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Sample Assignment Paper on Central Limit Theorem

Random process observed

To verify the truthfulness of the central limit theorem, I rolled two dice. I chose this random process because it is easy to conduct and verify its outcomes. Furthermore, I preferred it because the outcome of each roll does not depend on any other roll. Consequently, each outcome is independent as the theorem requires the process to be. Apart from this, it is possible to compute the expected value for the process and its variance (Pyzdek, 2014).   

Sampling my averages

In terms of sampling my averages, I rolled the die one after the other, recorded the outcomes and thereafter calculated the averages for the two dice using the excel program. I did this by first rolling the dice, recording their outcomes on the excel spreadsheet and later on computing the averages using the excel program. In order to obtain the frequency of each average, I used the COUNTIF function. This function produced the results shown on table 1 for each average. It was after I obtained those results I constructed the column chart 1.

Number of observations

In terms of the number of times I collected the averages, as instructed, I rolled the two dice each after the other, recorded the outcomes for the two dice and later on calculated the averages. I did this two hundred times for each die. This means that I collected the averages of the outcomes two hundred times. The frequency for the outcomes is as shown below.

Table 1

Possible sums

2

3

4

5

6

7

8

9

10

11

12

Average

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Frequency

0

10

20

28

39

41

21

17

13

6

5

 

Column chart

After collecting the outcomes, calculating the averages and plotting the graph, I obtained the following column chart using excel program.  

Column chart 1

Interpretation of results

As it can be seen from the above column chart, I was able to obtain a column chart that almost resembled a normal distribution even if it is not a perfect one. By this I mean that even if it is somewhat skewed to the left, one can see that the chart gives a curve that a normal distribution would give. This is in relation to the fact that as you move to the left hand side you can see that the probability of obtaining an average of the two dice equal to 1 is almost zero while an average of obtaining an average of the two dice equal to 1.5 is higher than that of 1. The same case applies to the right hand side because the probability of obtaining an average of the two dice equal to 6 is tending to zero as the number of random observation increases. To a great extent, I was able to do this because I repeated the exercise many times. Presumably, if I were to proceed on with the exercise, probably I would obtain a finer curve than the one I have right now. This notwithstanding, I have been able to verify the central limit theorem because my column chart shows that as the number of observation increases my random process assumes a normal distribution curve (Pyzdek, 2014). Therefore, I can confirm the central limit theorem based on the outcome of my random process.   

 

 

 

 

 

 

 

 

 

 

 

 

 

Reference

Pyzdek, T. (2014). Six sigma. New York: McGraw-Hill.

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