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2) Why is it necessary include some measure of dispersion in addition to meaSut = of location when analyzing of data?Under what circumstances might you decide t0 us...

Question

2) Why is it necessary include some measure of dispersion in addition to meaSut = of location when analyzing of data?Under what circumstances might you decide t0 use the median as mcasure of location rather than the mean when doing analysis of set of data?a) You scored 92 on your statistics final. If the mcan score on the test was 81 and - the standard deviation was 6, how did you do in comparison to the rest of the class?Your friend took statistics class from different professor. Your friend sc

2) Why is it necessary include some measure of dispersion in addition to meaSut = of location when analyzing of data? Under what circumstances might you decide t0 use the median as mcasure of location rather than the mean when doing analysis of set of data? a) You scored 92 on your statistics final. If the mcan score on the test was 81 and - the standard deviation was 6, how did you do in comparison to the rest of the class? Your friend took statistics class from different professor. Your friend scored 96 on his final, but his class had mean of 85 and standard deviation of 8. How did you score relative to your friend compared to the corresponding classes?



Answers

Use the same population of {4, 5, 9} that was used in Examples 1 and 5. As in Examples 1 and 5, assume that samples of size n = 2 are randomly selected with replacement. Sampling Distribution of the Sample Median a. Find the value of the population median. b. Table 6­3 describes the sampling distribution of the sample mean. Construct a similar table representing the sampling distribution of the sample median. Then combine values of the median that are the same, as in Table 6­4. (Hint: See Example 1 for Tables 6­3 and 6­4 that describe the sampling distribution of the sample mean.) c. Find the mean of the sampling distribution of the sample median. d. Based on the preceding results, is the sample median an unbiased estimator of the population median? Why or why not?

Hey there for this question we are given 50 did oppose, which are the one with distances from home to college for those 50 students on gay. In a first question eight, we need to construct a group frequency distribution of the data on Dhere. We already given once before as our first category or first class. That's We can easily imagine that the rest of these concepts will be we. Why 28 right? Because we have 1 to 4 four minutes wanted Street. Thus for next Category five plus recalls eight. And we have nine Basri because 12 the 13 plus three Because 16 17 plus three Because 20 and so on, we can see that the largest number of data set is 27. So that's just that up somewhere. Numbers on two week and 27. So we have 21 plus free. You goes 20 or then on the par three because 48 Okay, so I think we can't stop here, and it is our classes or our category. It's on. That's just computed frequency. Or how many points are there? Belonged to each. Rivers have been also each category. Okay, so that's first revealed it up was within 1 to 4. Let's just go with the points row by row. In our first real we have won 312111 street. So we have in total 40 up was here in the first row. The stubble? Yes, we're unless wants. The second way have is their numbers here that have between 1 to 4. And for our certain bro, we have 14 only. So it is one and you know force rule. We have 13 So is only one, you know, last room we have Okay. So again, with a while for here. So in total way have cell numbers, we'd be wonderful. Rest here it's pregnancy. This is okay. That's well, it's our second out of work while to eight. That's do reviews it up was a row by row. First role. We have far 66 So street in a points then you know second role we have five, 5858 So to get up force and the you know a certain role We have zeroed it up once, you know, before Stroll, we have 65 on selling that is rated up and industrial, we have 5878 So it's four numbers. Okay. Eso in total we have here Trump better boots, five glasses free eight before. So we have helped enough center for our circa 19. So that's first Will be our first nice. Okay, there's first. And you know Secondly, we have 10 9 10 9 So we have a little four months and, you know, sir, we have 9 10 cents. Will have no first force. Really have 9 11 10 So we have for my reports. There are not stroll. We have. Let's see. Noted up was right between ni and so zero again. So we have working. Been upwards inside the school. Okay, that's walks over. NextCard wished 13 to 16 1st rule. We have second rule. We have off working. 14 16 16. So it's street in our certain room. We have 15 only and, you know, forced roles. We have way. Don't have numbers here. A stroll. You have searching and searching. Okay, so I guess 88 All right. Our next category selling into 21st rodeo. Have any numbers in our second road? Uh, we do not have any number, you know, certain rules. We have plenty. Okay. Only one hunting stories. One you know, force rule. We have, uh, only 17 once again and our mass roles, we have plenty and 18 in this country. Okay, so for our next Edward for the one month before, you know, first truly one 11 So we have one only. So it's another one. You know, Force through way have nothing here in our monstro. We have 22. So it's another one in order. Last Khyber 48. I think we only have one that up months in our survey, which is 27 That belongs to this kind of. For the rest, they are all gold putting fire. So here we only I wanted just to double track. Let's add up all of these numbers to see if the attack 2 50 1914 equals soothe. Okay, first question or it is our group frequency distribution. Okay, that's no move on to the next. It is be It asks us to congregate and me on the standard deviation. Let's compute the mean first way have in total 50 their boats so mean it's simply 15 here this summer, all of numbers which is why, plus six plus five. Three Street right here, first road and lost second role for 5 16 on the survey. That's four strong way all of this shooting 50 numbers up and wanted by it. That's we get our mean another complication. E guess to be the meat off the dinner. Let's not compute this kind of deviation. It is actually the square root off the bears. So a computer beers and what is a very, uh, what is a form it off of your? It is simply expectation. Off acts or I'll get awards milers. Expectation of axe on. We have so for a temple in our first let's review our first it upon six. So So we have six minus. That means authorities outputs, which is 1.9 and right, so it is our first it up. What about our second war? Our second in the first round? It is five. So, again, five minus the meat or simple square off. We need to get the expectation or the mean off the all of these numbers I got. You offer media really formula here, So it's that easy, but way have in total picked it up was here. So the competition is kind of complicated, to be honest. So we just had a reason the rest onto our lots that have haunted just 18 minus and mean. IHS, it's important. Not this man. Okay, so we were It is a quantity inside a square root or the expectation off this summation here. So it is all very and after computation, get the foreign numbers, which is 30 salmon on 61. So our standard deviation is just the square root of mirrors, which is six points once re you seven approximately. Okay, now that's will mark your words, sir. Question. We need to compute axe. The mean off that's minus or plus to study. Deviations and the determining hole are the percent of that. Everything the rent off this So it's just, uh let's rewrite it. Being offense. Miners to us. Coma me a best in class as a brand of data it is. You call chew 10.9 minus two times 6.1 street to seven coma Temple night plus two times 6.13 to 7. It's just Well, that's it. That's brought it. You don't That's role here for more space. It's just minus one point before five for Oklahoma. 23 point. Once it's fuss. Why for Okay, so it is our wrench. And we need to compute the percent of their waiting this lunch at Crisfield you because you know that inside that this point is less, son less that there. But all of our data points are great. Zero. So we just need to do here in the upper point, which is 23 points. 16 here because we have, uh, this group frequency distribution way know that there's wanted a point that is above this number, but we need to check the numbers between below industry. So, you know, first rule we have their numbers here above for this read. Oh, sorry. I think we have wanted upon Evolved. You know, second rule, we have zero. About 23.16. Okay. You know, several. We have wanted upwards, which is 27 you know, fourth row and industrial. I don't think we have any more numbers. Evolved kind of three. So we have just for double check. I think so little. We have only two numbers evolved industry, so way know that 15 minus two because 48. Which means that 48 members all so but its report 16 and above, minus one for 36 called They're all boats. So 10% of data should be 48 or 50 because 96%. So they are not a six point there, 96% of their withing This this right here. What is the two standard deviation off? That's all this question.

So for this question, the population is 45 and nine 45 common I and the sample size is too, because you're picking so random, so s o n requested to. And then the first question I want you to figure out, um, the first question wants to figure out the value of the population Media. Andi, mind you, the media is always the media number. When I arrange, the data are not sending order. And from four, five and nine, you could definitely tell that night is meeting number. So definitely, um, the population media, the population media is you go to five. And that's the answer. The second question, You do remember that the two values a randomly selected replacements from 457 on. Um, we're going to write the samples off the possible, um, the possible send the samples of the possible I'm I'm gonna write the samples of the possible outcomes in order to describe the sampling distribution of the sampling. So the samples, the samples would be oh, four comma four, four comma, five for common night, $5 for five form of life and five common night also. No, I come up before night. Come up five and I come tonight so that their nine deny probabilities the problem. Total probabilities equal tonight, right? And then to construct the table for sampling distribution of media who need to find the sample meet media in order to my sugar table. And we know that it Number of probability, easy for tonight. And in order to find the sample media, we're gonna right the sample over a year. And then we're gonna have the samples. The nice samples, which is four come before. Welcome of five. Welcome a night 54 five from of five. Fire from my life. Yes, also night coming for like on the fly in nine Kalmunai on. And then we're gonna find the sample media for each of them in order to finish off the table. So the sample media and we know that the media is the, um the some off the number of divided by the actual number, the actual the total numbers we have. And since we have two figures because we have to, um, we're picking soon numbers the random so is too. And then we have, like, two samples for every samples. So four plus four developed by two is gonna be four. So that's gonna be the something something media four plus five they don't have the right to is going to be 4.5. That's going to be a sample media. The 4.94 point four plus nine when it bites, who is going to be six points? All right, that's sample media. Why, of course, for the bye bye. Service 4.5 on this one is fight. This one is seven seven. Just 16.5. This one several. And this one is night. And then the probability is gonna be the last thing we're gonna do. We know that they're night there, nine on samples. So each release is gonna be one of the night, and that's gonna be answered for the rest of the table. So I'm just gonna click the check. Yeah, and then the question also wants us to combine the values of the media, and they are the same. So combining the values and ideas are the same. We're gonna have a sample media sample media. Yeah, yeah, yeah. We're gonna have sample media. I mean, the other side, we're gonna have to probably t. Yeah. So obviously is gonna be, um, the sample medians from the top. You can see it for four. Um, 4.5, talking five, um, 5.0, 6.5, 7.0. And then I concert. Mhm. Yeah. Anybody with your fault is because one time one of the night. 4.5 focus twice. So overnight five or close ones one of the night. 6.5 occurs twice overnight. The seven because slice until the night. 90 Clockers ones, One of the nine. That's it. The next question wants us to find the mean off the sampling distribution of the sample media and the mean off the sample distribution of the sample begin is going to be is going to be this song. The some off the total, some of the total off the night sample media divided by the number of samples where by a number of samples. And this is gonna be equal. So you adding all of this together the 4456 months, five. The way. I don't have that together. It gives you 54 and the number of samples of nine. And that's Why my night? And that's six. That's the mean off the sampling distribution of the sample median. And the last question, Um, sorry about this. The last question say's, um, based on the preceding results is a sample media unbiased estimator for the population media. The answer is no. The sample media is not unbiased estimator of the population medium because the published So the answer is no, because the population media is no e. Quote the meat off the sub media. Remember, the population medium is the one we solved first, which is on five, and it's not a call. So the median actually the mean of the sample media, okay?

Uh huh. So in this problem we're comparing Reginald and and Sarah's 1st and 2nd test grades. And the reason why Sarah thinks she should get the higher grade is because it's all based on Z scores. So Reginald on the first test when he scored an 80 he fell right exactly at the average. So his Z score is zero Whereas Sarah she scored an 88, she scored two standard deviations above the main so her Z score is a two And on the second test regional scored an 85 which is one standard deviation above the mean. So his Z score is a one And Sarah scored a 65 which is a little bit below the mean. So her Z score we can find by doing her observed value minus demean divided by the standard deviation and that's going to give us a value of- .3 repeated. So yes, in terms of their scores on the test regional had 165. Sarah had 88 plus 65 which is only 1 53. So their scores, yes, regional totaled more. But if you look at their Z scores when you combine Z scores, Reginald is at a +10 plus one. Whereas Sarah's Z's combined Z scores two plus negative 20.3 repeated, which would make 1.6 repeated. So her combined Z score is higher. She scored higher on average above the means than he did. So that's why she thinks she should get the higher grade

In this question, we want a margin of error of two years with a 95% confidence, a standard deviation of 13.36 and we want to figure out the sample size that will do that. So I plugged in all of our numbers into this formula for margin of error. And now I'm going to solve for N. By using some algebra, I'm going to divide both sides by 1.96 which will leave 1.204 on the left, and then I'm gonna move that square root of an out of the denominator by multiplying both sides by the square root of an and that will cancel out the square root event over there on the right. Now, I'm going to get rid of this 1.2 away from the square root of end by dividing both sides by 1.204 Mhm. And that's gonna give me the square root of n is equal to 13.9 to 8. And then you get rid of the square root, I'll square both sides. And that will give me sample size of 171 42 Which will we will round up to the nearest whole number, 172 people as our sample size, or 172 million is the same. Millionaires are the sample size. And the reason that we use the 13.36 the sample standard deviation here instead of sigma. We're allowed to do that because the sample side, the original sample size was more than 30. It was actually 36.


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