Frank Wang November 16th, 2006 Stat3032 Project A) Probability Plot of C1 Normal - 95% CI 99 Mean StDev N AD P-Value 95 90 0.08395 0.006964 10 1.969 <0.005 Percent 80 70 60 50 40 30 20 10 5 1 0.06 0.07 0.08 0.09 0.10 0.11 C1 Histogram of C1 7 6 Frequency 5 4 3 2 1 0 0.080 0.085 0.090 0.095 0.100 0.105 C1 This distribution of the data is not symmetric. The distribution is skewed to the right, and the curve is not normal. Frank Wang November 16th, 2006 B) The mean from the data is 0.08395. The result shows that the median is 0.08165. The standard deviation for the dataset is 0.00696. The first quartile is 0.08070, and the third quartile is 0.08330. Descriptive Statistics: C1 Variable C1 Mean 0.08395 StDev 0.00696 Q1 0.08070 Median 0.08165 Q3 0.08330 IQR 0.00260 C) Analyst used four methods to find the 95% confidence intervals. They are t distribution, bootstrap confidence interval for mean, Wilcoxon Signed Rank confidence interval and bootstrap confidence interval for median. T distribution confidence interval This interval centers on the mean, and t1-α/2(StDev/n1/2) gives the standard error. Since the sample size is less than 30, the assumption for the t distribution is met. Bootstrap confidence interval for mean This interval centers on the mean. Analyst believes the sample distribution is not normal, therefore analyst choose this method to find the 95% confidence interval for the mean. Analyst generated 700 rows of data based on the original sample and randomly draws 10 numbers. Then analyst calculated the mean for each row, and analyst arranged the 700 mean in increasing order. The 2.5 percent quartile is at the position (0.025)*(700+1), and the 97.5 percent quartile is at the position (0.975)*(700+1). This interval is (0.08108, 0.08285). This method has no assumptions. Wilcoxon Signed Rank confidence interval There is another type of confidence interval for the mean that can be computed if the original distribution is not believed to be normal, but it is symmetric. Since the median is equal to the mean for a symmetric distribution, a confidence interval for the mean is also a confidence interval for the median. Let’s suppose that we have once again sampled ten people and asked their height. For example, let’s suppose the data was 60, 62, 62.5, 62.5, 63, 63.5, 64, 69, 70 and 72. To perform the wilcoxon signed rank confidence interval, you have to compute the average of every two heights in the data set, including an average of each height with itself. For instance, you would average 60 and itself. Then, you would have to average 60 and 62. Continuing, you would average 60 and 62.5. You would do this until every possible average was taken. Then, you would put the paired averages in order. By using a table and a formula in the book, you would determine which positions would be your lower and upper limits. Analyst is lucky that analyst have Minitab to compute them. This interval is (0.08008, 0.0920). This method assumes the data is symmetric, and the method does not meet the assumption. Bootstrap confidence interval for median Frank Wang November 16th, 2006 Analyst use the same procedure used in bootstrap confidence interval for mean to generate 700 rows of data based on the original data. Then analyst arranged the median from each row in increasing order. The 2.5 percent quartile is at the position (0.025)*(700+1), and the 97.5 percent quartile is at the position (0.975)*(700+1). This interval is (0.08070, 0.08260). This method has no assumptions. D) t distribution 95% confidence interval is (0.078969, 0.088931). We are 95% confident that the true mean is within the interval. However, the data is not normal or symmetric. This interval may not be accurate. Bootstrap 95% confidence interval for mean is (0.08108, 0.08285). We are 95% confident that the true mean is within the interval. Since there is no assumption for this interval, the true mean is 95% in this interval. Wilcoxon Signed Rank confidence interval for mean and median is (0.08008, 0.0920). We are 95% confident that the true mean is within the interval. Since the assumption for this method is not met, the analyst expects the mean and median are not within the interval. Bootstrap 95% confidence interval for mean is (0.08070, 0.08260). We are 95% confident that the true median is within the interval. Since there is no assumption for this interval, the true median is 95% in this interval. E) Median is the better measure of center than mean because the data is skewed to the right and not normal. F) Bootstrap 95% confidence interval for median should be used as an interval for measure of center for the data because other methods do not meet their assumptions, and because median is better measure of center. Frank Wang November 16th, 2006 Appendix ————— 11/14/2006 2:14:47 PM ———————————————————— Welcome to Minitab, press F1 for help. MTB > Save "C:\Documents and Settings\Frank Wang\My Documents\School\University of Florida\STA3032\MINITAB.MPJ"; SUBC> Project; SUBC> Replace. ————— 11/14/2006 2:22:50 PM ———————————————————— Welcome to Minitab, press F1 for help. Retrieving project from file: 'C:\Documents and Settings\Frank Wang\My Documents\School\University of Florida\STA3032\MINITAB.MPJ' MTB > Mean C1. Mean of C1 Mean of C1 = 0.08395 MTB > Median C1. Median of C1 Median of C1 = 0.08165 MTB > StDev C1. Standard Deviation of C1 Standard deviation of C1 = 0.00696360 MTB > Describe C1; SUBC> Mean; SUBC> StDeviation; SUBC> QOne; SUBC> Median; SUBC> QThree; SUBC> IQRange. Descriptive Statistics: C1 Variable C1 Mean 0.08395 StDev 0.00696 MTB > PPlot C1; SUBC> Normal; SUBC> Symbol; SUBC> FitD; SUBC> Grid 2; SUBC> Grid 1; SUBC> MGrid 1. Probability Plot of C1 Q1 0.08070 Median 0.08165 Q3 0.08330 IQR 0.00260 Frank Wang November 16th, 2006 MTB > Histogram C1; SUBC> Bar. Histogram of C1 MTB > Onet C1. One-Sample T: C1 Variable C1 MTB > SUBC> MTB > MTB > SUBC> MTB > MTB > SUBC> MTB > N 10 Mean 0.083950 StDev 0.006964 SE Mean 0.002202 95% CI (0.078969, 0.088931) Random 700 C3-C12; Discrete C1 C2. RMean C3-C12 C13. Sort 'Mean' 'Mean'; By 'Mean'. RMedian C3-C12 'Median'. Sort C14 C14; By C14. WInterval 95.0 C1. Wilcoxon Signed Rank CI: C1 C1 N 10 Estimated Median 0.0818 Achieved Confidence 94.7 Confidence Interval Lower Upper 0.0808 0.0920 MTB > Save "C:\Documents and Settings\Frank Wang\My Documents\School\University of Florida\STA3032\MINITAB.MPJ"; SUBC> Project; SUBC> Replace. ————— 11/16/2006 9:31:12 PM ———————————————————— Welcome to Minitab, press F1 for help. Retrieving project from file: 'C:\Documents and Settings\Frank Wang\My Documents\School\University of Florida\STA3032\MINITAB.MPJ' MTB >