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31 JANUARY 2005 MATH 311 LAB 3 – SAMPLING DISTRIBUTIONS Let’s imagine that you rolled a 34641-sided (trust me) die and let’s let X = the number of dots showing on the top face. Now let’s imagine that you rolled this die 5,000 times (note: this number is not important for the following. The bigger we can make it, however, the clearer the pattern we’re looking for will be. So if your computer can handle 10,000 or 100,000 – go for it!). Question 1: What do you think the mean value of X would be? Okay, enough imagining, let’s make Minitab do some work: Generate at least 5,000 rows of data with a lower endpoint of 1 and an upper endpoint of 34641 by following Calc>Random Data>Integer. Question 2: Make a Histogram and compute the mean and standard deviation of the data. Include these in your report. Are the data normally distributed? Test it. Okay, great. Now let’s imagine that instead of rolling the die just once you rolled it 4 times and let Y = the mean of the four rolls. Further, let’s imagine that you computed this value of Y 5,000 times (5,000 sets of 4 rolls). Question 3: What do you think the mean value of this new random variable Y is? Okay – so let’s make Minitab simulate this situation. Generate 5,000 rows and 4 columns (C1-C4) of Integer data with a lower endpoint of 1 and an upper endpoint of 34641. Each row represents one set of rolls. We can compute the mean of these 4 rolls as follows: Title column C5 “Mean. – 4 rolls” Select Calc>Row Statistics Click Mean In the Input variables: box enter C1-C4 In the Store result in: box enter C5 Select OK The mean of each of the 5,000 rows should appear in C5. Question 4: Make a Histogram and compute the mean and standard deviation of these data in C4 (the values of Y). Include these in your report. Are the data normally distributed? Alrighty, now imagine that you rolled the die 100 times (whew!) and let Z = the mean of the 100 rolls. And then you computed the value of Z 5,000 times. Tired yet? Well, make Minitab compute 5000 rows and 64 columns (C1-C100) of Integer data with a lower endpoint of 1 and an upper endpoint of 34641. Compute the mean of each row and store it in C101 (you should un-title column C5 and title C101 as “Mean – 100 rolls.”) Question 4: Make a Histogram and compute the mean and standard deviation of these data in C100 (the values of Z). Include these in your report. Are the data normally distributed? Have you noticed any patterns that the mean and standard deviations seem to be following for various sample sizes? Look closely at the numbers and, perhaps, round to the “tens-place.” Still don’t see anything? Try simulating 202 = 400 rolls. Then, maybe 332 = 1089 times (you might have to reduce the number of rows used – it depends on how much memory the computer you’re using has). Ready to generalize? Good, answer the last question: Question 5: Lastly, imagine that you rolled the die n times and let W = mean of the n rolls. Make a conjecture about the mean, standard deviation, and distribution of W.