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Variance of the sample 1, 3,4,5 is:Select one: 4.3333b. 2.916674.916675.12452...

Question

Variance of the sample 1, 3,4,5 is:Select one: 4.3333b. 2.916674.916675.12452

Variance of the sample 1, 3,4,5 is: Select one: 4.3333 b. 2.91667 4.91667 5.12452



Answers

The given values represent data for a sample. Find the variance and the standard deviation based on this sample.
12.1, 33.3, 45.5, 60.1, 94.2, 22.2

Part a tasks us with finding the sample variance of our data set using formula 2.5 in the textbook. So heart a the sample variance is going to equal. So using equation 2.5, the sum of squares over and minus one. So in this case, we're gonna subtract the mean of our values from each one of our date of each one of our values and squared. And then and then together, that's how we're gonna find R sum of squares and an equation two point. But now there are five step mortifying sample variance in this way. Step one. Let's add all the values together that would be seven plus six plus all the way plus 13 all the way down sweet at all our values. Together we get 70 to important In step two, we must find the means. So that's all the values that together divided by the number of values in our data set 10 that's going to equal 7.2 in step three. We're going to subtract the mean from each one of our values in the data set. So we're getting a seven minus 7.2 is negative point to six minus 7.2 is negative. 1.2 and so on. Five minus 7.2 Negative. 2.2 and 13. Minus 7.2 is five point. So you get all these values. Step four, we're going to get our summer squares are numerator. You're gonna take each value that we got in step three, square it and then add it to the next Value square and so on. So that would be negative. 0.2 squared plus negative 1.2 squared all the way to 5.8 squared, and then we square all of them and add them. Together. We get 70 three point six now, in step five, we're gonna put that all together here. I'll put step five over here. I'm gonna put that all together. So our sample standard a sample variance is going to equal 73.6 over the number of values in our data set subtracted by warned nine, which is approximately equal to eight. Coin 18 And this is our sample variance. Now for part B, we want to find sample variance using equation 2.9. So say the sample variance is equal to. So in this case, the equation will be the sum of each one squared minus the summation. Swear over and a your and minus flung. Now to solve it. In this way, there are four steps. Step one, just like before. We want to add all the values together. So we did that. That would be 72 in step two. We want to square. We want to some each square 77 squared plus six squared all the way to five squared plus 13 square square mall. Add them and you get 500 and 92. Step three. Yeah, we want to know. But we want to get the numerator for our variance. You're gonna get R sum of squares by. Look at our numerator. We want to replace the sum of X squared with 5 92 minus right place to some of X in our numerator with 72 square over 10 and this will get us 73 point sex. Our final step is to put this all together or sample of areas is going to equal r sum of squares which in this case is Step three which is 73.6 over n minus one, which is not which is going to be approximately e 0.18 now apart. See, we need to find standard deviation. Part C. What? Let's see. Let's racket this off here so we don't get confused. Part C. Standard deviation. So that's the sample. Saturday. Aviation is equal to the square root of ERM Areas in this case are variants is 8.18 So we take the square root of the 0.1 eat, and that is approximately 2.86 and that is our standard deviation.

In order to find the range of the data said, given we must subtract the lowest value from the highest value in our set. So a range is equal to or highest value nine minus our lowest value to which equals seven. So the range of our data said, given its seven in part B, we must use formula 2.5 in order to solve for sample variance. So that means our sample variance is equal to R sum of squares. In this case is going to be our the mean subtracted from each one of our data values in our set square A her and minus one. Now there are five steps in order to solve for sample variance using equation 2.5, Step one, some all the values together. So that's two plus four plus seven plus eat plus nine, which equals three Step two. We calculate the me of the data set, so that would be 30/5 30 divided by five, which is sex in step three. We want to subtract the mean from each one of our data values So that B two minus six, which is minus four four minus six which is minus two seven minus six, which is one eight minus six, which is two and nine minus six. Interest story. They remember all of these values, when added together should equal zero negative for minus two is minus 63 plus two plus 166 minus 60 So we're all good there. Known Step four. We wanna calculate R sum of square. So we want to square each one of the values we got in step three and then add them together. So that would be 16 plus four plus one plus four plus nine, which equals 34. Now, in step five, the final step. We put this all together to get our sample variance. So our sample variance is equal to their sum of squares, which we got in Step four. Just 34 over and minus one will satisfy the number values we have our data set minus one, which is four. My mother's 14 So 34 divided by four is equal to 8.5 and this is our sample varies now part C will put up here. We want to calculate for the standard deviation of standard deviation is equal to the square root of our sample variance, so that's equal to the square root of 8.5, which is approximately to 0.9 to.

In order to find the range of the given data set, we must subtract the lowest value from the highest. So here are range is going to equal E, which is our highest value minus three, which is our lowest value. So eight minus three equals five. This is gonna be our range for the data. Second, now Part B were asked to find sample of Arians using Equation 2.5 in her book. Now, Equation 2.5 states of the sample variance is equal to the sum of squares, which in this case, is going to be the sum of X minus X r squared. So we're gonna take the mean and subtracted from each one of our data values in our set square it and then added together, it'll ones here n minus one and they're five steps in order. Self sample variance is well, step one. We must find all we must some all the values in our data set together so that six plus eight plus seven plus five was three plus seven, which equals 36. Step two. We want to find the mean over a data set, so that would be our some, which is 36 divided by the number of values in our set, which is six. The people sex. So army is sex. Step three. We want to subtract. We want to subtract the mean from each one of our data values. So that would be six minus six, which is zero eight minus six, which is to seven minus six, which is 15 minus six, which is negative. One three minus six, which is negative. Three at seven minus six, which is more now instead, four we want to square and add the is we want to get r sum of squares so and stuff four We're gonna take each one of the values square it and added together so that zero squared 02 squared which is 41 squared which is one negative one squared which is one negative three squared which is nine and one squared which is one which is equal to 16 Now step five. We put this all together to solve for our sample variance or several areas s squared is equal to r sum of squares which we got in stuff for 16 over end minus one now and it's six. So this should be divided by by which is equal to three point to now apart. See, we want to find the sample standard deviation of sample. Standard deviation is equal to the square root of our variants of our sample variance. So that means our sample standard deviation is equal to the square root of 3.2, which is approximately one point 79 and that's our sample standard deviation.

For this problem. We're talking about a population standard deviation. Uh, so we know where populations standard deviation. Um, most of 25 units. It's important to recognize out of the population standard deviations. Might the word deviation for that here. So what happens? It says, what is the standard error of the mean if the sample samples of size 16 are selected Well, so we know we have a stand us standard deviation or standard error. And if we have a larger sample size is gonna make this end error go down. So we're gonna play this Formula seven, apply the formula, which says that this, uh, this standard error of the sample mean equals to the state Senator Gatien we know, divided by the square root of the sample size. The first part asks for sample size of 16. So we'll look at that. So, really, what we're doing is we're substituting in the 25. That's great 16 which we know is for And if I take 25 divided by four. I know that 6.25 is the, uh, standard error if the animals to 16. So let's do the same thing the next part asks if the sample size is a little bigger is in this case if the sample size is 36 someone used the same formula relationship. So you know that this equals two a. The 25 that it starts off with, and I call it in my class an adjustment so it gets adjusted by according to the sample size, so more sampling is better, which means less variation, less error. So that's 25 to about about square 16 which means 25 divided by six. That ends up being 4.16 repeating. Okay, so that's the 2nd 1 You notice it's smaller 1st 1 because it a larger sample size. And finally, the third part of this is what if the sample size is 100? So if we have a sample size, you have more samplings. Even better if our goal is to have less variation to represent the true population. So the standard error of the sample mean is that 25 adjusted by, um, and divided by the square root of the sample size, and so that's gonna end up giving us 2.5 even better. That's what it is I'm gonna make one little adjustment appear meant to write it for this 1st 1 So the overall relationship, we really to supply it waas The standard air equals two mouthful. So for any problem, the standard error of the sample me equals to the population standard deviation divided by the squared of them for any size. And we just showed three different examples of size and noticed that as the end got bigger, the stand.


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