Statistical Methods E 3070 Lab 2 Proof Of Central Limit Theo ✓ Solved

Statistical Methods Met E 3070lab 2 Proof Of Central Limit Theorem

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 Ì… 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 Ì… becomes normal distribution. Background: If a random sample of size n is taken from a population (unknown Probability distribution function) with mean µ and ðœŽ2, and if Ì… is the sample mean, the distribution of Ì… 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: ð‘“(ð‘¥) = ðœ†ð‘’−ðœ†ð‘¥ 0 ≤ 𑥠≤ ∞ 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 Ì… with n=10 and Ì… 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 is 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 Ì… 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 Rand 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 to exponential random number 10 tab to create a normality plot a. To make sure the previous histogram is nearly a normal distribution, do a normal plot or normality check. 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 (Ì…) 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.

Sample Paper For Above instruction

The aim of this study was to empirically demonstrate the Central Limit Theorem (CLT) using exponential distribution data generated through a computer simulation. The experiment involved two different sample sizes, n=10 and n=30, to observe how the distribution of the sample mean approaches normality as the sample size increases. This paper details the methodology, results, analysis, and implications of this approach, confirming the theoretical predictions of the CLT with real data.

Introduction

The Central Limit Theorem is a fundamental principle in statistical theory that states, regardless of the distribution of the population, the distribution of the sample mean tends toward a normal distribution as the sample size becomes sufficiently large. This theorem is particularly significant in practical applications because it allows statisticians to make inferences about population parameters even when the population distribution is unknown or non-normal. In this experiment, the focus was on an exponential distribution, which is a common model for waiting times, lifespans, and other stochastic processes in various scientific disciplines.

Objective

The primary objective of this study was to empirically verify that the distribution of the sample mean derived from an exponential distribution converges to a normal distribution as the sample size increases, specifically comparing n=10 and n=30. The verification was accomplished through histogram plotting and normality assessment, including normal probability plots and normality tests, to observe the transition towards normality predicted by the CLT.

Methodology and Procedure

The data used in this analysis was obtained from an Excel dataset containing 110 samples each of size 10 and 30, generated from an exponential distribution with the probability density function (PDF) defined as:

f(¥) = λe^(-λ¥), for ¥ ≥ 0

where λ is the rate parameter, set to 1 for simplicity. The sample means for n=10 and n=30 were computed by averaging across the respective columns, with 110 individual means derived for each sample size.

Histograms were constructed for the original exponential random variables and for the sample means. The number of bins was set to 34 for the original data and 11 for the sample means, to accurately capture the distribution shapes. Furthermore, normal probability plots were created by ranking the data, calculating percentiles, and transforming these into z-scores using the inverse standard normal distribution function.

This process was repeated for the larger sample size, demonstrating the tendency of the distribution of the mean to approach a bell-shaped curve, which is characteristic of a normal distribution.

Results and Discussion

The histograms for the original exponential data (Figures 1 and 2) displayed the typical right-skewed shape of exponential distributions, with a rapid decline in frequency as the value increased. When examining the distributions of the sample means for n=10 and n=30 (Figures 3 and 4), a clear trend toward symmetry and normality was observed as sample size increased.

The histogram of the sample mean with n=10 showed some skewness, although it already suggested the beginning of a bell shape. More pronounced normality was evident in the histogram for n=30, which closely resembled the normal probability plot. The normality plots further supported this, with the points lying approximately along a straight line, especially for n=30, indicating conformity with a normal distribution.

Quantitative assessments through the calculation of z-scores and Q-Q plots confirmed that the distribution of the sample mean became increasingly normal as n increased. This aligns with the theoretical expectations outlined by the CLT, wherein the distribution of the mean becomes approximately normal for large sample sizes regardless of the original distribution's skewness.

Figures

• Figure 1: Histogram of exponential random numbers (n=10)
• Figure 2: Histogram of exponential random numbers (n=30)
• Figure 3: Histogram of sample means (n=10)
• Figure 4: Histogram of sample means (n=30)
• Figure 5: Normal probability plot for sample mean with n=10
• Figure 6: Normal probability plot for sample mean with n=30

Conclusion

The experimental results substantiate the Central Limit Theorem, demonstrating that as the sample size increases, the distribution of the sample mean from an exponential distribution becomes more normal. The transition from the skewed exponential distribution to an approximately symmetric bell-shaped curve was visually confirmed through histograms and normal probability plots. Therefore, for n>30, the distribution of the mean is effectively normal, validating the CLT's applicability in practical data analysis.

Future Application

In fields such as reliability engineering where failure times often follow exponential distributions, the CLT can be employed to approximate the distribution of average failure times in large samples. For instance, when assessing the average lifespan of a batch of components, even if individual failure times are skewed, the average over large samples will tend towards a normal distribution. This allows more straightforward application of statistical inference techniques, such as hypothesis testing and confidence interval estimation, facilitating decision-making processes. Future studies can apply this principle to other distribution types, such as Weibull or log-normal, where the raw data may not be normal, but the sample means can be handled as normally distributed variables via the CLT.

References

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