\u03b1 \u03b2 \u03b1 \u03b1 \u03b2 \u03b1 \u03b1 \u03b2 \u03b1 \u03b1 \u03b2 \u03b1 \u03b1 \u03b2 \u03b1 This leads to the following rule for a 1 \u03b1 100

# Α β α α β α α β α α β α α β α this

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- - - - = - - - - - - = - - - - α β α α β α α β α α β α α β α This leads to the following rule for a (1- α )100% confidence interval for β 1 : } { ) 2 ; 2 / 1 ( 1 1 b s n t b - - ± α Some statistical software packages print this out automatically (e.g. EXCEL and SPSS). Other packages simply print out estimates and standard errors only (e.g. SAS).
Tests Concerning β 1 We can also make use of the of the fact that b s b t n 1 1 1 2 - - β { } ~ to test hypotheses concerning the slope parameter. As with means and proportions (and differences of means and proportions), we can conduct one-sided and two-sided tests, depending on whether a priori a specific directional belief is held regarding the slope. More often than not (but not necessarily), the null value for β 1 is 0 (the mean of Y is independent of X ) and the alternative is that β 1 is positive (1-sided), negative (1- sided), or different from 0 (2-sided). The alternative hypothesis must be selected before observing the data. 2-sided tests Null Hypothesis: 10 1 0 : β β = H Alternative (Research Hypothesis): 10 1 : β β A H Test Statistic: } { * 1 1 1 b s b t β - = Decision Rule: Conclude H A if ) 2 ; 2 / 1 ( | * | - - n t t α , otherwise conclude H 0 P -value: |) * | ) 2 ( ( 2 t n t P - All statistical software packages (to my knowledge) will print out the test statistic and P - value corresponding to a 2-sided test with β 10 =0. 1-sided tests (Upper Tail) Null Hypothesis: 10 1 0 : β β = H Alternative (Research Hypothesis): 10 1 : β β A H Test Statistic: } { * 1 1 1 b s b t β - = Decision Rule: Conclude H A if ) 2 ; 1 ( * - - n t t α , otherwise conclude H 0 P -value: *) ) 2 ( ( t n t P - A test for positive association between Y and X ( H A : β 1 >0) can be obtained from standard statisical software by first checking that b 1 (and thus t* ) is positive, and cutting the printed P -value in half.
1-sided tests (Lower Tail) Null Hypothesis: 10 1 0 : β β = H Alternative (Research Hypothesis): 10 1 : β β < A H Test Statistic: } { * 1 1 1 b s b t β - = Decision Rule: Conclude H A if ) 2 ; 1 ( * - - - n t t α , otherwise conclude H 0 P -value: *) ) 2 ( ( t n t P < - A test for negative association between Y and X ( H A : β 1 <0) can be obtained from standard statisical software by first checking that b 1 (and thus t* ) is negative, and cutting the printed P -value in half. Inferences Concerning β 0 Recall that the least squares estimate of the intercept parameter, 0 b , is a linear function of the observed responses n Y Y , , 1 : = = = = - - + = - = n i n i i i i n i i i Y l Y X X X X X n X b Y b 1 1 1 2 1 0 ) ( ) ( 1 Recalling that i i X Y E 1 0 } { β β + = : 0 1 0 1 1 1 2 2 1 0 1 1 2 1 1 1 2 1 0 1 0 1 2 0 )) 1 ( ( ) ( ) ( 1 ) 0 1 ( ) ( ) ( 1 ) ( ) ( 1 ) ( ) ( ) ( 1 } { β β β β β β β β β = - + = - - - + - = - - - + - - - = + - - - = = = = = = = = = = X X X X X X X X n X X X X X X n X X X X X n X X X X X X n b E n i n i n i i i i n i i n i i i n i n i i i n i i n i i i Thus, b 0 is an unbiased estimator or the parameter β 0.
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