Lab report

Statistical Methods (MET E 3070)

Lab # 2: Proof of Central Limit Theorem Fall 2020

Today’s Date: 9/25/2020

Objective:

The raw samples in a population follows the exponential distribution. The random variable �̅� denotes the sample mean of “n” samples taken from population. In this lab, we will prove that as the number of samples

taken increases, the distribution of �̅� becomes normal distribution.

Background:

If a random sample of size n is taken from a population (unknown Probability distribution function) with

mean µ and 𝜎2, and if �̅� is the sample mean, the distribution of �̅� becomes normal distribution, as per

central limit theorem, as you take more and more samples it becomes more and more normal. This is an

example : A dice follows a discrete uniform distribution. In the following, the histogram obtained after

averaging two dice, three dice, five dice and ten dice are shown. As the number of sample increases, the

distribution becomes closer and closer to normal PDF.

Figure 1: Distribution of average scores from throwing dice (Montgomery and Runger, Applied statistics and probability for engineers,

7th Edition, page 154)

Data set:

The excel sheet with exponentially distributed random numbers is posted. The exponential distribution

follows:

𝑓(𝑥) = 𝜆𝑒−𝜆𝑥 0 ≤ 𝑥 ≤ ∞

The first tab contains 110 rows and 10 columns. The second tab contains 110 rows and 30 columns. So

you can sum and average across rows to get �̅� with n=10 and �̅� with n=30. Follow the procedure outline

here to demonstrate the validity of the central limit theorem.

Procedure:

1. Create two tabs called “Histogram” and “Normality Plot”

a. This the place where you will place your histograms and normality plots

2. Go to “Exp Rand Num 10” tab

a. Select cell L2

i. Type =AVERAGE(A2:J2)

b. Drag the right bottom corner of cell L2 until you reach cell L111

c. Now you will have �̅� of 110 values of mean of exponential random numbers in column L

3. Go to the Histogram tab

a. Create a suitable histogram for columns A through J from Exp Ran Num 10 tab

b. Set the number of bins to 34

c. Describe the shape of the graph.

d. Create a suitable histogram for column L from Exp Rand Num 10 tab

e. Set the number of bins to 11

f. Is it normally distributed?

4. Go back exponential random number 10 tab to create a normality plot

a. To make sure the previous histogram is nearly a normal distribution, do a normally plot or

normality check. Here are some of the steps for normality plot.

b. Copy column “L” to “M”

c. Sort column M in ascending order

d. On cells N2 through N111

i. Create number 1,2,3,4…110

e. On cell O2 create the percentile values

i. Type =(N2-0.5)/110

ii. Drag the bottom right corner of the cell until you reach cell O111

f. On cell P2

i. Type =NORMSINV(O2) to get z-score

ii. Drag the bottom right corner of the cell until you reach cell P111

g. Plot column P (z score) on y-axis versus column M (sorted values) on x-axis

i. Is the data normally distributed?

5. Repeat the above steps for n=10 to n=30 because central limit theorem says for n>30, you will

surely get a normal distribution

a. Use the data exponential random data from the Exp Rand Num 30

Lab Report Format

1. Cover Page

2. General Introduction

3. Objective

4. Procedure and statistical principle

5. Results and Discussion

6. Figures

a. 2 x Histograms from the exponential random numbers

b. 2 x Histograms from the average values (�̅�)

c. 2 x Normality Plots

7. Conclusion

8. Future Application: Discuss a typical case in your field of study where the probability distribution

of the raw sample data may not be normal. Explain the study. Discuss how you will use the central

limit theorem in this study to do any analysis. Finally point out what you will conclude about this

study.