Business Statistics-the implications of the Central Limit Theorem (CLT)

Throughout this exercise, you will verify the implications of the Central Limit Theorem (CLT) introduced in the lecture. Recall that CLT does not require the population from which we draw our sample to be normally distributed. We will, therefore, assume that the population we study is exponentially distributed. The exponential distribution has a long history and it is mainly used to describe waiting times, e.g., time elapsed before an unemployed person finds a job. For example, the probability that one finds a job within “x” weeks is given by: F(x) =1− exp(−λx) λ is a positive parameter that solely describes exponential distribution. Both the population mean and the population standard deviation are equal to 1/ λ . 1. Download the excel file named “clt.xls” from Power Campus. Replace the name with your US state team name, e.g., “virginia.xls”. This is a template for you to create random observations and samples from an exponential distribution. 2. In sheet “Random Samples 1”, set λ = 0.2 in cell D1 and sample size, n, equal to 100 in cell B1. 3. Each row starting from the 4th row represents a “sample” and each column staring from “C” represents an “observation” in a sample. You are asked to generate 2,000 samples, each of which has 100 observations. For example, the 1st sample lies in C4:CX4. To generate a random exponential variable, use the following formula: −LN(1− RAND())/ λ 4. Calculate sample mean for each sample in the column A (A4:A2003). 5. Now, the numbers in (A4:A2003) represent a collection of 2,000 sample means from repeated samples of size 100. What does CLT say about its mean and standard deviation? Next, calculate the average and standard deviation of sample means using the excel functions =AVERAGE() and =STDEV() in cells A2004 and A2005, respectively, and compare them to the values implied by CLT. 6. Create a histogram in percentages for sample means using the cutoff values provided to you in sheet “Histograms” in the same file. To do so, select cells adjacent to the bin values, i.e., C2:C101, and array-enter (control and enter for PC and command and return for Mac){=FREQUENCY(data, bins)}. This will return you the exact counts for each cutoff value. In the next column, calculate percentages by dividing the exact counts by the total count. Insert a column chart in Sheet “Histograms” for the percentages you calculated and use bin cutoff values as your x-axis labels. Indicate sample size and lambda values in the title. Does its shape resemble that of a normal distribution? Comment on how we verified CLT by questions 1-5. 7. Now, move to sheet “Random Samples 2” and set the value of λ to 0.4. Repeat steps 3-6 above. What does CLT tell us about the standard deviation of sample means with regard to the population standard deviation? Verify this claim by comparing the histogram you plotted in this question to the histogram you plotted in question 6. 8. Move to sheet “Random Sample 3”. Change the value of λ back to 0.2, but this time increase the sample size to 225. Accordingly, increase the sample size for each sample and Central Limit Theorem 2 update average formulas. What does CLT tell us about the standard deviation of sample means with regard to the sample size? Verify this claim by comparing the histogram you plotted in this question to the histogram you plotted in question 6. 9. Upload both the excel and word files to Power Campus.

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