In this project, you will verify the Central Limit Theorem through experimentation. Document...

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In this project, you will verify the Central Limit Theorem through experimentation.

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ECON 310 EXTRA CREDIT PROJECT: The Central Limit Theorem th Due in class Wednesday, May 8 In this project, you will verify the Central Limit Theorem through experimentation. MATERIALS REQUIRED: • A standard deck of playing cards • A calculator -or- Microsoft Excel • Note paper and graph paper • Word processor (e.g. Microsoft Word) 1. Choose one of the following binomial experiments (Note: Choosing at random with replacement means you should shuffle the deck after each choice): A. Use cards Ace through 10 of one suit only (e.g., hearts, diamonds, spades, or clubs). Choose 5 cards at random, one at a time, with replacement. Note whether the card is higher than five (6, 7, 8, 9, or 10) or not. B. Use cards Ace through 10 of one suit only (e.g., hearts, diamonds, spades, or clubs). Choose 5 cards at random, one at a time, with replacement. Note whether the card is even (2, 4, 6, 8, or 10) or not. 2. Evaluate the expected value and standard deviation of your experiment A. Note the probability of success (p) and the number of trials (n) in the experiment of your choice. p = _____ n = _____ B. Compute the expected value (mean) and standard deviation of the binomial experiment. E(X) = np = ____________ SD(X) = (1-) = ___________ 3. Conduct the experiment 50 times: A. Record the outcomes for each trial by noting the number of success in each trial. For each card, write 1 if success and 0 if not success. Also note the average number of success in each trial. You might construct a table similar to this (NOTE: your table will have 50 trials):Trial 1: CARD ONE CARD TWO CARD THREE CARD FOUR CARD FIVE Average Trial 2: CARD ONE CARD TWO CARD THREE CARD FOUR CARD FIVE Average ? Trial 50: CARD ONE CARD TWO CARD THREE CARD FOUR CARD FIVE Average B. Report the distribution of trial means in a Frequency Distribution Table: Complete this table three times, one each for n = 10,...

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Robert · Accepted Answer

I choose the 1st experiment i.e. I draw 5 cards at random and checked whether it is higher than 5 or not. 
If it is higher than 5 I put a 1 else 0. 
The obtained results are as follows,
From the obtained output, 
P = 0.6, n =5.  
Theoretically, here p =0.5 thus theoretically E(X) = np = 2.5 and SD(X) = √          = √     = 1.118. 
But here we observed, 
P= 0.6 thus, 
E(X) = 5*0.6 = 3 and SD(X) = √          = √    = 1.0955. 
Card Outcome
1 0
2 0
3 1
4 1
5 1
 
The constructed table is given below, 
Trial Card Average Number of Success 
1 2 3 4 5 
1 1 1 1 1 0 0.8 
2 0 0 0 0 1 0.2 
3 1 0 0 0 1 0.4 
4 1 1 0 1 1 0.8 
5 0 1 1 0 0 0.4 
6 1 0 1 0 1 0.6 
7 0 0 0 1 0 0.2 
8 1 0 0 0 0 0.2 
9 0 0 0 1 0 0.2 
10 0 1 0 1 0 0.4 
11 1 1 1 1 1 1 
12 1 0 1 0 1 0.6 
13 1 0 1 1 0 0.6 
14 1 1 0 1 1 0.8 
15 1 1 0 0 0 0.4 
16 0 1 0 1 0 0.4 
17 1 0 0 0 0 0.2 
18 1 0 0 1 0 0.4 
19 1 1 1 0 1 0.8 
20 1 0 0 1 0 0.4 
21 0 0 0 1 0 0.2 
22 1 1 0 1 1 0.8 
23 0 0 0 1 1 0.4 
24 1 1 0 1 1 0.8 
25 0 0 1 1 0 0.4 
26 0 0 0 0 1 0.2 
27 1 1 1 1 0 0.8 
28 1 1 1 0 1 0.8 
29 1 1 1 0 1 0.8 
30 0 0 0 1 0 0.2 
31 0 0 1 0 1 0.4 
32 1 1 0 1 0 0.6 
33 0 0 0 0 1 0.2 
34 1 1 1 0 1 0.8 
35 1 0 1 1 1 0.8 
36 1 1 0 0 0 0.4 
37 1 0 1 1 1 0.8 
38 0 1 1 1 0 0.6 
39 1 0 0 0 1 0.4 
40 1 1 1 0 0 0.6 
41 0 0 1 0 1 0.4 
42 0 0 1 1 1 0.6 
43 1 0 1 0 1 0.6 
44 1 1 0 1 0 0.6 
45 1 1 0 0 1 0.6 
46 0 0 0 1 1 0.4 
47 1 0 1 0 1 0.6 
48 1 1 0 0 0 0.4 
49 0 0 0 1 0 0.2 
50 0 1 0 1 0 0.4
The required tables are given below.
For n=10,
For n=30,
Possible sample means Number of sample means observed
0.0 upto 1.0 4
1.0 upto 2.0 3
2.0 upto 3.0 1
3.0 upto 4.0 2
4.0 upto 5.0 0
Possible sample means Number of sample means observed
0.0 upto 1.0 5
1.0 upto 2.0 9
2.0 upto 3.0 8
3.0 upto 4.0 8
4.0 upto 5.0 0
 
For n=50,
Possible sample means Number of sample means observed
0.0 upto 1.0 10
1.0 upto 2.0 16
2.0 upto 3.0 11
3.0 upto 4.0 12
4.0 upto 5.0 1

Theoretically E(X) = 5.5 and SD(X)= 2.872281323. 
Observed, 
E(X) = 6 and SD(X) = 2.75680975. 
The outcome of 50 trials is given below. 
1 2 3 4 5
Face value 4 2 6 9 9
Card
Trial 
Card 
Average Number of Success 
1 2 3 4 5 
1 7 10 10 6 3 7.2 
2 1 5 4 2 8 4 
3 9 2 1 1 6 3.8 
4 7 9 2 6 6 6 
5 3 9 7 1 3 4.6 
6 7 5 7 2 7 5.6 
7 3 4 3 7 2 3.8 
8 9 4 1 5 4 4.6 
9 3 4 5 9 5 5.2 
10 1 8 2 9 1 4.2 
11 9 9 6 7 7 7.6 
12 9 3 10 4 9 7 
13 7 3 6 9 1 5.2 
14 6 10 3 10 6 7 
15 7 10 1 2 1 4.

In this project, you will verify the Central Limit Theorem through experimentation. Document Preview: ECON 310 EXTRA CREDIT PROJECT: The Central Limit Theorem thDue in class Wednesday, May 8 In this...

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