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Stat 100 MINITAB Project 4 Fall 2007 (With amendments for Version 14) Purpose: I. To use MINITAB to conduct a simulation experiment to illustrate the Central Limit Theorem. II. To use MINITAB to conduct simulation experiments to illustrate the concept of a confidence interval. Reading: Text, section 7.3 on the Central Limit Theorem and 8.3 on Confidence Intervals. Turn in: I. Session window and two histograms of the distributions of the sample means; II, a print out of the 90%, 95%, and 99% confidence intervals for the normal distribution, a print out of the 90%, 95%, and 99% confidence intervals for the uniform distribution, and the answers to the questions at the end of this assignment. Instructions: What follow are the MINITAB commands for producing histograms and basic statistics from randomly generated data. Words in capital letters followed by the symbol > indicate a sequence of menu items to be selected/clicked. I. The Central Limit Theorem. Begin by producing 20 random samples of size n = 6 from a uniform population on the interval [0,1]. The distribution has mean 0.5 and standard deviation 0.2887 (to 4 decimal places). CALC>RANDOM DATA>UNIFORM then type 20 for “rows of data” and store them in columns C1-C6, then OK. CALC>ROW STATISTICS select mean, input variables: C1-C6 and store in C7. Note, C7 contains x for each sample. STAT>BASIC STATISTICS>DISPLAY DESCRIPTIVE STATISTICS type C7 in variables. To create a histogram of the sample means in C7: EDIT>EDIT LAST DIALOG>GRAPHS select histogram of data with normal curve, OK, OK. Print the graph and session window. Now we repeat the procedure to produce 20 samples of size n = 100. Delete the data in the columns as follows: MANIP>DELETE ROWS type 1:20 in the “delete rows” from columns C1-C7. Now produce the samples by CALC>RANDOM DATA>UNIFORM 20 rows in columns C1-C100 CALC>ROW STATISTICS select mean, input variables C1-C100, store in C101, OK. STAT>BASIC STATISTICS>DISPLAY DESCRIPTIVE STATISTICS type C101 in variables, OK. II. Confidence Intervals. A. We simulate random sampling from a hypothetical normal population with 50 and 10 . Create 20 random samples of size 50: CALC>RANDOM DATA>NORMAL, 50 rows, store in C1-C20, set 50, 10 . To calculate 95% confidence intervals for the 20 samples: STAT>BASIC STATISTICS>1-SAMPLE Z, Sigma = 10. MINITAB will display in the session window a table comprising basic statistics and confidence intervals for the 20 samples (columns). For Version 14, the commands are STAT>BASIC STATISTICS>1-SAMPLE Z, Sigma = 10, test mean = 50, enter C1-C20 in “Samples in columns”. To observe the effect a change in confidence level has on the intervals, go to EDIT>EDIT LAST DIALOG, change 95% to 90%, OK. For Version 14, EDIT>EDIT LAST DIALOG>OPTIONS, change 95% to 90%, OK. MINITAB will display the 90% intervals below the 95% ones. Now repeat the above commands to produce 99% confidence intervals for the 20 samples. B. We repeat the simulation experiment replacing the normal distribution with the uniform distribution on [0,1]. This distribution has .5, .289 . CALC>RANDOM DATA>UNIFORM, 50 rows, store in C1-C20. STAT>BASIC STATISTICS>1-SAMPLE Z, Confidence Interval Level = 95%, Sigma = .289. For Version 14, STAT>BASIC STATISTICS>1-SAMPLE Z>OPTIONS, Confidence Interval Level = 95%, Sigma = .289. Finally, produce confidence intervals of 90% and 99% for the 20 samples using EDIT>EDIT LAST DIALOG and changing the confidence level appropriately. Questions: Answer precisely and concisely the following. 1. For part I, what are the mean and standard deviation for C7? For C101? Do they conform to the predictions of the Central Limit Theorem? What are the shapes of the histograms? Are they in agreement with the Central Limit Theorem? 2. For part II A. a) Determine the number of 95% confidence intervals that contain the population mean. Approximately how many would be expected to contain the population mean? b) Answer the above for the 99% confidence intervals. c) Answer the above for the 90% confidence intervals. 3. Answer question 2 for part II B. 4. What effect does increasing the confidence level have on the width of the intervals?