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Use the given data set to complete parts (a) through (c} below: (Use & = 0.05.)12.756.42Click here to view table of critical values for the correlation coeffici...

Question

Use the given data set to complete parts (a) through (c} below: (Use & = 0.05.)12.756.42Click here to view table of critical values for the correlation coefficient:Construct scatterplot Choose the correct graph below:b. Find the linear correlation coefficient; then determine whether there is sufficient evidence to support the claim of linear correlation between the two variables.The linear correlation coefficient is (Round to three decimal places as needed:)Using the linear correlation coeff

Use the given data set to complete parts (a) through (c} below: (Use & = 0.05.) 12.75 6.42 Click here to view table of critical values for the correlation coefficient: Construct scatterplot Choose the correct graph below: b. Find the linear correlation coefficient; then determine whether there is sufficient evidence to support the claim of linear correlation between the two variables. The linear correlation coefficient is (Round to three decimal places as needed:) Using the linear correlation coefficient found in the previous step, determine whether there sufficient evidence two variables_ Choose the correct answer below: support the claim of linear correlation between the



Answers

Explore provide two data sets from "Graphs in Statistical Analysis," by $F .$ J. Anscombe, the American Statistician, Vol. $27 .$ For each exercise,a. Construct a scatterplot. b. Find the value of the linear correlation coefficient $r$, then determine whether there is sufficient evidence to support the claim of a linear correlation between the two variables. c. Identify the feature of the data that would be missed if part (b) was completed without constructing the scatterplot.$$\begin{array}{l|c|c|c|c|c|c|c|c|c|c|c}\hline x & 10 & 8 & 13 & 9 & 11 & 14 & 6 & 4 & 12 & 7 & 5 \\\hline y & 9.14 & 8.14 & 8.74 & 8.77 & 9.26 & 8.10 & 6.13 & 3.10 & 9.13 & 7.26 & 4.74 \\\hline\end{array}$$

Here we have a set of data and using what we know about linear regression, coefficient of correlation and hypothesis testing, we're going to determine whether or not we can conclude from the set that there is a significant linear correlation. If we were to input this data into our graphing calculator and graph the scatter plot of it, we would get something that looks similar to this, which in essence, is fairly linear. So it seems as though using the coefficient of correlation will indeed be a good measure of correlation so long as it's significant, considering that the coefficient of correlation is best used in linear functions going ahead and also in your graphing calculator running the linear Regression T test, we'll find that we have in our value So our coefficient of correlation, which is equal to 0.816 That's a relatively strong and positive correlation here. But in order to determine whether or not that significant, we also need to look at our P value That was out that our calculators gave us. So here it has given us a P value equal to 0.2 If we were testing this at a significance level of 0.5 we could indeed include that. This is significant. So our coefficient coefficient of correlation equal 2.816 is indeed a good measure of the correlation here. And if we had completed this without first using a scatter plot, we may have tried to use our r squared value, which may not have been accurate or if we had instead seen that are scatter. Plot was not linear. If it had, you know, a curve to it, a non linear graph, then we would have erroneously concluded that the coefficient of correlation was a good measure. But in this case, we actually did not miss anything. So that is good news, and we can conclude that there is a strong and positive correlation between X and y here.

In problem 48. We're going to be considering some given by various data of X and Y, and we're going to use this deter to construct this cut of Bagram. After that, we're going to walk out this PM on rank correlation coefficient. Then we won't got the Pearsons correlation coefficient. And lastly, we compare results for those two coefficients now. So the first thing we need to do to get a scatter diagram Mr Selected to no said so deter and then impressing front. Then a good chance and pick this counter. Bagram. So this is a scatter diagram for the to the by. Very deter. So good into this. Get a diagram. You can see that it looks like a reflection off. One. The values look like like the Y axis looks like it's a mirror. It's actually cutting this too into to huffs that are equal so but this doesn't show any specific. Any interesting relationship between the two are any positive linear relationship between the two. Next Input B three. Calculate the Spearman's rank correlation coefficient, and this is a formula that in two years, so we need to get the rungs for X on the wrong small. Why, then the D squared. So let's get their runs equals Young, which or X then, like the entire column. Impressive. Four. Then see Drew. And those are the rankings for from the X, Not for Why do the same thing drops of Rich by into reference. Press, therefore, and zero wreck it and that's it. So way have good unique rankings. 1.51 and three point fact, because we have to taste next. We get, describe and describe. It is equal. It's the part two and from North. So now we need to get this some so that we substitute the values into this equation. The some off, um, the sample dese Go ahead. So the some of this crowd here is obtained from from some all those values men Tom his name. Then we want to get therefore the answer for the entire part here. So it's going to be one minus one minute. Six times nine divided by four times forced Gladys 16 minus 1 15 So it's going to be one minus 54 divided by 60. Okay, good luck. That out here so equals one minus before, So that's going to be zero point one. It's not with Yeah, So it's exactly 0.1. Next we can We can't let the Pearsons correlation coefficient. And this time we're going to be using the rankings as X and why not the actual value? So at this time, this will be excellence will be. Why? So we substitute the values into the formula. So what? We'll need a few more columns you need the column for X squared next business half to You'll also need for rice. Go ahead. Off to that Would lead a column for X Y. Okay, thumps Hummel Rex uh, some of why and also the some off the violence that we're going to introduce here. Okay, so let's touch with extra two equals four racing evolved to down and then y squared equals one from five rigs off to X Y. You calls four times 1.5 medical on the so that we can start from getting the value off the sample scares. So, Mr Tell It or mix any good to the song as my and that's thanks by it's toast as Vex three with some off X squared, which is 30 minus some of X 10 squared feted by n five As off Why, with the equal to some of wise grand just 29 minus the sum of Why Scrat Bed by n Now we go to the summer scraps of X y So it's equal some off X. Why just 25 minus um four x So that's going to be these 10 times. And I did buy and peaceful yes, have everything that we need to our coats, the PS once correlation coefficient. And in this case, you see that the some off this almost because for X Y is zero. And that means that even in our fraction, I will be zero because zero divided by any number will still be zero. So we don't need to key in the money's here. Just take it to be zero. So the Pearsons correlation coefficient r equals zero. While the Spearman's rank correlation coefficient equals 0.1, they're not equal, but they're very similar. 0.1 shows a very weak positive correlation between X and Y, and zero shows are no correlation between X and y. So the question on another question we need to answer is what weather the missions of coalition measure the same thing So wild. UH, Pearson Frankel Relation coefficient. End of the Spearman's rank correlation coefficient. Spearman Correlation coefficient Both check for a relationship between two variables. You'll find that the Spearmon Runk correlation coefficient determines the the strength off a mono tonic relationship, which means that it could be in the same direction. The relationship could be in the same direction, but not the constant Street. So yes, that's right has detected that there's a small relationship. But when we look at the Pearson correlation coefficient, it looks for both as the linear relationship which is focusing on both redirection and concentrate so on. That is right. It has a value very blue scope at lower than the one for this payment correlation coefficient because it focuses on both fixed direction and fix. Treat this Spearmon drunk Correlation focuses on direction on Lee, regardless off the rate

We have this set of six data points. And if we plot them, you can see that we get this distribution here. And you know, well, it's kind of not really good. There seems to be a positive correlation here. I would say that there's some positive correlation between X and Y. So that generally as X increases, why increases? So you'd expect our line are linear fit to go kind of through here. We can see that the parameters for the linear fit, the slope is actually one, so it is positive. So that as a positive correlation and then our is 0928, which is not too bad, but again, not great. And as you can see that this this kind of some weirdness going on here. Now, if we put our trend line in there linear trendline and you can see that, you know, it tries to do what it can with these two data points, but there's something weird going on here. And so we have this is a trend and r squared is 0.86 which is just, you know, this value squared. And that's, you know, that's starting to get on the verge of baby. Maybe there's a lot of noise in the data or, um, maybe there's just really not a linear, a linear function between these two variables and the model. She needs to be, um, more complicated, more sophisticated than that. Uh, nothing much more to say there. But the Senate is definitely a positive correlation here.

Well, given six data points. And let's see here, they ask us to look at the correlation here and then calculate it. So what we can see as if we plot these six points, they look completely uncalled. There's really seems like there's no linear correlation whatsoever. Um You know, there's, you know, these ones look maybe, but then what do you do with these looks more like a V. So if you look at the, you know, we can obviously fit a line through here as best we can, and that actually has a positive slope. So, you know, the line is saying that there's kind of some positive correlation. But if you look at um you look at the R value, which is the core of the correlation between the data here, um you know, it's very close to zero, which means that it's that the linear function, that the linear model is a very poor approximation of this data. And we can actually we can put the linear trendline on here and see how where it goes, and so there it is. And, you know, it tries. But again, it's you can see this r squared value is very low, which means the same is just the square. This value, which just means that it's very it's a very bad fit. So any correlation you can get out of this because of, you know, this is being positive or this being positive really is meaningless. There's basically no correlation here between the X and the Y.


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