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Based on a normal distribution, what proportion of our data fallwithin the following range (4 < x < 12), with a populationmean of 10 and population standard d...

Question

Based on a normal distribution, what proportion of our data fallwithin the following range (4 < x < 12), with a populationmean of 10 and population standard deviation of 6? PLEASE SHOWSTEPS.

Based on a normal distribution, what proportion of our data fall within the following range (4 < x < 12), with a population mean of 10 and population standard deviation of 6? PLEASE SHOW STEPS.



Answers

For a population with a mean of 75 and a standard deviation of 12, what proportion of sample means of size $n=16$ fall above 82 ?

In problem 60. We have a normal distribution with a mean of 70 you equal 70 and the standard division of 12, which means Sigma equals 12. We want to determine the limits that includes the middle 65% of the cases. We want to calculate the limits. Four. 65% of the case. Let's recall the normal distribution, the normal distribution like that we have here. The mean it's centered about the mean 70 and has a Norman or a standard division of 12, which means we want now to determine elements that lower limit X one, for example, the upper limit x two. These limits include 65% of of the cases, which means this area showed equal to 0.65 Then what is this area? The allotted area. Here we can calculate the area as it equals the total area. Under the normal distribution, which is one minus the area between the limits, which is 4.65 Divide by two because one minus 4.65 gifts the total area to the left of X one and to the right of X two. But we want to get only the dotted area here which equals one minus, opened 65 by by two. So we divide by two. Then this area equals 4.175 Now we can use the standard normal distribution tables to get. Does it value corresponding to X one? The standard values is one is zero Sorry. Zero. This is a standard value for the mean and we have here that one. And here we have that too. So we can get that one using the normal distribution tables, the standard normal distribution tables by entering and getting the corresponding that value to this area. Let's enter the table. We have here the areas to the left of the school. We will search the tables for 4.175 We have here 2.1 seven four we have here. Oh boy. In seven 4.176 Which means we can choose this value to represent 4.175 because it's the closest one and the corresponding Z score or that value is minus 4.9 01 23 Then that one equals minus 0.9 three. This corresponding to the minus a 0.93 and because of symmetry, the two will be 4.93 We can use the formula by transform transforming from X to set, which says that that one equals x one minus mu divided by sigma or we can get X One equals that one multiplied by sigma and to blow it by sigma Plus you. And of course, we can get X two by multiplying exit to buy Sigma plus me, then X one equals minus 0.93 multiplied Boy Sigma, which is 12 plus 17, which equals 58.84 and X two is all 0.93 complied by 12 plus 70 it equals 81.16 which means the limits, or 58.84 and 81.16 which includes 65% of the cases.

Okay, so we have our unbearable X and X is following some distribution, but we don't know what exactly the distribution is, but what we know is that the mean off X 25 in the centre deviation of packs of six. So now we're interesting in the sample size and such that the center deviation call summation off. Excess? Yes, we call to 42. Um, well, the standard deviation. Oh, Summation of access. It's actually the square. Root off the variations. Summation Next. So it's no hard to calculate the various off Summation X, which is going to be good Teoh and times variance off hex. So which is sometimes square of six. And we know that, uh, we use this question to so for n Well, we have at that. We have awarded Teoh. You go to square root off and time six So scared of then it's a good years on end. We got in in close to 40 night

Okay, so let's look at x x minus. Export and parentheses. X months export close parentheses squared. And now we're looking at 2345 and six. How first to calculate our range. We know this is going to be our biggest number six minus our smallest number to you, which is going to equal forward like your last problem. Now then, our range is gonna going to be for in our average is going to be for as well, not three. Yep. 20. Divided by five is four. So that's true. And we know that this is gonna be two minus two. Started to minus four. It's going to be three minus 44 minus four, five minus four and six minus four, which equals negative to negative. 101 and two. This is similar to what we had on our last problem, where the squares are going to be 4101 and four again. So, in effect, we also know that the sum of X minus X bar squared is going to equal 10. And looking at our formula again, we have a square root of X minus X bar squared over in minus one for sample. Senator Deviation. Now, when we plugged this in for us, it's going to be square root of 10 over four again, which is a square root of 2.5 now for Sigma. If this was population, it's going to be 10/5 because we're not subtracting the one this time, which is going to be square root of two. So the only thing that changed was our range Here, our sample and population standard deviation remained the same.

Okay, so the question here is to find the variance and standard deviation of this set of data right here, um and we're given that it is a population, not a sample, so because um these numbers are a bit hard to work with and it would be too time consuming to go through the entire process on my part of finding the entire mean, finding the mean, than taking each and every difference to the mean, especially since we find out later that the mean isn't even a whole number, then we take the square of the differences, sum them up etcetera. Um I went ahead and actually used an online calculator, I apologize I for cheating a little bit, um but it would just be way too time consuming on my part, but I can go ahead and describe what this calculator is doing, I think that I think it's best if I go ahead and describe what this calculator is doing and maybe do like a few, take a few of each just to show you like, if you have to do it manually then like this is how to do it. Okay, so here goes, so what I did was I entered the numbers into the text box marked numbers as seen here and what this calculator does, it counts the number of numbers. So in this case there are 15 numbers It sums them up. So basically the sum of all 15 of these numbers is 147 and takes the mean, which is 147 divided by 15, so we get this 9.8 right here. So now in order to keep up with the formula, we take the difference of each number and the mean. So for instance, this first number, you can see this 1.2 right here is just our first number 11 minus the mean. And similarly, you see this negative 3.8 right after the 1.2 is just our second number six minus are mean. So, So yeah, um that's how the calculator is getting the numbers in this text box right here um and it's doing it for all 15 numbers. Again, it would be a little time consuming on my part to do all 15 on screen. So I apologize if I'm cheating a little bit. But yeah, it's but the process, as you can see is incredibly straightforward after that. Once you have all 15 of your differences, you square each one. So in this case um Again, you just take each number each of the 50 numbers and you square them. So let's take the first number, for instance, 1.2 and square it. And that's how you get the first number in the differences squared box at 1.2 squared is just 1.44, similarly minus 3.8 squared is just 14.44. If you're getting a negative number, when you're taking the square of the numbers in the differences box, then something's going wrong, a square of any number is positive or zero. So just be wary when you do that. Um Now, lastly, what you do is you take the numbers in this text box right here, you add them all up and you should get Um the number in this text box right here. 1 24.4. Um and then lastly what you do to calculate the variance is just take 1 24.4 and then divide it by Uh your number of things in your population. In this case it's 15. So if you divide 1 24 .4 x 15, you should get this variance right here, 8.29. So this is your answer for variance. And then to get standard deviation, you just take the square root of variance. So 8.29. The square root of 8.29 is 2.879. As seen here. And those are your answers


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