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.)

a. Find the variance of the population {9 cigarettes, 10 cigarettes, 20 cigarettes}.

c. For each of the nine different possible samples of two values selected with replacement, find the variance by treating each sample as if it is a population (using the formula for population variance, which includes division by n); then find the mean of those nine population variances.

Accepted Answer

Welcome back, everyone. In this problem, our population consists of 4 hours, 7 hours, and 15 hours of study time per week based on a student survey. Select samples of 2 values with replacement from this population and list all possible peers. For each of the nine possible samples, calculate the variance by considering each sample as an entire population and applying the population variance formula, which includes division by N. Finally, compute the average of these nine population variances. A says the average is 32.25, B 25.75, C19.54, and the D 11.22. Now, what are we trying to find here? Well, we could split it up in basically 3 things. We want to list all possible peers from our population, OK? Then we want to calculate the variance for each peer for each sample by considering it as an entire population to the population variance formula, and then compute the average of these 9 population variances. So before we do that, we need to establish what is our population variance. Well I recall That the formula for the sample variants, OK, is equal to the sum of the deviations for each data point squared, OK, divided by our sample size N, where XI represents the sample values and X bar is the sample mean. Now, if we're going to look at samples of two values, OK, OK, then for our samples. Of two values that is where n equals 2 then. Our population variance is going to be equal to the sum of deviations squared. Divided by 2. So, what we'll need to do is, first, we'll need to find the sample, we'll need to find our samples, OK? Then we'll need to find the mean of each sample, the deviation for each point from the mean in the sample, and use that to find the population variance using this formula where N equals 2. OK? So let's go ahead and do that. Let's make a table, right? So now we're gonna have our samples. And then we're gonna have our sample means, which is X bar. And then we're gonna have our deviations, that is the first value minus the mean squared, and then the second value X2 minus the sample mean squared, OK? And then we want to use that to find our population variance, OK? So let's first start by thinking. Remember we were told, let me just complete draw the table here. We were told that we have 4 hours, 7 hours and 15 hours. OK, that's those are the, the values in our or the values that we were given. So no, from that, if we're going to list all the possible peers, how could we combine all of those values with our replacement? Well, starting with 41 combination could be 44, another could be 47, and another could be 415. For if we start with 7, then a combination could be 74, 77 and 715. And then if we start with 15, the combination could be 154, 15 7. And 1515. So if we think about it, these are all the possible ways. OK, these are all of the possible ways that we could combine our data points. Let me just make our table a bit longer here. So now that we have the values in our sample, we can go ahead and find the sample mean and then our deviations. Now for our first sample 4 and 4, the mean here would be 4 + 4 divided by 2, which equals 4. For our deviation for the first point, it would be 44 X1 minus or mean 4 squared, and that's 0 squad which equals 0. For the second one, it would also be 4 minus 4 squad, which equals 0. I know that we have these, we know that to find our variances we would sum both of these values and divide by 2. So this would be 0 + 0 divided by 2, which would be equal to 0, OK? So these are, or this is the population variance for our sample 44. Essentially, we're just gonna repeat this for the rest of values. So for 47, the sample mean is 5.5, 4 minus 5.5 equals -1.5, and when we square that, we get 2.25. 7 minus 5.5 is. Uh, 2.5, and when we square that we get 6.25. And then 2.25 plus 6.25 divided by 2 would be 4.25. Next, for 4:15, the mean is 9.5. The deviations are 30, the deviation squared are 30.25 and 30.25. So our population variance is also gonna be 30.25, OK? For 7 and 4, OK. The mean is 5.5, you gonna have 6.25 and 2.25 for our square deviations, and our variance 4.25. For 7 and 7, it's gonna be 700 and 0.7 and 15, we're gonna have 1116, 16 and 16. 15 and 4. It's gonna be 9.5, 30.25, 30.25, and 30.25, OK, and, and you'd be working this out on your own. For 15 and 7 it's gonna be 1116, 16 and 16. Finally for 1515, we have 15, 00 and 0. So now these values are all of our population variances. So we've solved that part of the problem in finding our population variances. All we need to do now. It is to calculate the mean of the population variances, and we know that the mean of the population variances, OK. Is gonna be equal to the sum of all our population variances divided by the number of samples. Now we know that our sum here, OK. If we add all of these values, OK. We would get 101 and we divide that by 9, OK, to give us 11.22 to 2 decimal places. Thus, the mean of the population variances is 11.22. If we look back at our answer choices, that means D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Population Variance

Sampling with Replacement

Mean of Variances

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