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Lab 8 Suppose we have a box that contains tickets with values that range from 0 to 1. There are many, many tickets in this box and each ticket has its own unique value (no 2 tickets are identical). Each ticket has the same chance of being drawn from the box as any other ticket. This description describes the uniform distribution. 1. If you could compute the average of all of the tickets in the box, what would the average be? 2. Is the average of all of the tickets in the box a population parameter (population average) or a sample statistic (sample average)? Let’s create the probability histogram for the tickets in the box. 3. What is the name of that probability histogram? Type Net from http://www.ats.ucla.edu/stat/stata/ado Net install clt Clt, samples(50000) n(1) dist(u) [draw 1 ticket with replacement from the box of tickets which has a uniform distribution, 50000 times] 4. Describe the shape of the distribution. Does it look like a normal curve? Is the mean of that probability histogram the same value as the population average? If not exactly, why not? Jot down the standard deviation of the distribution. Let’s imagine that we are going to draw some tickets at random with replacement from this box and compute the average value of those tickets. 5. Is the average of the tickets in the sample a population parameter or a sample statistic? Now let’s imagine that we are going to repeat the process of drawing some tickets at random with replacement from this box an infinite number of times, compute the sample average each time, and create a probability histogram of all of those sample means. 6. What is the name of that probability histogram? We can approximate a theoretical sampling distribution of sample means from the box described above by taking many SRSs, like 50,000. And we can choose to repeatedly draw a few as 2 tickets at random with replacement, or we can choose to repeatedly draw many more tickets at random with replacement. Unfortunately, we are limited to repeating the process 50,000 times rather than an infinite number of times. Type clt, samples(50000) n(2) dist(u) normal [repeat the process 50,000 times, draw 2 tickets at random with replacement, compute the sample average each time, plot the histogram of the 50,000 sample averages] 7. Describe the shape of the distribution. Does it look like a normal curve? What is the mean of this distribution? Is it very close to the population mean? If it’s off a bit, why? Compute the standard deviation of this distribution using the information about the standard deviation from the population distribution. Do you get the same standard deviation as the computer program got? Type clt, samples(50000) n(20) dist(u) normal [repeat the process 50,000 times, draw 20 tickets at random with replacement, compute the sample average each time, plot the histogram of the 50,000 sample averages] 8. Describe the shape of the distribution. Does it look like a normal curve? What is the mean of this distribution? Is it very close to the population mean? If it’s off a bit, why? Compute the standard deviation of this distribution using the information about the standard deviation from the population distribution. Do you get the same standard deviation as the computer program got? Type clt, samples(50000) n(100) dist(u) normal [repeat the process 50,000 times, draw 100 tickets at random with replacement, compute the sample average each time, plot the histogram of the 50,000 sample averages] 9. Describe the shape of the distribution. Does it look like a normal curve? What is the mean of this distribution? Is it very close to the population mean? If it’s off a bit, why? Compute the standard deviation of this distribution using the information about the standard deviation from the population distribution. Do you get the same standard deviation as the computer program got? 10. At what sample size did the sampling distribution of sample means look like the normal curve? Now let’s repeat the whole process using a different population distribution. This time imagine that if you created a probability histogram of the tickets in the box, that the distribution would look like a normal curve with mean = 0 and standard deviation = 1. 11. If you could compute the average of all of the tickets in the box, what would the average be? Let’s create the probability histogram for the tickets in the box. 12. What is the name of that probability histogram? Type Clt, samples(50000) n(1) dist(n) normal [draw 1 ticket with replacement from the box of tickets which has a normal distribution, 50000 times] 13. Describe the shape of the distribution. Does it look like a normal curve? Is the mean of that probability histogram the same value as the population average? If not exactly, why not? Jot down the standard deviation. Now let’s imagine that we are going to repeat the process of drawing some tickets at random with replacement from this box an infinite number of times, compute the sample average each time, and create a probability histogram of all of those sample means. 14. What is the name of that probability histogram? Type clt, samples(50000) n(2) dist(n) normal [repeat the process 50,000 times, draw 2 tickets at random with replacement, compute the sample average each time, plot the histogram of the 50,000 sample averages] 15. Describe the shape of the distribution. Does it look like a normal curve? What is the mean of this distribution? Is it very close to the population mean? If it’s off a bit, why? Compute the standard deviation of this distribution using the information about the standard deviation from the population distribution. Do you get the same standard deviation as the computer program got? Type clt, samples(50000) n(20) dist(n) normal [repeat the process 50,000 times, draw 20 tickets at random with replacement, compute the sample average each time, plot the histogram of the 50,000 sample averages] 16. Describe the shape of the distribution. Does it look like a normal curve? What is the mean of this distribution? Is it very close to the population mean? If it’s off a bit, why? Compute the standard deviation of this distribution using the information about the standard deviation from the population distribution. Do you get the same standard deviation as the computer program got? Type clt, samples(50000) n(100) dist(n) normal [repeat the process 50,000 times, draw 100 tickets at random with replacement, compute the sample average each time, plot the histogram of the 50,000 sample averages] 17. Describe the shape of the distribution. Does it look like a normal curve? What is the mean of this distribution? Is it very close to the population mean? If it’s off a bit, why? Compute the standard deviation of this distribution using the information about the standard deviation from the population distribution. Do you get the same standard deviation as the computer program got? 18. At what sample size did the sampling distribution of sample means look like the normal curve? 19. Please describe: a) how this lab illustrates the central limit theorem b) the conclusion that you draw about the value of the population average and the average of the sampling distribution of sample means. What to turn in: all the numbered sections.