80 even better is if we allow a computer package such

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Even better is if we allow a computer package such as Stata to compute a p -value. It automatically uses the appropriate t distribution. 81
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EXAMPLE : Effects of job training grant on worker productivity. Look at change from 1987 to 1988. Only 19 firms were both given a grant and reported a productivity measure in both years. The response variable is actually the scrap rate – number of items out of 100 that must be scrapped – which falls if productivity increases. 82
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. des Contains data from jtrain.dta obs: 157 vars: 4 19 Nov 2009 06:07 size: 2,669 (99.9% of memory free) ------------------------------------------------------------------------------- storage display value variable name type format label variable label ------------------------------------------------------------------------------- fcode float %9.0g firm code number grant byte %9.0g 1 if received grant clscrap float %9.0g change in log(scrap rate) cscrap float %9.0g change in scrap rate ------------------------------------------------------------------------------- Sorted by: fcode 83
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. list in 1/20 --------------------------------------- | fcode grant clscrap cscrap | |---------------------------------------| 1. | 410032 0 . . | 2. | 410440 0 . . | 3. | 410495 0 . . | 4. | 410500 0 . . | 5. | 410501 0 . . | |---------------------------------------| 6. | 410509 0 . . | 7. | 410513 0 . . | 8. | 410517 0 . . | 9. | 410518 0 . . | 10. | 410521 0 . . | |---------------------------------------| 11. | 410523 0 -.1823215 -.01 | 12. | 410529 0 . . | 13. | 410531 0 . . | 14. | 410533 0 . . | 15. | 410535 0 . . | |---------------------------------------| 16. | 410536 0 . . | 17. | 410538 0 .037179 .0999999 | 18. | 410540 0 . . | 19. | 410544 0 . . | 20. | 410546 0 . . | --------------------------------------- 84
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. tab grant 1 if | received | grant | Freq. Percent Cum. ------------ ----------------------------------- 0 | 121 77.07 77.07 1 | 36 22.93 100.00 ------------ ----------------------------------- Total | 157 100.00 . ttest cscrap  0 if grant One-sample t test ------------------------------------------------------------------------------ Variable | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] --------- -------------------------------------------------------------------- cscrap | 19 -1.303158 .5437056 2.369958 -2.445441 -.1608747 ------------------------------------------------------------------------------ mean mean(cscrap) t -2.3968 Ho: mean 0 degrees of freedom 18 Ha: mean 0 Ha: mean ! 0 Ha: mean 0 Pr(T t) 0.0138 Pr(|T| |t|) 0.0276 Pr(T t) 0.9862 85
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. ttest clscrap  0 if grant One-sample t test ------------------------------------------------------------------------------ Variable | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] --------- -------------------------------------------------------------------- clscrap | 19 -.3744936 .1460807 .6367511 -.6813978 -.0675894 ------------------------------------------------------------------------------ mean mean(clscrap) t -2.5636 Ho: mean 0 degrees of freedom 18 Ha: mean 0 Ha: mean ! 0 Ha: mean 0 Pr(T t) 0.0098 Pr(|T| |t|) 0.0195 Pr(T t) 0.9902 . log close 86
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The preceding example falls under what is sometimes viewed as a paired test of the equality of means from two populations. But one can always set up a paired test as standard test of the mean from a single population by defining X i as the difference between pairs. For example, suppose that the outcomes for pair i are Y i 1 , Y i 2 with 1 E Y 1 and 2 E Y 2 . We are interested in testing hypotheses about 2 1 , but this is E X i with X i Y i 2 Y i 1 So just define E X i 2 1 and test hypotheses about . 87
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In the job training example, we can take X i Y i ,88 Y i ,87 where Y i ,88 and Y i ,87 are the (log) scrap rates for firm i in the two years 1988 and 1987. Nothing is gained by viewing these as being from two different populations but where the observations are paired. We are interested in E X i directly. 88
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On the Language of Hypothesis Testing When the null H 0 : 0 is rejected at, say, the 5% significance level, we often say that “ X ̄ is statistically significant at the 5% level” or X ̄ is statistically different from zero at the 5% significance level.” If instead we reject H 0 : 1 against a two-sided alternative we say “ X ̄ is statistically different from one at the 5% significance level.” 89
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If we reject H 0 : 0 in favor of H 1 : 0 we might say “ X ̄ is
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