Constructing Data Sets In Exercises 25–28, co...

Question

Constructing Data Sets In Exercises 25–28, construct a data set that has the given statistics.

N = 6

μ = 5

σ ≈ 2

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says to select the data set of N equal to 6 numbers, that has a sample mean X equal to 7, and a sample standard deviation of S approximately equal to 2.4. And we give 4 possible choices as our answers. For choice A, we have 45. 689, and 10. For choice B, we have 35779, and 11. For choice C, we have 2468, 10, and 12. And for choice D, we have 56778, and 9. So, we need to calculate the sample mean and sample standard deviation of all of our answer choices here, and determine which one has the values given in the problem. So, recall to calculate your sample mean, X bar, that that is going to be equal to the sum of X of I. Divided by N. Where X by here is an individual data point and and here is the total number of data points we have in our data sets, which in this case is equal to 6. The other quantity that we need to calculate is the sample standard deviation S, and so we called for that formula that 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 in quantity squared divided by the quantity of n minus 1. So, we're going to start off by calculating the mean for each of our sample data sets. So, starting off with option A. We'll have that X bar is going to be equal to, and here in the numerator, I just need to add up all my data points. So we'll have the quantity of 4 + 5 + 6 + 8 + 9 + 10 in quantity, and then it's going to be divided by N, which is 6. And when we evaluate this expression, we see that X bar is going to be equal to 7. We'll do the same thing for option B. We'll have that X bar is equal to, and here in the numer we'll have the quantity of 3 + 5 + 7 + 7 + 9 + 11 in quantity, and that's going to be divided by 6. And when we evaluate this expression, we again get that X bar is equal to 7. Looking at option C, we will have that X bar is equal to the quantity of 2 + 4 + 6 + 8 + 10 + 12 in quantity, and that's going to be divided by 6. And again, when we evaluate this expression, we get that this is equal to 7. And then finally for option D. We'll have that X bar is equal to the quantity of 5 + 6 + 7 + 7 + 8 + 9 in quantity, and that is all divided by 6. And again, evaluating this expression, we'll see that X bar is equal to 7. So, the sample mean is equal to 7 for all 4. Answer choices. So that means we'll need to calculate the sample standard deviation also for all of our options. So, starting with option A. We're going to have that S it's going to be equal to you, the square root. Of the quantity, and here we'll have in the numerator the quantity of 4 Minus 7 in quantity squad, plus. 5 the quantity of 5 minus 7 in quantity squared plus the quantity of 6 minus 7 in quantity squared plus the quantity of 8 minus 7 in quantity squared plus the quantity of 9 minus 7 in quantity squared plus the quantity of 10 minus 7 in quantity squared. And that whole quantity is going to be divided by. The quantity of N minus 1, which is going to be 6 minus 1. And when we evaluate this expression, this is going to be approximately equal to. 2.4. So, that matches up with what we're asked to find a problem. Let's see if one of the other options also has the same sample standard deviation. So, we'll look at option B. And so we'll see that S is going to be equal to the square root. Of the quantity of the quantity of 3 minus 7 in quantity squared, plus the quantity of 5 minus 7 in quantity squared. Plus the quantity of 7 minus 7 in quantity squared plus the quantity of 7 minus 7 in quantity squared plus the quantity of 9 minus 7 in quantity squared plus the quantity of 11 minus 7 in quantity squared. And then that whole quantity is going to be divided by. The quantity of 6 minus 1. And evaluating this expression, this is going to be approximately equal to. 2.8. So, doing the same for option C, we'll have that S is going to be equal to the square root. Of the quantity and here the numerator will have the quantity of 2 minus 7 in quantity squared plus the quantity of 4 minus 7 in quantity squared plus the quantity of 6 minus 7 in quantity squared plus the quantity of 8 minus 7 in quantity squared. Plus the quantity of 10 minus 7 in quantity squared plus the quantity of 12 minus 7 in quantity squared. And then all of that is going to be divided by. The quantity of 6 minus 1 again. And evaluating this expression, this is going to be approximately equal to. 3.7. Then finally for option D. We'll have that S is equal to the square root. Of the quantity and the numerator will have the quantity of 5 minus 7 in quantity squared plus the quantity of 6 minus 7 in quantity squared plus the quantity of 7 minus 7 in quantity squared plus the quantity of 7 minus 7 in quantity squared plus the quantity of 8 minus 7 in quantity squared plus the quantity of 9 minus 7 in quantity squared. And then all that is going to be divided by the quantity of 6 minus 1. And evaluating this expression, this is gonna be approximately equal to. 1.4. So now if we compare our values to those values that we're looking for, we see that only option A has a mean of 7 and a standard deviation of 2.4. So that means that option A, is going to be the correct answer choice for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Constructing Data Sets In Exercises 25–28, construct a data set that has the given statistics.

N = 6
μ = 5
σ ≈ 2

Key Concepts

Population Size (N)

Mean (μ)

Standard Deviation (σ)

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