Business statistics
Case Study Case Study 7.1 Central Limit Theorem: In this case analysis, you will generate random samples from several probability distributions. You will be able to observe for each distribution the effect of sample size; and, you will also come to better understand the central limit theorem which has been proclaimed as one of the most important theorems in statistics and perhaps the most important result of statistical theory. The Central Limit Theorem states that no
matter what the shape of the distribution is, the distribution of the sample means ( x ) approaches a normal distribution as the sample size increases. For a large sample ( 30n ), the distribution almost always approaches a normal. In this case, you will be able to observe this result. We will first use EXCEL to generate 30 columns of uniformly distributed random numbers and store them in the first 30 columns of a spreadsheet. Each column will contain 300 rows, representing 300 samples. Starting from column 31, we will calculate and store the Means of column 1, columns 1‐5, columns 1‐10, columns 1‐20, and columns 1‐30. This will give us the means of sample size 1, 5, 10, 20, and 30, or n=1, n=5, n=10, n=20, and n=30. We will then construct the histograms of each of samples and observe the how the shape of the distribution changes as the sample size is increased. To do this follow the steps outlined here. Steps: (i) Label the first 30 columns and the next five columns of the EXCEL worksheet as shown
A B C … AC AD AE AF AG AH AI [1] [2] [3]…. [29] [30] n=1 n=5 n=10 n=20 n=30
(ii) Generate 30 columns of 300 random numbers using a uniform distribution with (continuous)
values between the endpoints 0 & 1.0, and store in Columns A through AD. You will have 30 columns and 300 rows of generated data stored in columns A, B, C,…,AD. To generate the random numbers, select the following command sequence:
Data Data Analysis ‐>Random Number Generation
Complete the Random Number Generation dialog box as shown in the figure below.
(iii) Determine the Means of sample sizes n=1, n=5, n=10, n=20, and n=30 by calculating the
following averages: In cell AE, type: =AVERAGE(A2) and hit the enter key. Copy this function to down to row 301 In cell AF, type: =AVERAGE(A2:E2), hit the enter key and copy to row 301 In cell AG, type: =AVERAGE(A2:J2), hit the enter key and copy to row 301
In cell AH, type: =AVERAGE(A2:T2), hit the enter key and copy to row 301 In cell AI, type: =AVERAGE(A2:AD2), hit the enter key and copy to row 301
The above steps will calculate and store the means for sample sizes n=1, n=5, n=10, n=20 and n=30 in columns AE through AI. Construct Histograms of the values in columns AE to AI, which represents a sampling distribution with n=1, n=5, n=10, n=20, and n=30. To construct the histogram for n=1 (column AE), perform the following steps using the menu bar:
Data Data Analysis Histogram and complete the histogram dialog box as shown.
When the histogram is created, double click on the histogram and bring the gap width to 0 by moving the pointer to the left. The value in the output range box can be any blank cell where you want to display the histogram.
(iv) Repeat step (3) and construct the histograms for AF to AI or for columns labeled n=5, n=10, n=20,
and n=30. The histograms for n=1 and n=30 are shown below.
Note how the shape becomes symmetrical, or normal as the sample size is increased from n=1 to n=30. For n=1, the distribution is uniform (a rectangular shape where each value between a low and high value has equal probability of occurrence).
Assignment: (1) Repeat the steps (i) to (iv) above to generate random numbers from a Uniform Distribution and create
histograms for the sample sizes n=1, n=5, n=10, n=20, and n=30.
0 5 10 15 20 25
F re q u e n cy
Bin
Histogram (n=1)
Frequency
0 10 20 30 40 50
0 .3 4 2 1 6 5 3 9
0 .3 8 1 2 4 5
0 .4 2 0 3 2 4 6 0 9
0 .4 5 9 4 0 4 2 1 9
0 .4 9 8 4 8 3 8 2 8
0 .5 3 7 5 6 3 4 3 8
0 .5 7 6 6 4 3 0 4 8
0 .6 1 5 7 2 2 6 5 7
0 .6 5 4 8 0 2 2 6 7F re q u e n cy
Bin
Histogram (n=30)
Frequency
(2) Repeat the steps (i) to (iv) above but this time generate your random numbers using an Exponential Distribution or any other distribution .Create histogram for n=1, n=5, n=10, n=20, and n=30.
(3) Repeat the steps (i) to (iv) above but this time generate your random numbers using a normal distribution. Create histogram for n=1, n=5, n=10, n=20, and n=30.
Write a brief report including all the histograms you created. Discuss your understanding and the importance of the central limit theorem.
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