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PRINCIPLES OF STATISTICS Qal 44 Home My courses PRINCIPLES OF STATISTICS | J 8 General 2020-08-26 .L~ytty+jeQuestion 113 Notyut onsweredIf the median of the observa...

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PRINCIPLES OF STATISTICS Qal 44 Home My courses PRINCIPLES OF STATISTICS | J 8 General 2020-08-26 .L~ytty+jeQuestion 113 Notyut onsweredIf the median of the observations 0, 3,X, 12 is 5,then the mean of these observations will beKalked oul ot 100Tlog questionA) 5.75 B) 5 C) 4.75 D) 5.25 E) 5.5Type nere t0 search

PRINCIPLES OF STATISTICS Qal 44 Home My courses PRINCIPLES OF STATISTICS | J 8 General 2020-08-26 .L~ytty+je Question 113 Notyut onswered If the median of the observations 0, 3,X, 12 is 5,then the mean of these observations will be Kalked oul ot 100 Tlog question A) 5.75 B) 5 C) 4.75 D) 5.25 E) 5.5 Type nere t0 search



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A random sample of 49 measurements from one population had a sample mean of $10,$ with sample standard deviation $3 .$ An independent random sample of 64 measurements from a second population had a sample mean of $12,$ with sample standard deviation $4 .$ Test the claim that the population means are different. Use level of significance 0.01. (a) Check Requirements What distribution does the sample test statistic follow? Explain. (b) State the hypotheses. (c) Compute $\bar{x}_{1}-\bar{x}_{2}$ and the corresponding sample distribution value. (d) Estimate the $P$ -value of the sample test statistic. (e) Conclude the test. (f) Interpret the results.

In this problem, we have a sample and the sample size is 80 the mean is 139, and the standard deviation is 13 and nothing else is known. So that means we do not know the shape of the distribution. And since we do not know the shape of the distribution, that means we could apply chevy shelves theorem and chevy steps here, um is that at least one minus one over K squared of the data lies in a certain interval. So let's look at our information. So since we don't know the shape of the data, we're just gonna draw some crazy shape and we're going to place our mean in the center And are mean was 139. And then we're going to talk about within one standard deviation. So that means we are Either 13 above that, or 13 below that, putting us at 1:52 And 1 26. And then we want to talk about being two standard deviations above and below the mean. So that would put us at 165 and 113. And if we talked in terms of three standard deviations above and below the mean, We would be at 178 and 100 in this particular problem part A is asking us what can be said about the number of observations that lie in the interval From 1 26 to 1 52 and the answer to that is going to have to be nothing. And the reason it has to be nothing is because when we are talking in terms of 126 and 152 were one standard deviation away from the mean. So it's as if Kay was equal to one and if we substituted that into Chevy Chefs theorem, we would get an undefined value because you would be dividing Or you get an answer of zero. If we did one minus 1/1 squared, we'd get one minus one or zero. So Chevy Chase theorem isn't telling us anything. Part B. We want to know what can be said about the number of observations that lie in the interval from 113 Up to 165. So 113 is two standard deviations below the mean And 165 is two standard deviations above the mean. So that means K would be too. So when we apply Chevy Chefs theorem, we'd get one minus 1/2 squared, which would be one minus 1/4 Or 3/4. So we could say 3/4 of our data lies in that interval. Now keep in mind we had 80 pieces of data, so that means 3/4 of the 80 pieces of data or 80 observations, which is going to give us 60. Now keep in mind when we're using chevy chaves theorem, it's an at least amount. So technically we should say at least three quarts of quarters of the data lies in that interval. So we could then say at least 60 pieces of data or the state A was observations lies in the interval From 1 13 to 1 65. Let's take a peek at part C, part C. What can be said about the number of observations that exceed 165. So let's think about what we just found. We just found that at least 3/4 of the data falls in between 1 13 and 1 65. So if at least three quarters of the data is in there, then at most One quarter of the data could be above the 165. So at most one quarter of the data can exceed 1 65. And we had 80 pieces of data, So one quarter of 80 would be at most 20 observations exceeds 1 65. And then finally part D. We want to know what can be said about the number of observations that exceed 65 or are less than 113. So again, if you look up here at our picture, we said at least three quarters has to be in there, so at most one quarter is not in there, and this is not in their below 1 13 and this is not in their above 1 65. So we could say again, at most one quarter of the data can exceed 1 65 or be less than 1 13. So again, one quarter of the data would be 20 observations, so we could repeat what we said in part C. So at most 20 observations can exceed 1 65 Or be less than 1 13

In this problem you are provided with some sample data and I like to work in a more vertical chart than a horizontal chart. So I'm going to turn it on its side. So we were given X. Values of 26 27 28 29 30 31 and 32. You were also given frequencies of each of those numbers. So we had 3 26 is 4 27 16 28 12 29 six thirty's to 30 ones and only 1 32 and part A is asking you to calculate the mean and to find the mean. And actually this was sample. So we'll say X. Bar equals. And to find X. Bar we're going to some up X. F. And divide by N. So we're gonna need to create an X. F. column. Since there were 326 is they would add up to 78. Since there were 4 27 they would add up to one oh eight. There were 16 28th. If I were to add those up, I would get 448, there were 12 20 nines. For a total of 348. There were 6 30's Which added up to 180, There were 2:31 which added up to 62 And there was 1 32. So when I add up the X. F column, I'm going to get the sum of all the data. So when I add up the X. F column, I will get 1256 and to find n, I'm going to sum up the f column Because there were three of one number and four of another number. So there was a total of 44 pieces of data in this sample. So therefore my average turns out to be 28 and 6 11th, Which as a decimal that is 28.54, repeating for part B, we need to determine our sample standard deviation and the formula for sample standard deviation is the square root of the variance. So we're gonna have a square root symbol. And the variance of grouped data is found by taking x minus x bar squared, multiplied by F And divide by the end -1. So we already know that N was 44. So we are going to divide by 43 and now we've got to add on to our chart. So we're going to create an x minus X bar column. We're going to create an x minus X bar squared column and then we're going to create an X minus X bar squared times F column. And in order to assist with all of this, I'm going to use my graphing calculator. So I'm gonna bring in my graphing calculator. I'm sorry about that. And as you can see I've already placed the data into my calculator. I know there's The numbers 26- 32 and L two is my frequencies. I've calculated my average out to be 28 6/11. So in such in column three enlist three I'm going to tell it to take every X. Value And every x value was found in list one And I'm gonna subtract 28 and 6 11th. So I'm gonna open up a parentheses and I'm gonna say 28 plus 6/11. And in doing so I'm gonna get these ugly long decimals but I want those but in my chart I'm not going to necessarily record all the decimal places. So I get about a negative 255 Across from the 26. I get about a negative 1.55. About a negative .55. And again I'm going to end up using the full decimal places. In my calculations. I'm just recording them to fewer decimal places for the sake of east. So now the next column is calling for me to square the column I just created. So I just created L three. So I'm going to move over and I'm going to tell L 42 square everything in L three. Again, I'm getting some ugly decimals. I'm not going to record every decimal place but I will use them in my calculations. So I get about a .6.48. I get about a 2.39. About 30.212 point 1 to 6.2 and 11.93 for our last column. We want to take this column that we just created and multiply it by the frequency. So We just created column L four. And we want to Multiply by L two because the frequencies were in list too. These were the numbers in list four. So I'm going to scoot over in my calculator and I'm gonna say take everything in list four. I am multiplied by everything in list too. Again, I'm going to get some ugly decimals. I'm only gonna record to a couple decimal places but in the long run I'm going to use this entire number. So I'm going to record those as about a 19438 9554 476, 0 2462479, 12.694, 12.050 And 11.934. And our formula is calling for us to add up that column. So again, that column was list five. So I need to tell my calculator to add up list five. So I'm gonna quit out of my lists and I'm going to hit 2nd stat. I'm gonna move over to the math menu and I want to sum up Everything that I have enlist five. And in doing so I'm getting 72 and 1011th, so 72 and 1011th. Is that some? I have to now divide that by a 43 and I have to take the square root of that in order to find bye, Standard deviation. Now, I don't like that answer. I'm going to get math decimal and get my head around that number and I'm getting about a 1.30 to as my sample standard deviation. Let's go on to part B in part B. It's asking about the empirical rule and in the empirical rule, I'm going to draw a bell curve because the empirical rule governs distributions that are bell shaped. And when you're using that empirical rule X bar would go in the center and then within one standard deviation of that mean, which means that I'm going one standard deviation above and one standard deviation below That should account for approximately 68% of the data. If I go two standard deviations away from the mean, then it should approximately be 95% of the data. And if I go three standard deviations away from the mean, We should account for approximately 99 7% of the data. So therefore about how many of those measurements does the empirical rule predict. So from X bore minus S. Up to X bar plus S. We expect 68% or approximately 68%. So 68 Of the 44 numbers would be 29.92. So we should expect approximately 30 pieces of data between two standard deviations out. So x minus two S And x bar plus two s. We're expecting about 95% of the data. So if I calculate 95% of the 44 pieces of data I'm going to get 41.8. So I'm expecting about 42 pieces of data. And for three standard deviations out, I'm expecting about 99 7%. Which would be approximately 43.868 or 44 pieces of data. Now in part C we have to actually compute the number of measurements that are actually in each of the intervals and compare those to what we just predicted. So these are our predicted values. So let's generate our number line for our data. Remember we got an average To be about a 28-54 repeating and we got a standard deviation of about 1302. So I'm going to place the 28 .54 repeating in the center. So I'm gonna bring in my calculator, so I'm gonna do 28.545454 and I'm going to add 1.30 to, So I'm getting about 29.847 And then I'm gonna add 132 again And I'm getting about a 31.149. And then I'm going to add the standard deviation again And I'm getting 32 451 ISH. And then I'm going back and I'm doing the same problem, but I'm subtracting the standard deviation. So back here, I'm getting about a 27.243, I'm subtracting another standard deviation And I'm back here at about 25 941 And I'm subtracting a 3rd standard deviation And I'm back here at about 24.639. So now when we look at our data, There are 326 is So 26es would fall right in here and there's three of those, There are 427s four, fal like right in here, There are 1628th, so 28, they're gonna fall right in here and there are 16 of them. There are 12, 29, 29 is gonna fall somewhere in here and there are 12 of those. There are 630s, 30 is gonna fall somewhere right in here and there's six of those 31 there are 2 30 ones, so 31 is about right here And there is 1:32 and 32 is going to fall somewhere right in here. So when you think of our curve, one standard deviation out, We've got 16 and 12. So we've got 28 numbers within one standard deviation of the mean, two standard deviations of the mean, we've got seven plus 16 plus 12 plus eight, Which is 43 pieces of data and three standard deviations out. We've got the 43 pieces of data plus the one more here. So there's 44 pieces of data. So how did we compare? We were right on the money for this last one they matched, We were off by just one for two standard deviations away from the main and we were off by two in the first. Within one standard deviation of the mean. So again, keep in mind these are approximate values. These are approximate percentages, so therefore it doesn't have to be perfect. So we were close in every aspect

This problem is based on Chevy Chefs serum. This is based on several ships serum. So what exactly does this theory? Um, say this? Is that at least one minus one by Z square. At least at least one minus one by Z square off the data values must be within Z standard. Deviations off the data. Values off the data. Values must be within Z standard deviations Z standard deviations, Z standard deviations off the mean disease. Any value greater than one? These enter deviations off the mean, very important. Mm. End off the mean Now, this condition is very important. Where Z, where Z is any value greater than one. This is the concept that we are going to use to solve these questions. Moving onto party. What do you want to determine in this question? If you want to use the membership serum to determine the percentage of data value within each of the rangers Okay. So part a is range 20 to 40. Part A is range. 20. Do 40. If I look at the number line, this is 30 over here. I have 20 over here. I have 40. Okay, so, uh, in this question, it is given to us that our immune very important, are immune is 30 and our standard deviation is given to us as five. All right, so I can see that here. There's a spread off 10 10 upwards and then downwards. Now, if five is one standard deviation, 10 will be two standard deviations. Right? So putting this to in our formula when minus one by to square What do I get? This is nothing but one minus one by full. Or this happens to be three by four. Or this is 0.75 Okay, so what can I say? I will say that 75 percent off data points off. Data points lie within. Lie within three standard deviations. Right? How many standard deviations was this? This was too sorry. Live within two standard deviations. Two standard deviations off. The mean off the mean. This is the answer to our but E right now, moving on the back. B, what is the range that we have this time? It is 15 to 45. It is 15. Do 45. Okay, So if I look at this number line, this is 30 over here. This is 15. And this is 45. So I have the range off 15 played a spread of 15. So 15 means, actually three standard deviations, if five is one standard deviation 15 is three standard deviations. So again, putting in the formula one minus one by three square, which is nothing but one minus one by nine, which happens to be, what, eight by nine. And if I use a calculator for this, this thing actually happens to be 88.89 Or I can say that this is 0.8 889 So this is 88.89%. So I can say that 88.89% off. Values off. Data values off. Data values lie within. Lie within three standard deviations. Three standard deviations off the means off. The me. Okay, moving on the part. See? What is this going to be? 22 to 38. 22 2 38. So again, taking the head of the number line. This is 30 over here. I have 38 over here. I have 22. So this is a spread off. Eight. Five means one standard deviation. So eight is going to mean 1.6 standard deviations. This is 1.6 standard deviations Putting this in the formula. I have one minus one by 1.6 square. Right. This means what this is going to be. If I use a calculator for this, this is going to be somewhere close to 60. This is 0.6093 if I'm not wrong. Yes, this is 0.93 are how will write this? This is going to be 60.93% off. Data points off. Data values lie within. Lie within 1.6 Standard deviation off the mean. Okay, moving on about D. Now I have 18 to 42. Now I have 18. 2. 42 again if I take the help off a number line. This is 30. This over here is 42. This over here is 18. So this is a spread off. 12. 12 means what? 2.4 standard deviations, right? Five means one standard deviation. So 12 is going to mean 2.4 standard deviations. Putting this in the formula. I have one minus one by 2.4 square. Right. And what will this turn out to be? This turns out to be 0.8 2840.82 84 Okay, so what? We're right. I will write 82.84 percent off. Data values percent off. Data values lie within. Lie within 2.4. Standard deviations off the mean moving on the part E. What were eight? This is from 12 to 48 from 12. 2. 48. This is 30. This is 48. This is 12, so I can see that the spread is off 18. Right? 18 means what 80 means, 3.6. Standard deviations. So what? We're right in my formula. This is going to be one minus one by 3.6 square. And if I use a calculator for this, this actually turns out to be this turns out to be 0.9 to do it. 0.9228 Or I can say that 92 0.28 percent off data values off data values. Yeah, lie within. Live within 3.6. Standard deviations off the mean, And these would be my answers. Right? We had another part E D c B. And yes, these are my answers


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