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Numerade Educator

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Problem 16 Medium Difficulty

a. Use a computer to draw 500 random samples, each of size $20,$ from the normal probability distribution with mean 80 and standard deviation 15.
b. Find the mean for each sample.
c. Construct a frequency histogram of the 500 sample means.
d. Describe the sampling distribution shown in the histogram in part $c,$ including the mean and standard deviation.

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Video Transcript

this question asks us to take 500 samples of size 20 from a normal distribution. Now, I'm going to use a computer to do this because it wasn't a real other way to do it. Um, and I'm using the programming language are, which is a statistical programming language is perfect for stuff like this. I'm gonna start with this four loop, which will take 500 samples from this normal distribution of using the function aren't arm to do this, which just essentially picks 20 random numbers from a normal distribution with a mean of 80 and a center deviation of 15. Like the question tells us to each of those samples, lists of 20 numbers will be stored in the list called Samp's, so I'll go ahead and run that and we'll see what that looks like. We now have a list of 500 and each of those has each element. In that list is a list of vector of 20 with, uh, elements from our normal distribution. No, the question asks us to find the mean of each of the samples that were given, So we're gonna end up with 500 sample means I'm gonna do that with this function, which will find them mean for each element in that list. Now we have a vector. Sample means that has length 500 and each element in that vector is a sample mean, let's go ahead and plot this and hissed a gram to see what it looks like. We can see pretty clearly that it's, uh, the unit motile. It has one peak and also seems to be pretty symmetrical. Uh, the tails on either side of the peak are about the same, seems to be centered around value 80 and has a spread from has probably 65 all the way up to 90. That's not the only description of our sampling distribution we can use. We can also talk about the mean of the sampling distribution and the standard deviation. So what's the mean of our sampling distribution? In other words, what's the grand mean? We can use our to figure that out if we look at the mean of sample means we get that our sample mean is 80.1. That's pretty close to what we said. We said it was sitting centered around 80 and so that's that's a pretty good estimate. Our grand mean is 80.1. What's your standard deviation percent? A deviation of this. A sampling distribution is about 3.28 Again, it will be expected with a range from 65 to 90. So this is how we can describe this sampling distribution of sample means from a random, uh, from a normal distribution.