth because I forgot things at my dorm or such. 

I was a lot cooler than Mara.  Although it’s stated that the SD is .7, I was a full 1.06 degrees cooler. Then again, she exercises and is probably more active than I am. In addition, I am not very representative of the male population. I am more towards the introverted extreme, spending as much time as I can inside, in front of a keyboard. 

97.06F – 32F * 5/9 =36.14 C  

Mara’s Temperature Results 

Calculator 

Mode: My data was tri-modal, meaning I had three modes; 95.9, 98.4, and 98.9.  These are the most commonly occurring temperatures in my data.

 Median:  98.55.  The median is the middle number when the data are listed in ascending order.  This is the middle-most number in the data. 

Arithmetic Mean:  98.226.  The arithmetic mean is the average of all of the temperatures.  If asked to give one number to represent the person’s overall temperature, it would be the arithmetic mean.  

Standard Deviation: 1.57402741.  The standard deviation is the average departure from the mean.

Variance: 1.339581105

SPSS

Statistics

N	Valid	34
	Missing	0
Mean			98.2294
Median			98.5500
Mode			95.90(a)
Std. Deviation			1.16060
Variance			1.347

 Multiple modes exist. The smallest value is shown.

Outlier temperatures would most affect the arithmetic mean because it is included in the calculation.  The outlier would make the mean more extreme than it truly is.  It wouldn’t disrupt the mode because it is unlikely to repeat many times and therefore would be disregarded.  The median is a number in the middle of the data and outliers lay beyond this boundary.  Therefore, they would not affect the median. 

Outliers may occur because of systematic events such as taking your temperature outside.  The extreme value may also be the result of faulty equipment or a random event.  Either way, these events are rare and are not reliable data values.  The normal curve shows that most data points should be clustered about the mean. 

Randomness plays a part in creating outliers.  Since randomness is not predictable, then outliers that are influenced by randomness are not reliable.   98.6 degrees F is not the correct body temperature.  It is actually 98.25 degrees F.  The confusion is attributed to unreliable thermometers and an insufficient account for random fluctuation.  As the instrument of measure becomes more precise, our mean because more accurate.  Also, the original estimate did not take into account daily fluctuations in temperature; in part, randomness. 

(My arithmetic average temperature) – (Correct Average Temperature) = SD

98.226 degrees F - 98.25 degrees F = -.024 degrees F

My average was .024 degrees F colder than the average correct temperature.  This is significantly closer to the correct temperature than the sample SD of .73.  In other words, I am particularly average.  One reason that my temperature was so close to average could be that I used a Vick’s Comfort-Flex thermometer, a $17 thermometer that is highly accurate.  Overall, my data are fairly representative. The mean of my data is probably lowered by my morning temperature, which was consistently around 95.9 F.  Extending the length of the experiment would make it more accurate because of the Law of Large Numbers, which states that the more data that is used, the closer the answer will be to the mean.  

98.226 degrees F= 36.792 degrees Celsius

Degrees Celsius= (Degrees Fahrenheit-32) x 5/9

MaraMike

Mike AVG	97.06176
Mara AVG	98.35429

A random event is impossible to predict and occurs without preference over other events.  Random events interfer with our ability to predict the future because their occurence is unpredictable.   

A systematic event is one that creates a bias.  As statisticians we must be aware of these biases and their impact on our data.

Due to the interplay of random events, it is not possible to accurately and consistently predict future events.  However, randomness may work to your favor and you may accurately guess a few predictions.  However, in the long run, randomness with work to disrupt your predicting ability.  The best guess for a prediction is the mean because it will come closest to the real value than any other option. 

Sorry Mara but i’m taking over your post. If Rebecca is reading this, can ya do  me a favor and make me an admin on this blog. Apparently i can only edit on this one, and i can’t post on my old groups. Must’ve gotten angry when i got switched around.

Anyway! Here’s our graphs so far.

1) A random event is impossible to predict and occurs without preference over other events.  Random events interfer with our ability to predict the future because their occurence does not follow a pattern.   A systematic event is one that creates a bias. Systematic events are not random, but can also interfere with the integrity of data.  As statisticians we must be aware of these biases and remove them from our data.

2)  Due to the interplay of random events, it is not possible to accurately and consistently predict future events.  However, randomness may work to your favor and you may accurately guess a few predictions.  However, in the long run, randomness with work to disrupt your predicting ability.  The best guess for a prediction is the mean because it will come closest to the real value than any other option. 

3)  I (Mara) had some systematic effects including slight fluxuation in the timing of taking birth control which would effect hormone levels.  I also took some extremely hot showers for extended periods of time over the five day period. 

7 sources of random variation:

1) Variation in the temperature of Chandler

2) Outside temperature

3) Briskness of walking between classes

4) Hotness of food (hot tea vs. ice tea)

5) Time of Eating (in relation to temperature-taking)

6) Exercise- causes an increase in temperature

7) Sickness- causes an increase in temperature

 

Strengths:

Both of us were fairly reliable about remembering to take our temperature and also to not take reading too close together.  The use of a digital thermometer increases the accuracy of the readings. 

Weakness:

Few conclusions can be drawn from this experiment because it covers such a small sample size.  Randomness would play too strong of a role in this size experiment to assign significance.  The sample size needs to be much bigger in order to reduce the effects of randomness.  Assigning significance to this data would falsely apply the law of large number to a set of small numbers (Hastie and Dawes 2001). 

 Systematic effects must be eliminated to ensure there is no bias.  As psychology majors, we are aware that millions of Americans are on antidepressant and antipsychotic medications that can reduce their bodies temporal regulation (McKim 1999).  If we were to conduct this experiment on a larger scale, these individuals would need to be screened for an eliminated from our data to remove this possible systematic effect.

As for our mini-experiment, we may want to impose rules such as “do not take your temperature within twenty minutes of eating, showering, exercising…ect.”  This would create more uniform data, but would likely reduce compliance.

 Referenced Sources:
Hastie, R., & Dawes, R. E. (2001). Rational choice in an uncertain world. The psychology of judgment and decision making.

        Thousand Oaks, CA: Sage, Inc.

MacEwen, B. (2008).  Randomness- lecture.  Chandler 304

McKim, WA (1999). Drugs and behavior: An introduction to behavioral pharmacology (4^{th  Edition). New York:  Prentice Hall}

^{I hereby declare upon my word of honor that I have neither given nor received unauthorized help on this work.}

^{Mara Rice}

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