Reference no: EM132586767
Assignment - This assignment explores the sampling distribution of the sample mean by looking at the sampling distribution of the sample mean, and also sample sum.
You roll a die. If it comes up a 6, you win $20. If not, you get to roll again. If you get a 6 on the second roll, you win $10, if not you roll again. If you get a six on the third roll, you get a $5, if not you lose $10. Let the random variable X be the amount of money you win or lose by playing this game. It is clear that the distribution below is discrete, and only four possible values ever occur. The graph on the right depicts the information provided in the table. The mean and standard deviation are provided for you and you can verify on your own that these values are correct (No need to show. The standard deviation has been rounded to four decimal places).
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X ($)
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-10
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5
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10
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20
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P(X)
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0.5787
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0.1157
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0.1389
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0.1667
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1. State the Central Limit Theorem as best as you can in your own words. You are graded based on correctly mentioning the key components.
2. Create histograms for the 1000 sample averages, as well as the 1000 sample sums. Instead of manipulating the class widths, let the computer manipulate the class widths for both. Copy and paste the graphs on this page. What do you notice about the shape of both distributions?
3. Create normal probability plots (also known as QQplot, Quantile-Quantile plot) for the 1000 sample averages, as well as the 1000 sample sums. The easiest thing to do is either use GeoGebra or StatCrunch to create these graphs. There is no button on Excel that does this, you have to work harder to do this on Excel. Copy and paste the normal probability plots here. What do these plots indicate? Do the graphs support the Central Limit Theorem, explain?
4. The following questions are not about the simulation results (so don't put down results from your Excel sheet)
a. What is the value of the mean of the distribution you were sampling from?
b. What is the mean of the distribution of sample averages consisting of averaging 64 numbers (what theory says it should be)?
c. What is the reason for the relationship between the two numbers?
5. SIMULATION RESULTS FOR THE SAMPLE MEAN - The purpose of asking you to do the sample sum is to allow you the opportunity to understand that a sum is a type of summary. The second purpose is to allow you to see that the sample average is just the sample sum divided by n. That is why the distributions of both of these graph are identical, one is just a transformation of the other. From now on though we concentrate on the distribution of the sample (sample averages), as this is a more common summary.
A. What is the mean of all your 1000 sample means (not sums, the sample averages of 64 numbers) that you got in your simulation?
b. Does your result validate what theory says you should get?
Now I want you to look at your histogram and data of the sample means. Understand that in practice only one sample mean is generated to eventually estimate E(X). The next questions ask you to look at the extremes of what could occur, even if the sampling process is unbiased, as is the case here; the program is built to produce unbiased results. Yet, due to random variation during sampling you will have unusual values. Like a friend that happens to roll throw five dice and all land on six; rare but not impossible. This next three questions are just forcing you to consider what is possible.
c. Out of your 1000 sample means, write down a low value you generated for an average; not the lowest, though that is fine.
d. Out of your 1000 sample means give a high value you generated for an average.
e. Give a range that contains the most common sample averages out of your 1000 values (answer vary, this is your opinion).
6. COMPARING SIMULATION RESULTS OF THE STANDARD DEVIATION SD(X-BAR) = WITH THEORETICAL EXPECTATIONS.
a. What is the standard deviation of the population you are sampling from (Look at the table on page 1)?
b. According to theory (not your simulation results) what should be the standard deviation of the sampling distribution of the sample mean for the case where each sample mean consist of averaging 64 numbers?
b. What is the standard deviation of the 1000 sample averages you got in your simulation (simulation results)?
7. What did you learn from this simulation? Did anything surprise you?
Attachment:- Assignment & Data Files.rar