Week8_Notes

# Week8_Notes - Psych 100A Week 8 Discussion Notes Dawn Chen...

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Psych 100A Week 8 Discussion Notes Dawn Chen November 20, 2009 Correlation When it is impractical or unethical to manipulate one variable (e.g., gender, level of smoking) to see if it causes a change in another variable, we cannot determine if there is a causal relationship between the two variables. However, we can still measure the two variables and determine whether they are simply related by calculating the correlation coefficient between them. The sample correlation coefficient , denoted by r , tells us the direction and strength of the linear relationship between two variables X and Y in a sample of ( x , y ) values 1 . r can take values in the range [–1, 1]. Below are some possible (though sometimes unlikely) scenarios for the sample data and their corresponding r values. Note that r can be 0 even if there is a relationship between X and Y , if that relationship does not have a linear component (as in scenario (d)). Hypothesis testing also applies to r . In this case, we usually want to know whether r is significantly different from 0, or in other words, whether there exists a linear relationship between X and Y . This hypothesis is expressed in terms of , ρ the population correlation coefficient. 0 : 0 and : 0. A H H = The equation for r given in lecture was: ( 29 ( 29 1 1 1 n i i i x y x x y y r n s s = - - = - Doing some algebraic manipulations, we have: ( 29 ( 29 ( 29 ( 29 1 1 1 1 1 1 n i i n i i i i x y x y x x y y x x y y r n s s n s s = = - - - - = = - - ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 1 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 n n i i i i i i n n n n i i i i i i i i x x y y x x y y n n x x y y x x y y n n n = = = = = = - - - - = = - -  - - - -   - - - 1 In these notes, an upper-case letter (such as X ) denotes a variable and the corresponding lower-case letter (such as x ) denotes a particular value of the variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) r = 1 (b) r = –1 (c) r = 0 (d) r = 0 (e) r = 0.8 .

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( 29 ( 29 ( 29 ( 29 ( 29( 29 1 2 2 1 1 n i i XY i n n X Y i i i i x x y y CP SS SS x x y y = = = - - = =  - -   In the last formula, ( 29 ( 29 1 n XY i i i CP x x y y = = - - is called the cross products of X and Y , ( 29 2 1 n X i i SS x x = = - is the sum of squares for X , and similarly, ( 29 2 1 n Y i i SS y y = = - is the sum of
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Week8_Notes - Psych 100A Week 8 Discussion Notes Dawn Chen...

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