The exponential distribution is an example of a skewed distribution. It is defined by the density function

with parameter l. The mean of an exponential distribution

The exponential distribution is used to model the failure rate of electrical components. For many electrical components, a plot of the failure rate over time appears to be shaped like a bathtub. The exponential distribution is used widely in reliability engineering for the constant hazard rate portion of the bathtub curve, where components spend most of their useful life. The portion of the curve prior to the constant rate is referred to as the burn-in period, and the portion after the constant rate is referred to as the wear-out period. Another interesting property of the exponential distribution is the memoryless property. This says that the time to failure of a component does not depend on how long the component has been in use. For example, the probability that a component will fail in the next hour is the same whether the component has been in use for 10, 50, or 100 hours, and so on. In our discussion of summary statistics, we have
seen that the mean is used as a measure of the center for data sets whose distribution is (roughly) symmetric with no outliers. In the presence of outliers or skewness, we used the median as the measure of the center because it is resistant to extreme values. However, for skewed distributions, it is often difficult to determine which measure of center is more useful. In some cases, the mean, though not resistant, is easier to find than the median or is more intuitive to the reader within the context of the problem. Here we look at the sampling distributions of the mean and median for two different populations, one normal (symmetric) and one exponential (skewed).
1. Using statistical software, generate 250 samples (rows) of size n = 80 (columns) from a normal distribution with mean 50 and standard deviation 10. 

2. Compute the mean for the first 10 values in each row and store the values in C81. You might label this column "n10mn" for normal distribution, sample size 10, mean. 

3. Compute the median for the first 10 values in each row and store the values in C82. You might label this column "n10med" for normal distribution, sample size 10, median. 

4. Repeat parts 2 and 3 for samples of size 20 (C1-C20), 40 (C1-C40), and 80 (C1-C80), storing the results in consecutive columns C83, C84, C85, and so on. 

5. Compute the summary statistics for your data in columns C81-C88. 

6. How do the averages of your sample means compare to the actual population mean for the different sample sizes? How do the averages of your sample medians compare to the actual population median for the different sample sizes? 

7. How does the standard deviation of sample means compare to the standard deviation of sample medians for different sample sizes? 

8. Based on your results in parts 6 and 7, which measure of center seems more appropriate? Explain. 

9. Construct histograms for each column of summary statistics. Describe the effect, if any, of increasing the sample size on the shape of the distribution of sample means and sample medians. 

10. Using statistical software, generate 250 samples of size n = 80 from an exponential distribution with mean 50. This could represent the distribution of failure times for a component with a failure rate of λ = 0.02. Note: The threshold should be set to 0. Store the results in columns C101-C180. 

11. Compute the mean for the first 10 values in each row and store the values in C181. You might label this column "e10mn" for exponential distribution, sample size 10, mean. 

12. Compute the median for the first 10 values in each row and store the values in C182. You might label this column "e10med" for exponential distribution, sample size 10, median. 

13. Repeat parts 11 and 12 for samples of size 20 (C101-C120), 40 (C101-C140), and 80 (C101-C180), storing the results in consecutive columns C183, C184, C185, and so on. 

14. Compute the summary statistics for your data in columns C181-C188. 

15. How do the averages of your sample means compare to the actual population mean for the different sample sizes? How do the averages of your sample medians compare to the actual population median for the different sample sizes? 

16. How does the standard deviation of sample means compare to the standard deviation of sample medians for different sample sizes? 

17. Based on your results in parts 15 and 16, which measure of center seems more appropriate? Explain. 

18. Construct histograms for each column of summary statistics. Describe the effect, if any, of increasing the sample size on the shape of the distribution of sample means and sample medians. 

19. Can the Central Limit Theorem be used to explain any of the results in part 18? Why or why not?

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