Two-sided $t$-test

Null and alternative hypotheses:

\[\begin{split} H_0: \hat\beta = \beta \\ H_1: \hat\beta \neq \beta \end{split}\]

Test statistic:

\[t=\frac{\hat\beta-\beta}{SE(\hat\beta)} \sim t_{n-k-1}\]

$P$-value: Also called the significance probability, the probability of observing a statistic at least as different from $\beta$ as the actually estimated value in your sample, assuming the null hypothesis is true.

Stated mathematically,

\[p\text{-value} = 2P(T>|t|)\]

Because $\hat\beta$ is approximately normally distributed in large samples,

\[p\text{-value} = 2\Phi(-|t|)\]

A $p$-value of less than 5% provides evidence against the null hypothesis in the sense that, under the null hypothesis, the probability of obtaining a value of $\hat\beta$ at least as different from $\beta$ as the actually estimated value is less than 5%. If so, the null hypothesis is rejected at the 5% significance level.

One-sided $t$-test

Q: Why should a one-sided test be used?
A: In practice, one-sided alternative hypotheses should be used only when there is a clear reason for doing so. This reason could come from economic theory, prior empirical evidence, or both. Or it could be we are only interested in an effect in one direction, not the other.

  • The direction we are interested should be in the alternative hypothesis.
  • One-tailed test improve the power of a test, that is correctly reject $H_0$ if the null hypothesis is truly false.

Ex1: we compare the mean strength of parts from a supplier (102) to a target value (100). We are considering a new supplier only if the mean strength of their parts is greater than our target value. There is no need for us to distinguish between whether their parts are equally strong or less strong than the target value — either way we’d just stick with our current supplier.

$H_0$: new supplier = target value
$H_1$: new supplier > target value

Ex2: We want to know if the battery life is greater than the original after a manufacturing change.

$H_0$: new battery life = original life
$H_1$: new battery life > original life

If we have reason to believe that the true value of $\beta$ should be negative, then we can perform a one-sided $t$-test. The null and alternative hypotheses are:

\[\begin{equation}\label{one-sided-hypothesis} \begin{split} H_0: \hat\beta = \beta \\ H_1: \hat\beta < \beta \end{split} \end{equation}\]

Because the null hypothesis is the same for a one- and a two-sided $t$-test, the test statistic is the same as well.

The only difference is how you interpret the $t$-statistic. For the one-sided $t$-test in $\eqref{one-sided-hypothesis}$, the null hypothesis is rejected against the alternative hypothesis for large negative values, but NOT for large positive values, of the $t$-statistic. That is, the hypothesis is rejected if $t < c_{\alpha}.$ For large samples, the hypothesis is rejected if $t<-1.64.$

The $p$-value for the one-sided left-tail test is obtained as

\[p\text{-value} = P(T<t)\]

For large samples,

\[p\text{-value} = \Phi(t).\]

If the one-sided test is right-tailed, the $p$-value is the right-tail probability:

\[p\text{-value} = P(T>t)\]

For large samples,

\[p\text{-value} = 1-\Phi(t).\]

Commonly used critical values for standard normal distribution:

  • $c_{0.5\%}=2.58,$ corresponding to a 99% confidence level or a 1% significance level
  • $c_{2.5\%}=1.96,$ corresponding to a 95% confidence level or a 5% significance level
  • $c_{5\%}=1.64,$ corresponding to a 90% confidence level or a 10% significance level