Steve Smith Tuition: Maths Notes

Transcription

1 Steve Smith Tuitio: Maths Notes e iπ + 1 = 0 a 2 + b 2 = c 2 z +1 = z 2 + c V E + F = 2 Fidig Equatios of Regressio Lies Cotets 1 Itroductio A Quick Overview of Correlatio Positive Correlatio Negative Correlatio No Correlatio The Correlatio Coefficiet A Quick Overview of Regressio Correlatio ad Regressio Equatios The Product Momet Correlatio Coefficiet The Regressio Equatios Fidig Regressio Lie Equatios You Are Give S xy, S xx, x ad y You Are Give x 2, xy, x, y ad You Are Give the Data Poits Whe You Have a Facy Calculator

2 Prerequisites I am assumig that you have covered the essetials of correlatio ad regressio. I this documet we are goig to use what we kow about correlatio ad regressio to fid the equatio of the regressio lie (the best-fit straight lie) through some data. Notes Noe. Documet History Date Versio Commets 8th December Iitial creatio of the documet. 2

3 1 Itroductio Here we are just goig to go over the very rudimets of correlatio ad regressio, to remid ourselves what these terms mea. 1.1 A Quick Overview of Correlatio Two variables (x ad y, say) are said to be correlated if there is some patter, some relatioship, betwee the values of the two variables i a give set of (x, y) data Positive Correlatio So for example, there is said to be positive correlatio betwee the variables x ad y if a set of (x, y) data looks like Figure 1. y x Figure 1: Positive Correlatio It s a positive correlatio because, i geeral, y icreases as x icreases, so that if you were to draw a best-fit straight lie through this data, it would have a positive gradiet. 3

4 1.1.2 Negative Correlatio Ad there is said to be egative correlatio betwee the variables x ad y if a set of (x, y) data looks like Figure 2. y x Figure 2: Negative Correlatio It s a egative correlatio because, i geeral, y decreases as x icreases, so that if you were to draw a best-fit straight lie through this data, it would have a egative gradiet No Correlatio Ad there is said to be o correlatio betwee the variables x ad y if a set of (x, y) data looks like Figure 3. y x Figure 3: No Correlatio There is o correlatio because there is o relatioship betwee the x ad y values i this set of data. 4

5 1.1.4 The Correlatio Coefficiet Mathematicias like to compare thigs usig umbers. Like this ruler is loger tha that oe because the legth of this ruler is 1 m ad the legth of that ruler is 30 cm. Well, so it goes with correlatio. I order to compare the correlatios of two sets of data, mathematicias assig a umber to each set of data. They the just have to compare the umbers to compare the correlatios. Ad the ame of the umber that they give to a correlatio is called the correlatio coefficiet. The correlatio coefficiet is usually give the symbol r. There will be more about this later, but here, you just eed to kow that the value of the correlatio coefficiet is ay umber i the rage 1 to +1. A set of data that has perfect positive correlatio will have a correlatio coefficiet of +1; a set of data that has perfect egative correlatio will have a correlatio coefficiet of 1; ad a set of data that has o correlatio will have a correlatio coefficiet of A Quick Overview of Regressio I the above sectios I metioed a imagiary straight lie that we were drawig through our data: the best-fit straight lie. Well, give a set of data, regressio is just the process of fidig the best-fit lie through the data 1. So, for example, if we were to take the data from Figure 1, the we could draw a best-fit straight lie as i Figure 4. y x Figure 4: Positive Correlatio With Lie of Best-Fit 1 This best-fit lie does ot have to be straight, it ca be ay fuctio: a quadratic, cubic, expoetial, logarithmic, etc. But i this documet we are oly worried about straight lies through our data. That s bad eough! 5

6 2 Correlatio ad Regressio Equatios 2.1 The Product Momet Correlatio Coefficiet The bad ews: there are as may differet correlatio coefficiets as I ve had hot diers. The (relatively) good ews: there is a stadard correlatio coefficiet that is used by all mathematicias, ad it s called the product momet correlatio coefficiet (PMCC). More bad ews: you calculate the PMCC like this: r = S xy Sxx S yy (1) where ad where S xx = x 2 ( x) 2 S yy = y 2 ( y) 2 S xy = xy ( x)( y) (2) (3) (4) x = the sum of all the x values; (5) y = the sum of all the y values; (6) x 2 = the sum of the squares of all the x values; (7) y 2 = the sum of the squares of all the y values; (8) xy = the sum of: the x y values for all the data poits. (9) Horredous, or what? Oe sliver of good ews is that you do t have to remember ay of these equatios. They re all i the formula book for your exam. Phew! But you do have to kow how to use them. 2.2 The Regressio Equatios There s actually plety of good ews here! Oce you have goe through the pai of workig out S xx, S xy, ad S yy as described i Sectio 2.1, the actually fidig the equatio of the best-fit straight lie through your data is relatively easy. Here s how to do it. Remember that the geeral equatio of a straight lie is y = mx + c ad so it order to determie the equatio for our straight lie, we eed to fid the m (the gradiet) ad the c (the y-itercept). First, the m: m = S xy S xx (10) 6

7 Secod, the c. Oce we kow the gradiet, the to fid the c, we eed a poit that the lie goes through. Ad from the theory of all this, it turs out that the best-fit straight lie will always go through the poit (x, y) where x x = = the average of all the x values; (11) y y = = the average of all the y values. (12) Now remember that the way to fid the equatio of a straight lie if you kow the gradiet m ad a poit that the lie goes through, (x 1, y 1 ), is to use the equatio y y 1 = m (x x 1 ) So the equatio of the best-fit straight lie will be y y = S xy S xx (x x) (13) The formula for the equatio of a regressio lie is: y y = S xy S xx (x x ) Figure 5: Regressio Lie Formula 7

8 3 Fidig Regressio Lie Equatios Here we re goig to fid the equatios of regressio lies i a few differet situatios. How easy this will be depeds etirely o what iformatio you are give You Are Give S xy, S xx, x ad y OK, let s say that i a questio you were give this iformatio: S xx = 10 S xy = 8.6 x = 3 y = 2.76 Well i that case we ca just use the regressio formula equatio, Equatio (13), sice we kow everythig we eed from that formula: y y = S xy S xx (x x) so, puttig the umbers i: or, simplifyig a bit: Now if we multiply out the brackets: y 2.76 = (x 3) y 2.76 = 0.86 (x 3) y 2.76 = 0.86x 2.58 ad fially, addig 2.76 to both sides: Ad there s the equatio of the lie. y = 0.86x It s very rare to be give this kid of iformatio i a exam questio. You are much more likely to get somethig like... 8

9 3.2 You Are Give x 2, xy, x, y ad This sort of iformatio is very commoly give i exam questios. So if you were give: x 2 = 55 xy = 50 x = 15 y = 13.8 = 5 what do you do with that? Well, from the example from Sectio 3.1 what we eed are the values for S xx, S xy, x ad y. Well to get those we just use the equatios (7), (9), (11) ad (12): S xx = x 2 ( x) 2 = = 10 ad S xy = xy ( x) ( y) = 50 = ad to get x ad y we use x x = = 15 5 = 3 ad y y = = 13.8 = Ad ow we just follow the example of Sectio

10 3.3 You Are Give the Data Poits What if the oly thig that you were give was the data poits themselves? Say you were give: Table 1: Data Poits x y Now what? Well, from the example of Sectio 3.2, we ca see that the first step i the process is to fid x 2, xy, x, y ad. We ca do that directly: = 5 because there are 5 data poits. Now, x 2 = = 55 ad xy = (1 1) + (2 1.6) + (3 3.2) + (4 3.8) + (5 4.2) = 50 ad x = = 15 ad y = = 13.8 Ad ow we just follow the example of Sectio

11 3.4 Whe You Have a Facy Calculator I do t wat to boast, but I have a CASIO fx-cg 20. It s the best calculator you ca get. It does amazig thigs. Amogst the may amazig thigs it does is to perform a wide variety of statistical calculatios. For example, if I was give the data from Table 1, I ca eter the data ito a pair of lists i my calculator by goig to MENU 2 to get me ito the Statistical fuctios of the calculator, the just puttig the data ito List 1 (the x-values), ad List 2 (the y-values). I ow press CALC to do a calculatio o this data, the to do a regressio calculatio, the the ad it gives me (amog other thigs): REG X ax+b a = 0.86 b = 0.18 r = where the a ad the b are the gradiet ad y-itercept of the regressio lie respectively, ad r is the product momet correlatio coefficiet. Job doe. Oh, ad if you go back to where the Lists are displayed, ad press to draw this data, the GRAPH GRAPH1 the calculator will draw the data o a scattergraph. The press CALC the the ad it gives me (as before): X ax+b a = 0.86 b = 0.18 r = but the, press DRAW ad the calculator draws the lie of best fit o the scattergraph of the plots. 11