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The data shown represent the scores on a national achievement test for a group of 10th-grade students. Find the approximate percentile ranks of these scores by constructing a percentile graph. a. 220 b. 245 c. 276 d. 280 e. 300 $\begin{array}{ll}{\text { Score }} & {\text { Frequency }} \\ \hline 196.5-217.5 & {5} \\ {217.5-238.5} & {17} \\ {238.5-259.5} & {17} \\ {259.5-280.5} & {48} \\ {280.5-301.5} & {22} \\ {301.5-322.5} & {6}\end{array}$ For the same data, find the approximate scores that cor- respond to these percentiles. f. 15th g. 29th h. 43rd i. 65th j. 80th

And this problem, we have six test scores only are asked to find the percentile that corresponds to each you'll be using this formula. P equals number of numbers lower plus 0.5 over end. Where n is the total number of numbers Times 100 to convert 12%. So we need to identify how maney numbers we have. And we said this is six test scores. So for this problem and equal six, all right. So to calculate the first percentile for the corresponds to test score five. How Maney data values are lower than five and that answer is zero. So we ads. You're a 0.5 divide by six and multiplied by 100. That turns out to be 8.33333 etcetera. So 8.3 repeating, and that working around them corresponds to about the eighth percentile. So for the data value of 12 we have one data value lower than 12. So we're gonna have one plus 10.5, divide by six and multiply by 100. So 1.5 divided by six uh, and then multiply by 100 is 25. So that corresponds then to 25th percentile. Say I'm going to repeat that same pattern. And for 15 we have two values lower. So 2.5 divided by six times 100 and that gives us 41.6, which is going around then to the 42nd percentile. And we're going to continue in that same fashion for the last three numbers. So we've got three numbers less than 16 and when you compute, that comes out to be 58.3, which rounds to the 58th percentile and then four plus 40.5. I'm just gonna go ahead and write 4.5, divided by six and multiply by 100 and that gives us the 75th percentile. And then our last number. We have five numbers less than that. So five plus 50.5 is 5.5, divided by six multiplied by 100 and that corresponds to or that equals 91.7, which is the 92nd percentile. So, um, you may notice that there's a little bit of a pattern that these air evenly spaced, and that makes sense because if we look at 100% split six ways, we've got six data values. It turns out to be 16 0.6 repeating percent. So if you keep adding 16.6% you'll make your way through that. And that's the pattern. It's in that table, All right. For the last part of the problem, we are asked to find the 33rd percentile. So we need to use the other formula. C equals end times p over 100 where n is equal to the number of numbers and P is the percentile. So in this case, 33 make that a multiplication sign there, um, divided by 100 and that comes out to be 1.98 So that means we're looking for the number in the 1.898 position. Um, since that's not a specific position, we round up when we always round up. So that means we're looking for the number that's in the second position. So if we go back and look at our data, then number 12 is right here. That's in our second position. Our second number when we look at the data organized from smallest to largest. So the 33rd percentile Oops, um, is what was that? 12

And this problem. We have information about three people who took a test, and we have some information about the mean and standard deviation of the overall group of people who took that test. So we're given that the mean of everybody who took the test is 1 25 and the Sandra Deviation for everybody who took the test is 18 and they were given different pieces of information. About three of the test takers were told that Tom's Ward 1 58 on this test. We're told that Dick scored in the 98th percentile, and we're told that Harry has a Z score of two. So he was two standard deviations above the mean eso. All three of these measures are on different scales, so it's a little bit hard to tell, but our objective is to put them in order from lowest sport to highest score. So if we start with the 90th percentile, Dick is probably going to have the highest score because the 90th percentile is pretty good. That means he scored better than 98% off the other students in the class. Um, and Tom and Harry are on, um, opposite scales Tom is on the data scale and Harry's on a Z score spill, but let's convert them to the same school, so they're easier to compare now. We don't know what the Z score for Dick is going to be, but it's probably gonna be greater than two. So if Tom has a Z score greater than two, then he did better than Harry on. We can arrange the three of them in order. So let's find the Z score for Tom. And we know the formula for calculating a Z score is X minus X bar over s where X is the data value expert is the mean of the sample and s is the standard deviation. So for Tom, it's gonna be 1 58 minus 1 25 over 18. So Tom did better. But was it two standard deviations better? And it turns out that this see score is 1.83 So Tom did not do better than Harry Ah, and so we can place them in order. Then the person who did the worst or have the lowest score was Tom in the middle. What is the score of two was Harry, and the person who did the best on the test was stick

And this question were given seven data values and asked to find the percentile rank of each. So we have the data values sorted in order from smallest to largest on. We need to use this formula that tells us what the percentile ISS so P equals the number of numbers that are lower than any other than whatever number we're looking to find. The percentile loves a number of numbers lower. We add 0.5 and divide by the total number of numbers that we have and then multiplied by 100. So in this particular problem, we have seven data values. So on is equal to seven. All right, so is we got through to find each percentile, we need to look at the number of numbers lower. So when I look at my smallest data value, there are zero data values lower than that. So zero plus 0.5 divided by seven and multiplied by 100 and that gives us 7.14 Since we don't have a seven point fourth per position, we go ahead and round that and this number would round down. So this would be approximately the seventh percentile and we continue on in that fashion for each. Each additional number that we find there is one more number that is lower than that. So it's sort of this pattern. So for 28 there is one number lower than 28 in our data set, and we're gonna divide that by seven, multiplied by 100 and that gives us 21 point for until again we round, and that's going to be the 21st percentile. And then 35 has two numbers smaller than it, so two plus zero point 5/7. Multiply that by 100 and we find out that's equal to 35.7, so that would round to the approximately 36 percentile. Now notice that the value of the number doesn't really have any relevance to it. It's just the position of the number that helps us identify percentile. So the 42nd are the value 42 is going to have three numbers less than it. When I add 30.5, I get 3.5 under by seven and multiply that by 100. That translates to be 50th percentile. It's the 40 said the data value 47 has four numbers when we add four numbers lower than it. When we add 40.5, divide by seven multiplied by 100 that becomes the 64th percentile, or we can see that that is the 64th. Percentile. 49 has five numbers lower than that. When we add fought 50.5 divide by seven multiplied by 100. That be we can see that that is the 79th percentile. And finally, the largest number has six numbers smaller than it. So 6.5, divided by seven multiplied by 100. And that is the 93rd percentile. So you may notice that there is a bit of a pattern, so seven plus 14 is 21 plus another. About 14 is 36 plus 14 plus 14 a little bit more than 14 plus 14. Like there's this pattern of adding 14 on. There is a reason for that. So if we take 100% of the numbers and we divide by seven, you'll see that that's a little over 14. It's 14.3, so each sing. Each one of those data values represents about 14.3% of the data, so That's why you see that pattern in the percentiles. Just a fun fact. All right. Next part of the question says, then, find the value of the number in the six or find the buy of the 60th percentile. So for finding the percentile, then we have to use this other formula. C is equal to end times p divided by 100. Where n is the number of numbers that we're looking for and P is the percentile that we're looking for, so C equals. We have seven numbers times. We're looking for the 60th percentile, divided by 100. So when we multiply about that that out, it comes out to 4.2. Well, we don't have a number in the 4.2 position because we only have whole numbers of positions the 1st 2nd 3rd etcetera. So we would round this and we always round up. So this is going to be the number that's in the fifth position. So if we go back over and look at our numbers that are in order, you've got 12345 This number is in the fifth position, so the 60th percentile is 47. Don't know where that line came from, so the 60th percentile is 47

Let's look at the statistics problems about percentile. But before I start I recommend students to pause the video, do the question themselves and come back to see if they've got a right or not. That is the true way of knowing if you fully understand this stuff. So let's start with what we're given. Were given a student's score on and examine the statistics class break And it is the 64th person type and recall that percentile is a value of like 100 right? It doesn't have to be because we're not given the number of students, it doesn't have to be 100 students. It could be 180 200. But if you draw a scale here, what else Let's say this is zero and this is the 100%ile then um student would be somewhere here, 64th%ile but it could be like 100 students, 180 students for example, It's just percentile is out of 100 because if you think about the work percent %,'s 100%. So and I feel like a box plot would help a lot in this situation. So let's look and act. So we have the options that his score could fall between the minimum and first court Tyler. Right? But if you look over here, so the first quartile is the uh huh. The fourth quarter is the 25th%ile. That means like there is 25% in this region. So it wouldn't make sense. And then there's another 25% between the first quartile and the median. So that adds up to 50%. Right? And then when you come from the media into the upper quartile is another 25%. So that adds up to 75%. And we can see from the image over here, it's written the Q three or the upper quartile, it's the 75%. So we can expect the student's score to be somewhere around here, right? It would be greater than the medium because the medium is the 50th%,ile, but it would be between the from the upper quartile Or the 3rd quartile and the media. So the final answer is because The 64 is between 50th%ile, of so it has to be between the third quarter out and the media. And I hope that was


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