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Suppose:

$H_0: \beta \le 0.5$

$H_1: \beta > 0.5$

$\widehat \beta = 0.49$

If $\widehat \beta = 0.49$ and I want to test the alternative hypothesis that $\beta > 0.5$ (versus the null that $\beta \le 0.5$). Won't it be the case that a one tailed t statistic will always fail to reject the null, given that the t-stat will be negative and to reject the null, the t-stat would need to be positive? This seems counterintuitive that we would never reject the null even if $\widehat\beta$ had a high standard error. Furthermore wouldn't a constructed confidence interval then suggest there are null hypotheses where beta $> 0.5$, which we would fail to reject (contrary to the calculated t-value)?

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  • $\begingroup$ Need context. Because you mention a t statistic, I suppose your data is normal. Then what is $\beta?$ the population mean? $\endgroup$
    – BruceET
    May 18 '17 at 20:55
  • $\begingroup$ The higher the standard error, the less likely you would be to reject the null hypothesis. $\endgroup$ May 21 '18 at 21:25
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Overall, the answer to the question in the title is: It depends. It depends on the distribution of $\hat \beta$ and the level of significance $\alpha$. But the general sense you are observing holds.

Now, to be specific, suppose $\frac{\hat \beta - \beta_0}{\text{SE}(\hat \beta)}$ has a central $t$ distribution (which is symmetric about 0). Then if $\hat \beta - \beta_0$ is negative in the case of the alternative hypothesis $\beta > \beta_0$, or if $\hat \beta - \beta_0$ is positive in the case of the alternative hypothesis $\beta < \beta_0$, the $p$-value will be at least 0.5. Unless we are willing to tolerate a large type-I error rate (i.e., an $\alpha$ even bigger than the $p$-value which already is higher than 0.5), yes, we will fail to reject $H_0$.

Philosophy of hypothesis testing gives more credence to $H_0$ in the sense that unless strong evidence (completely clearing "reasonable doubt") is presented by the data, we will not question $H_0$ which might be the current standard. This can be seen also from P(Type-I Error) being controlled first.

One last note: One cannot truly compare a 2-sided confidence interval to a 1-tailed hypothesis testing decision. But, to clear your intuitive thinking, if confidence interval overlaps then the hypothesis testing decision theory essentially says we cant be sure - so let's just not jump the gun (i.e., stick with $H_0$, our current belief state). Confidence interval will have to "completely clear the $H_0$ area" (loosely speaking) for us to feel strongly about the evidence provided by the data in the direction of the alternative hypothesis. Again, keep in mind, 2-sided CI cannot be exactly compared to 1-tailed test.

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