∙ nothing is gained by viewing these as being from

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Unformatted text preview: ∙ 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 On the Language of Hypothesis Testing ∙ When the null H : 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 : 1 against a two-sided alternative we say “ X ̄ is statistically different from one at the 5% significance level.” 89 ∙ If we reject H : ≤ 0 in favor of H 1 : 0 we might say “ X ̄ is statistically greater than zero at the 5% significance level,” with the qualifier “greater” indicating the one-sided alternative. Similar language can be used for different nulls, different alternatives, and different significance levels. 90 ∙ If, at the chosen size (significance level) and alternative, we reject H , the language is unambiguous. For example, “We reject H at the 5% significance level in favor of H 1 .” We might emphasize whether H 1 is one-sided or two-sided: “We reject the null hypothesis that the population mean is zero at the 5% level against the alternative that it is greater than zero.” 91 ∙ When we fail to reject H (again, at the chosen size and the specific alternative), one might say “we accept H .” This is less desirable than “we fail to reject H (at the 5% significance level against a two-sided alternative).” ∙ We may not reject H because we just do not have enough evidence; perhaps the sample size is small. That does not mean H is true. 92 ∙ In virtually any application, there will be many null hypotheses we cannot reject. But we cannot “accept” them all. ∙ In the job training example, consider the nulls H : − 1 and H : − 1.5 Neither of these can be rejected at anything close to .05, even using a one-sided alternative. 93 . ttest cscrap -1 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 -0.5576 Ho: mean -1 degrees of freedom 18 Ha: mean -1 Ha: mean ! -1 Ha: mean -1 Pr(T t) 0.2920 Pr(|T| |t|) 0.5840 Pr(T t) 0.7080 . ttest cscrap -1.5 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------------------------------------------------------------------------------...
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