PROBABILITY AND STATISTICS. Least Squares Regression

Transcription

1 PROBABILITY AND STATISTICS Leat Square Regreion

2 LEAST-SQUARES REGRESSION What doe correlation give u? If a catterplot how a linear relationhip one wa to ummarize the overall pattern of the catterplot i to ue a Leat-Square Regreion line. Leat-Square Regreion A method for finding a line that ummarize the relationhip between two variable

3 REGRESSION LINE Definition A traight line that decribe how a repone variable change a an explanator variable x change. Ued a a predictor Ued to predict the value of for a given value of x. Mathematical Model of the data Need to have an explanator and repone variable.

4 WHY USE LEAST-SQUARES REGRESSION TO FIND A LINE OF BEST FIT? If we all had a catterplot in front of u and a ruler to draw the line of bet fit for the data. How man line of bet fit would we get?

5 LEAST-SQUARES REGRESSION LINE We want to make the vertical ditance between our predicted - value and the actual -value to be a mall a poible.

6 LEAST-SQUARES REGRESSION LINE The line that make the um of the quare of the vertical ditance of the data point from the line a mall a poible. Equation: ˆ = a + bx

7 ŷ a bx Slope: b r x Intercept: a bx

8 EX 3.9 CONSTRUCTING THE LEAST-SQUARES EQUATION ŷ a bx where and a P153 If = , = , x = , = , and r = bx x b r x Calculate the leat-quare equation

9 EX 3.9 CONSTRUCTING THE LEAST-SQUARES EQUATION ANSWER ŷ a bx where and a P153 If = , = , x = , = , and r = bx x b r x b r x a bx o, ˆ x

10 WHAT DOES THE SLOPE AND Y-INTERCEPT TELL US? Slope? Remember back to our algebra da! -intercept?

11 USING THE EQUATION AS A PREDICTOR. Jut plug in a value for x and olve for.

12 EX 3.10 SMALL r 2 p158 Conider a imple example: Enter into calculator x Aociation appear poitive but weak Ue 2 Var Stat to find the mean x 3, 4 Ue 4 for the height Geometric quare contructed

13 EX 3.10 CONT D SST tand for total um of quare about the mean 2 SST ( ) Ue L3 to calculate. What do ou find? Calculate the LSRL (Leat Square Regreion Line) ŷ a bx where and ˆ 3 )x ( 1 3 a It ha a intercept at 3 Pae through the point ( x, ) (3,4) bx b r x Notice how it matche up?

14 EX 3.10 CONT D SSE tand for um of quare for error. SSE ( ˆ) 2 Ue L4 to calculate. Be careful with parenthee. You ma have to write it out on paper before putting into our calculator. You ma want to calculate one b hand to check if entered correctl. Finall, to find the % variation in, quare of correlation coefficient 2 SST SSE r SST 56 We a that 3.57% of the variation in i explained b leat-quare regreion of on x.

15 EX 3.10 LARGE r 2 p158 Your turn: Conider a imple example: Enter into calculator x Aociation appear poitive and trong Ue 2 Var Stat to find the mean x 5, 5 Ue 5 for the height Geometric quare contructed

16 EX 3.10 CONT D SST tand for total um of quare about the mean 2 SST ( ) Ue L3 to calculate. What do ou find? Calculate the LSRL (Leat Square Regreion Line) ŷ a bx where and ˆ x a It ha a intercept at 1 Pae through the point ( x, ) (5,5) bx b r x Notice how it matche up?

17 EX 3.10 CONT D SSE tand for um of quare for error. SST ( ˆ) 2 Ue L4 to calculate. Be careful with parenthee. You ma have to write it out on paper before putting into our calculator. You ma want to calculate one b hand to check if entered correctl. Finall, to find the % variation in, r 2 SST SSE SST We a that 84.2% of the variation in i explained b leat-quare regreion of on x. 6

18 ROLE OF r 2 IN REGRESSION r 2 i the coefficient of determination The fraction of the variation in the value of that i explained b the leat quare regreion of on x.

19 WHAT DOES r 2 TELL US? It i a meaure of how ucceful the regreion wa in explaining the repone. r 2 for thi graph i.5857 which mean that 58.57% of the variance in core on the exam for Coure 2 i accounted for or explained b the relationhip with core for Coure 1

20 FACTS ABOUT LEAST-SQUARES REGRESSION Fact 1: The ditinction between explanator and repone variable i eential in regreion. If ou were to witch the explanator and repone variable. You would get an entirel different LSRL. Remember: The LSRL i tring to be the bet predictor of the repone variable. Fact 2: There i a cloe connection between correlation and lope. A change of one S.D. in x correpond to a change of r S.D. in. Fact 3: The leat-quare regreion line alwa pae through the point( ) x,

21 FACTS CONTINUED Fact 4: The correlation r decribe the trength of the traight-line relationhip. In the regreion etting, thi decription take a pecific form: The quare of the correlation, r 2, i the fraction of the variation in the value of that i explained b the leat-quare regreion of on x. In other word, r 2 i the meaure of how good of a predictor i the LSRL of the repone variable.

22 MORE ON THE COEFFICIENT OF DETERMINATION(r 2 ) When ou report a regreion, give r 2 a a meaure of how ucceful the regreion wa in explaining the repone. When ou ee a correlation, quare it to get a better feel for the trength of the aociation. Turn to page 145 to refreh ourelf on the r correlation. The r 2 i the % of variation in a explained b leat-quare regreion of on x. So, if r = -0.7 or 0.7, then r 2 = 0.49 which mean that ½ of the variation in i explained b the leat-quare regreion of on x.

23 WORKTIME 3.38,