Why Divide by ? Let a population consist of the values 9 cigar...

Question

Why Divide by ? Let a population consist of the values 9 cigarettes, 10 cigarettes, and 20 cigarettes smoked in a day (based on data from the California Health Interview Survey). Assume that samples of two values are randomly selected with replacement from this population. (That is, a selected value is replaced before the second selection is made.)

b. After listing the nine different possible samples of two values selected with replacement, find the sample variance (which includes division by ) for each of them; then find the mean of the nine sample variances s2.

Accepted Answer

Welcome back, everyone. In this problem, a population includes the values 4 hours, 7 hours, and 15 hours of study time per week based on a student survey. Randomly select samples of 2 values with replacement from this population and list all possible peers. After we do that, we want to calculate the sample variance Squared for each sample using the formula that divides by N minus 1. Finally, we want to determine the average of all computed sample variances. A says the average of all computed sample variances is 18.67, B, 22.4, C 25.75, and D 30.12. Now, ultimately, what are we looking for here? But I think it's safe to say there are basically three parts to this problem. First, we want to find all of the possible peers or all of the possible ways that we can combine 47, and 15 into a pair, OK? After we do that, we want to find the sample variants for each of those samples, and then we want to determine the average of all computed sample variances. So we're finding the average of all of these sample variances that are given. So, the first place to start is to, let's just try to do a quick table, right? Let's try to compute or put together what we're really trying to find. So first we'll need our values, OK? Or rather, we'll need our sample, basically, OK? And that sample is going to contain all of our values, right? Next, we'll need to find the sample mean. That's pretty important because we'll need that based on what we know about the formula for sample variances because recall that the sample variance is equal to the sum of each data point minus the mean squared divided by N, the sample size, OK? So that's why we'll need the sample in X-bar. Next, once we have the sample mean, we'll need to find how far each of those values in the sample are away from the mean. In other words, the deviations. So we'll need the deviation of the first point X1 minus X bar, and we need to square it. And then we'll need X2 minus X bar also squared. And once we have both of those, then we can use that to calculate the population variance Squad. Now, what's pretty important about this here is that because we're talking about two values in our sample, for our sample variants, N will be equal to 2. So we could write that in our in blue here. So when N is equal to 2 for our sample variants. Then the sample variance is going to be equal to the sum of X I minus X bar or quad divided by 2. OK. So, we're gonna add these two values and divide by 2 to get our sample variant. So let's first start by trying to figure out our sample. That's our plan. Now, let's see if we can come up with a uh uh uh the actual values, OK, for our strategy. So now we're given 47, and 15. How many different ways can we combine that? Well, we could get 4 and 44 and 74 and 15, OK. We could get it as 7 and 477 and 715, because remember we're using our values without replacement. And then we could also get it as 154, OK, 15 and 7 and 1515. So these are all the possible ways we could combine 47 and 15 in a pair. Now that we have that we can find our sample means. Now, the mean, let, let's, you know what, let's actually do it, uh, line by line, OK? So just to get a hang of what exactly we're trying to find. So for 44, the sample mean is going to be 4 + 4 divided by 2, which is 4, OK? For the deviation, since the first point is 44 minus 4 is 0 and 02 is 0, while our second point X2 is 44 minus 4 is 0, and 0 quad is 0. So that means our population variance is going to be 0 + 0 divided by 2, which equals 0 as well, OK? So, these are the values for our first sample. We're basically going to repeat that idea for the second, OK? So, you know what? I should actually just write that out here. So, we had 0 minus 02, which equals 0, the same thing here. 0 minus 0 squared which equals 0. You know, for our variance, it's going to be um the sum of both of these 0 + 0 divided by 2, which equals 0. So we're following the same idea for the rest. No, for a sample 4 and 74 + 7 divided by 2 is 5.5, 5.5 minus 4 squared is 2.25, and 5.5, 5.5 minus 7 squared is going to be 6.25. Now, when we add both of those and divide by 2, then we get the population variance to be 4.25. For 4 and 15, the mean is 9.5. The first uh deviation is 30.25. OK. The second is also 30.25 and the variance of those two is 30.25. For 7 and 4, OK, it's 6. Well, the mean is actually 5.5, OK. Of deviations are 6.25 and 2.25, OK. So our population variance is 4.25. For 7 and 7, OK. The mean is 7, and our deviations are both 0 which make our variant 0. For 7 and 15, uh, we have 1116, 16, and 16, respectively. For 15 and 4, 9.5, uh 30.25 for both. Which means 30.25 is also our variant. Uh, 15 and 7, it's 1116 and 16, which means our variance is 16. And finally, for 15 and 15, the mean is 15, which means that our deviations are 0 and 0, thus our variance is 0 as well. OK? So now, what have we done? We've found the sample variances for each. Uh, or rather we found the variants for each sample. Now that we have the variance for each sample, we can compute the mean of the population variances because all we're going to do is to find the sum of the variances and then divide them by the number of samples. Notice we had 9 samples here, so thus. What this means then is that our population variant sigma squared will be equal to the sum, OK, of sigma squared divided by the number of samples N. So that's going to be the sum of all our values here which we we would get to be. 101, OK. And 101 divided by 9 is equal to 11.22 to 2 decimal places. Therefore, the meaning of the population variances is 11.22. D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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Key Concepts

Sample Variance

Random Sampling with Replacement

Mean of Sample Variances

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