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SMAM 319 Computer Assignment 3 Use Minitab to do this problem that demonstrates The Central Limit Theorem Do the computations in part A and answer the questions on page 2 stapled to the front of the computer output that should follow. A.This part is to be done by hand. You will use the probability mass(distribution) below with n is the number of letters in your last name. 1 , x = 1,2,3,…, n n For example if your last name is Doe your probability mass function is f (x) = 1 1 1 f (1) = ,f (2) = ,f (3) = 3 3 3 For your value of the number n find the mean and the standard deviation of the distribution. For your value of n you should get an answer with numerical value. n +1 n2 − 1 E( x = sd(X) = 2 12 Thus, for John Doe () EX=2 sd(X)= 2 / 3 For me n = 6. 1 1 1 1 1 1 21 7 EX = + 2 ⋅ + 3⋅ + 4 ⋅ + 5⋅ + 6 ⋅ = = 6 6 6 6 6 6 6 2 1 1 1 1 1 1 91 EX = + 22 ⋅ + 32 ⋅ + 42 ⋅ + 52 ⋅ + 62 ⋅ = 6 6 6 6 6 6 6 91 49 35 σ2 = − = 6 4 12 σ = 1.708 You will now simulate your distribution 100 times. calc>randomdata>integer In dialog box Minimum value 1 Maximum value 6 Generate 100 rows of data Store in columns c1-c100 OK Enable command language enter the commands rmeans c1-c100 c101 Make a stem and leaf display for c101 using the pull down menu or the command stem-and-leaf c101 describe c101 Welcome to Minitab, press F1 for help. MTB > rmeans c1-c100 c101 MTB > stem-and-leaf c101 Stem-and-Leaf Display: C101 Stem-and-leaf of C101 N = 100 Leaf Unit = 0.010 1 1 2 10 25 45 (24) 31 15 6 1 29 30 31 32 33 34 35 36 37 38 39 0 6 23356799 011125567788889 00000123345666788888 000111111223344556677777 0000134455556899 022334578 03469 5 MTB > describe c101 Descriptive Statistics: C101 Variable Q3 C101 3.6375 Variable C101 N N* Mean SE Mean StDev Minimum Q1 Median 100 0 3.5105 0.0176 0.1761 2.9000 3.3925 3.5100 Maximum 3.9500 Make a normal probability plot c101. Answer the following questions on the sheet stapled to the front of your computer output. 1. Based on the stem and leaf display and the normal probability plot for c101 does the data appear to be normally distributed? Explain your answer. The data appears to be normally distributed because the stem and leaf display is bell shaped and the normal probability plot is close to a straight line with the possible exception of one point. 2. What is the mean and the standard deviation obtained in the describe command for c101? Mean = 3.5105 Sd =.1761 3. What should the mean and standard deviation be in theory for c 101?[Hint: the mean is (n+1)/2 for your value of n .The standard n 2 −1 /10 12n The mean should be 3.5. The standard deviation should be .1708 4. Compare the mean and standard Deviation in questions 2 and 3 by finding the percentage error? deviation is %error = value in describe command - theory value x100 theory value 3.5105 − 3.5 x100 = .3% 3.5 Standard deviation Mean %error = %error = .1761− .1708 x100 = 3.1% .1708 5. State the Central Limit Theorem carefully and explain how the results you obtain in c101 validate it for your problem. The central limit theorem states that as n gets large the sample mean tends to a normal distribution with the population mean and the population sd divided by the square root of the number of observations. For this data when n=100 the stem and leaf is bellshaped indicating normality. Also the mean and standard deviations are reasonably close to the values of the central limit theorem. The solution will be different for students who work independently but the conclusions should be similar. If you get a large percentage error for your mean and standard deviation check your calculation of the theoretical mean and standard deviation. Chances are pretty good that you made a mistake.