Develop a sample of size n = 8 such that x̄ = 15 and s = 0.

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Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says which of the following data sets meets these criteria. And we're getting the criteria of N is equal to 5, X is equal to 12, and S is approximately equal to 3. And we're going 4 possible choices as our answers. For choice A, we have 9, 1112, 13, and 15. For choice B, we have 1012, 1212, and 14. For choice C, we have 8, 1012, 14, and 16. And for choice D, we have 1112, 1212, and 13. So, we're asked to determine which of our data sets that we're given as our answer choices meet the criteria given to the problem. So, the first criteria is that N is equal to 5. So that means the number of data points in our data set is going to be equal to 5. And if we look at all of our answer choices, they all have 5 data points in them. So that means that N equal to 5 criteria is met for all 4 options. So that means we're gonna have to rely on the other two criteria, our sample mean and standard deviation. So, recall for your formula for the sample mean, which is going to be X bar, that that is going to be equal to the sum. Of X of I divided by N. Where X of I here is an individual data point in our data set, and N is the total number of data points, which in this case is equal to 5. Also recall your formula for the sample standard deviation, which is going to be S, and that is going to be equal to the square root of the quantity of the sum of the quantity of X of i minus X bar in quantity squared, and then it's going to be divided by the quantity of N minus 1. So, the first thing that we'll need to do is to calculate our sample mean for each of our data sets. So, we'll do this for each of ours. So, starting with option A. We'll have that x bar. It's going to be equal to and in the numerator here I'll just sum up all of our data points. So we'll have the quantity of 9 + 11 + 12 + 13 + 15 in quantity, and that is going to be divided by the number of data points that we have, so in which is equal to 5. And when we evaluate this expression, we see that X bar is going to be equal to 12. Now, we'll do the same thing for option B. And so we'll have that X bar is equal to the quantity of 10 + 12 + 12. Plus 12 + 14 in quantity, and that is divided by 5. And again, when we evaluate this expression, this is going to be equal to 12. If you look at option C, We'll have that X bar is going to be equal to the quantity of 8 plus 10 + 12 + 14 + 16 in quantity, divided by 5. And again evaluating this expression, we see that this is going to be equal to 12. And looking at option D, we'll have that X bar is going to be equal to the quantity of 11 plus 12 plus 12 + 12 + 13 in quantity divided by. 5. And again, when we evaluate this expression, this is going to be equal to 12. So, all four of our options have a sample mean of 12. So, that means that we're going to have to calculate the standard deviation for all of our options. So looking at option A. We'll calculate the sample standard deviation, so we'll have that S is going to be equal to the square root of the quantity and the numerator will have the quantity of 9 minus 12 in quantity squared, plus the quantity of 11 minus 12 in quantity squared, plus the quantity of 12 minus 12 in quantity squared. Plus the quantity of 13 minus 12 in quantity squared. Plus the quantity of 15 minus 12 in quantity squared. And we're gonna divide all of that by the quantity of N minus 1, which in this case is going to be 5 minus 1. And evaluating this expression, this is going to be approximately equal to. 2.2. For option B, we'll have that our sample standard deviation S is going to be equal to the square root. Of the quantity and the numerator will have the quantity of 10 minus 12 in quantity squared. Plus the quantity of 12 minus 12 in quantity squad, plus the quantity of 12 minus 12 in quantity squared plus the quantity of 12 minus 12 in quantity squared. Plus the quantity of 14 minus 12 in quantity squared. And then all of that is going to be divided by. The quantity of 5 minus 1. And evaluating this expression, this is going to be approximately equal to. 1.4. So calculating our sample standard deviation for option C, we'll have that S is going to be equal to the square root. Of the quantity, and in the numerator we'll have the quantity of 8 minus 12 in quantity squared, plus the quantity of 10 minus 12 in quantity squared, plus the quantity of 12 minus 12 in quantity squared, plus the quantity of 14 minus 12 in quantity squared, plus the quantity of 16 minus 12 in quantity squared. And all that is going to be divided by. The quantity of 5 minus 1. And evaluating this expression, this is going to be approximately equal to. 3.2. And finally, for option D, we'll have that S is going to be equal to. The square root of the quantity and the number we have the quantity of 11 minus 12 in quantity squared plus the quantity of 12 minus 12 in quantity squared plus the quantity of 12 minus 12 in quantity squared plus the quantity of 12 minus 12 in quantity squared plus the quantity of 13 minus 12 in quantity squared. And again, all that is going to be divided by the quantity of 5 minus 1. And evaluating this expression, this is going to be approximately equal to 0.7. So, when we round our sample standard deviations to the nearest whole number, we see that option C rounds to a value of approximately 3. And it also has a. Mean our sample mean of 12. So that means that uh answer choice C is going to be the correct answer for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Develop a sample of size n = 8 such that x̄ = 15 and s = 0.

Key Concepts

Sample Mean (x̄)

Sample Standard Deviation (s)

Constructing a Sample with Given Statistics

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