Statistically greater than zero at the 5 significance

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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
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If, at the chosen size (significance level) and alternative, we reject H 0 , the language is unambiguous. For example, “We reject H 0 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
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When we fail to reject H 0 (again, at the chosen size and the specific alternative), one might say “we accept H 0 .” This is less desirable than “we fail to reject H 0 (at the 5% significance level against a two-sided alternative).” We may not reject H 0 because we just do not have enough evidence; perhaps the sample size is small. That does not mean H 0 is true. 92
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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 0 : 1 and H 0 : 1.5 Neither of these can be rejected at anything close to .05, even using a one-sided alternative. 93
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. 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 ------------------------------------------------------------------------------ mean mean(cscrap) t 0.3620 Ho: mean -1.5 degrees of freedom 18 Ha: mean -1.5 Ha: mean ! -1.5 Ha: mean -1.5 Pr(T t) 0.6392 Pr(|T| |t|) 0.7215 Pr(T t) 0.3608 94
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The Rule of Two When the df are reasonably large – say df 30 or larger – a simple rule-of-thumb is often used to reject a null hypothesis: reject H 0 if the absolute value of the t statistic is greater than two. This rule comes from (1) Using a 5% size test and (2) Specifying a two-sided alternative. For such a test, the cv in the t distribution is about 2.04 when df 30 and reaches 2.00 at df 60. For larger df , the critical value is less then two, and converges to 1.96 – the critical value for the standard normal distribution. 95
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In fact, t df Normal 0,1 as df . With even modest sample sizes, one is safe obtaining critical values or computing p -values from the standard normal distribution. 96
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Practical versus Statistical Significance In almost all applications it is important to distinguish between the size of the estimated effect – say, of a job training program – and whether the estimate is statistically significant.
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