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Suppose you were to generate several random samples, all the same size, all from the same normal probability distribution. Will they all be the same? How will they differ? By how much will they differ?

a. Use a computer or calculator to generate 10 different samples, all of size $100,$ all from the normal probability distribution of mean 200 and standard deviation 25.

b. Draw histograms of all 10 samples using the same class boundaries.

c. Calculate several descriptive statistics for all 10 samples, separately.

d. Comment on the similarities and the differences you see.

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{'transcript': "this question wants us toe make 200 samples of size 10 from the normal distribution. Now, you can see that I'm not on the white board. In fact, I'm using a programming language called R. R. Is a statistical programming language that's perfect for stuff like this creating a bunch of samples of random variables. Um, So what we're gonna do is I am using a for loop to make 200 samples. The way I'm gonna sample from the normal distribution is I'm gonna use this function are norm which create which will create random numbers from the normal distribution that would give it It will create 10 random numbers of vector of 10 random, normal numbers and it will take it from the normal distribution with a mean it 100 a standard deviation of 20. Just like the question asked us to do. So let's go ahead and run this and it will quickly create 200 samples each of length 10 weaken. Take a look at this and it will see that we have here a list of 200 samples and each element in this list is a sample of like 10 so now, it also the question asked us to find the mean of each sample. To do this, I'm gonna use this function that will find the mean of every element in our list of samples. So we'll do that. And now we have a vector of length 200 that has sample means for each, uh, interest sample in our list of samples. Finally, it asks us to plot all of these sample means in a hist a gram. So using the function hissed will go ahead and do that, and we can see that this is the resulting distribution. So let's go ahead and describe this sampling. Distribution of sample means we can see pretty clearly that it's centered around probably around 100 or somewhere a little bit less than 100. It's clearly, you know, motile it only has one peak, and it seems to be pretty symmetrical. It might be skewed a little bit toothy left that is ahead a little bit longer tail on the left than does on the right. But we don't really have enough evidence for that. It looks pretty symmetrical, so we can say its image. This, uh, distribution also has a range from what seems like 80 to 1 15 So we'll say that it's uniforms symmetrical, centered around 100 with a range of 80 to 1 15 And that is the description of our sampling distribution of sample means."}

University of Oklahoma

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