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Answers

Following are the American College Test (ACT) scores attained by the 25 members of a local high school graduating class: $$\begin{array}{lllllllllllll} \hline 21 & 24 & 23 & 17 & 31 & 19 & 19 & 20 & 19 & 25 & 17 & 23 & 16 \\ 21 & 20 & 28 & 25 & 25 & 21 & 14 & 19 & 17 & 18 & 28 & 20 & \\ \hline \end{array}$$ a. Draw a dotplot of the ACT scores. b. Using the concept of depth, describe the position of 24 in the set of 25 ACT scores in two different ways. c. Find $P_{5}, P_{10}$ and $P_{20}$ for the ACT scores. d. Find $P_{99}, P_{90},$ and $P_{80}$ for the ACT scores.

And this question were given 25. 80 c a c t scores of graduating seniors and were asked to make a box plot. So this is part a of the question. So I've already drawn my scale, noting that my lowest number is 14 and my highest number, My high scores 31. So when I looked at the numbers and I arrange them in order from smallest to largest, I saw that there was one score of 14 some in a place an X there for a dot plot. You can use dots or exes, but they all just show up a little bit better, I think in this platform, using an X so ah, one score. 14. No scores of 15 1 score 16 three of 17. So I'll just stack those up, trying to take about the same amount of space for each x one of 18 four at 19 three at 20 three. At 21 no scores a 22 two scores, a 23 one a 24 three a 25 to a 28 and one at 31. So those exes represent each of our 25 different scores. So for Part B were asked to identify in two different ways. Where the number 24 where the data value of 24 um, where it is in terms of depth. In other words, how deep into our data is that looking for the position that so we're look right here. Here's our number 24 and we want to count up in two different ways. So from the lowest and down from the highest. So if we count up one, 2345 6789 10 11 12 13 14 15 16 17 18 We find that it is the 19th number from the lowest number. So the 19 and then if we do the same thing but count down from the highest we see one, 234567 We see that it is the seventh number from the highest, so we can identify it by counting up or by counting down from either of the extremes. Part C asks us to identify the percentile, so we're gonna need to know this formula and times K over 100 is how we can identify what percentile where K is the case percentile and n is the number of numbers. So we know that there are 25 scores. So in this case and equals 25 K is gonna be the little sub script that we have there on P, which stands for percentile. So the first question asks us to identify the fifth percentile. So we have 25 numbers times five, which is K and divide by 100. So we're essentially doing is saying Okay, five over 100 is 5% as a decimal 50.5 I'm multiplying that by 25. So that's going to give us than 5% of 25 which is 1.25 In other words, there 1.25 numbers lower than the fifth percentile. So we're going around that up then to the second position. So if we look on our are set of data here, we see that the second position right there, it's 16. So the pit, the fifth percentile is 16. The next percentile that were asked to identify is the 10th So we'll repeat the same procedure and we see that this is 2.5, which means we are looking for the number that's in the third position. So 123 it's gonna be that number 17 and then were asked to identify the 20th percentile. So again, 25 times 20 over 100. And that turns out to be five. So that means there are five numbers less than that. So we're not looking for the number in the fifth position, but the number that has five numbers less than that. So that means we don't want to go all the way to the sixth number. We want to go halfway between the fifth and the sixth. So we have to identify both the fifth and the sixth number and then take their average. So if we got 123456 my fifth number is 17 and my six number is 18. So I'm gonna take the average of 17 and 18 and intuitively in the middle of that is going to be 17 0.5. So the fifth percentile is 17.5. Okay. For part D were asked to find three more percentiles, but now at the upper end of the percentages, So higher numbers. The 1st 1 that were asked to find is the 99th percentile. The process doesn't change those, so we're still going to use the same formula. So 25 is still end. We're looking for the 99th percentile, and that comes out to be 24.75 So, in other words, were looking for the number in the 25th position. And that makes sense that our largest number would be in the 99th percentile because 99% of the other data values would be lower than that. Now we're asked to find the 90th percentile. Same procedure. That's a nine all nines there, cleaning it up, just a smidge, and that number turns out to be 22.5. So again we'll round up. So we're looking for the number in the 23rd position. So when we're looking for these percent house at the upper end, this 23rd represents references the 23rd number from the lowest. But that's also the third number, down from the highest, which might be easier to identify. So if we kind of count backwards, if that's 25th this will be 24th and this would be then the 23rd number, so that would be 28 and finally were asked to find the 80th percentile. So same procedure. And this turns out to be 20. So again, we've got a number that is Ah, whole number. So that means we're looking for the number that is between the 20th and the 21st. So if we look at that, we'll see. OK, this is 25 24 23 22 2120. These are the 20th and the 21st numbers, and both of them are 25. So halfway between there is gonna be 25.

Using Chevy Chest Theorem, you're going to look at, ah, couple of different means and standard deviations and solved to see what proportions of what percentages would fall within certain standard deviations. So for this example, we're gonna look at averages of test scores, and our mean score is going to be 20 0.8 for a mean and our senior deviation is going to be 4.6. So let's say that the score that you would get that averages 20.8 points and the standard deviation is 4.6. So the variability between the means So what would it look like? Or what percentage of these scores would be between 11? So we're looking for between 11.6 and 30 points, okay? And we're gonna look at at least what percentage falls within those 11.6 and 30 points. So using Chevy Test serum, the equation uses one minus one over K squared. So we're gonna look at what it would mean for at least so two standard deviations. So it would be one minus 1/2 squared or one minus 1/4 and out equal 2.75 So at least 75% would fall within two standard deviations off the mean so at least 75% would be between the sex. And then you would have the left hand side. So the negative to center deviations and the positive two standard deviations of the right hand side. So on the negative here on the side doesn't always mean negative. It just means on the lower end of the mean, so that would be the 11.6 and then at the top of the positive or above the mean we would have 30. So 75% would fall within those No, if we know juxtaposed Chevy Tabs. If we know that this is normally distributed, what percentage then can we say would fall within? So if we remember what normal distribution would look like, it would be this bell curve, and it's symmetrical and are mean or mu be right smack in the middle there. Okay, So based off of that clear based off of what we know for normal distributions, we know that within who's gonna draw this all out? The standard deviations. But we have listed out right there within two that we've already found within these two, based on normal distributions and the pure horrible, we know that 95% falls within two senior deviations. So we can conclude. Since we already know that it's two standard deviations from the mean. 95% would score within 11.6 and the 30 points so again to write that out, it would be 11.6. And then on the positive side of two standard deviations, it would be 30. So you can see the juxtaposition of the differences between Chevy Chevez, which is saying 75% um, versus what we're seeing in the normal distribution of 95%. Okay, so that's what I would look like for the two different theories.

It just said here that the mean for the average score for this group prices from 25 to 30 and that is only a 0.3% chance of getting those results by chance. So yes, looking at this statement, this result is statistically significant. Does the course appear to be effective when the sample size is small? But in real life, it is quite possible that the score might rise. So yes, it is. We can say that it is also practically significant, but again, we have to take note that the sample size in this case is very small.

First step, we stayed off. Problem. We need to find out who has better study habits and ridiculous that worth learning. Women or men, please. On this course we have for 18 women, number of women equals. Eating on number of me equals 20. We have 20 scores off Ben, and these scores high scores indicate go to study habits and attitudes. It work learning. This is our problem. Second, we're going to use statistical tools to solve this problem best. It'll for this is creating the mean value of both. You know, the average, the school of women and as and every school off men Number three, we do our calculations for women. It's got great high school of women, equals some nation off. All these numbers provided, boy, they're going If we can create a submission 2000 559 and work with the current, which is this? Gifts 141 score off each woman For men, export equals the submission off all that which gives 2000 and 400 25 avoided. Boy 20 equals 121. The fourth we conclude using these numbers as we see the mean value of women or greater than mean very with men. This in the kids that the debt has aboard the belief that woman have better study habits and attitudes toward delivering then.


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