I have done a one-tailed one-sample t-test with the following setup:

$H_0:\overline{x} = \mu = \$151$

$H_A: \overline{x} < \mu $

$n = 24$

For $\alpha = 0.05$, the t-critical value is $-1.711$ (from the t-table, using degrees of freedom $= 24$), and I calculate $\overline{x} = \$126$ (skipping the details of the calculation). I get t-statistic $=-2.5$ so I reject the $H_0$. So far, so good.

Now, the problem asks to calculate the confidence interval using $\alpha = 0.05$.

I know that the margin of error $\overline{x} = \pm t{\text -}critical \times standard \ error$.

The answer to the problem uses the two-tailed t-critical value, which is $2.064$, for $95\%$ confidence interval.

I am confused about why they use the two-tailed t-critical value rather than the one-tailed one?

Is it because calculating the confidence interval is a separate task, and has nothing to do with the original hypothesis, and they are interested to see the deviation from the mean in both directions?


Your use of notation is strange. The hypothesis should be written $$H_0 : \mu = 151 \quad \text{vs.} \quad H_a : \mu < 151.$$ The hypothesis is a statement about the value of the unknown parameter, in this case the mean $\mu$. We make an inference about its value based on the data, from which we calculate a test statistic, which under the null hypothesis is $$T \mid H_0 = \frac{\bar x - 151}{s/\sqrt{n}},$$ where $\bar x$ is the sample mean and $s$ the sample standard deviation. A statistic is never included in the hypothesis because it is something that is calculated from the data; there is no uncertainty about the value we get for it.

Given that $\bar x = 126$ and the value of your test statistic is $T = -2.5$, I get $s \approx 48.9898$ (the exact value of which may differ due to the limited precision you reported for the test statistic).

As for the calculation of a confidence interval, this need not be one-sided even if the hypothesis is one-sided. The two concepts are related, but a confidence interval is nothing more than an interval estimate. So, rather than using the sample mean as a simple point estimate for the true mean, we can incorporate the variability observed in the data to give a more sophisticated estimate of the true mean. You can calculate a two-sided interval or a one-sided interval. The question as stated is not specific about which one is intended.

A two-sided $95\%$ interval is computed as $$\bar x \pm t_{n-1,\alpha/2}^* \frac{s}{\sqrt{n}},$$ where $s/\sqrt{n}$ is the standard error of the mean, and $t_{n-1,\alpha/2}^* $ is the upper $\alpha/2$ quantile of the student $t$ distribution. In your case, it is $$t_{23,0.025}^* = 2.06866.$$ Thus the two-sided interval is $$[105.31, 146.69].$$ The one-sided interval that is in the same direction as the hypothesis test is $$\left(-\infty, \bar x + t_{n-1,\alpha}^* \frac{s}{\sqrt{n}}\right].$$ Here, $t_{23,0.95}^* = 1.71387$ so the upper confidence limit is $143.14$, which is smaller than the upper confidence limit of the two-sided interval. This makes sense because we selected the one-sided upper confidence limit in such a way that the probability the interval does not contain the true mean is $\alpha$; i.e., the upper tail probability of the student $t$ distribution is the full $0.05$, rather than the equal-tailed, two-sided interval.

  • $\begingroup$ Is the decision to calculate one-sided or two-sided CI left purely to the statistician, depending purely on what they are interested to see? $\endgroup$ – Saskia Nov 23 '19 at 13:12
  • $\begingroup$ @Saskia one-sided CIs are rarely reported because one-sided hypotheses are uncommon in practice. To frame it another way, a two-sided test at significance $\alpha$ is the same as a one-sided test at significance $\alpha/2$. So you might as well test the two-sided hypothesis or construct a two-sided CI. $\endgroup$ – heropup Nov 23 '19 at 14:49
  • $\begingroup$ I am confused about something else: if we run a one-tailed test, shouldn't we calculate the one-tail CI as well? Just for consistency. I was confused that in the solution, they used a one-tailed test, while then calculated a two-tailed CI. Seemed inconsistent to me, while I understand that it might be ok to do that in general. However, I think it would incorrct to have it another way round: to calculate a one-tailed CI for two-tailed test. $\endgroup$ – Saskia Nov 23 '19 at 16:35
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    $\begingroup$ @Saskia As I stated in my answer, the question is not specific about whether to construct a one- or two-sided CI. I can't tell you more because I don't have more information. All I can say is that two-sided intervals are far more common. There is no inconsistency because you are assuming that if a hypothesis is one-sided, the interval estimate must also be so; and this is not the case. It is true that the inversion of the one-sided hypothesis is analogous to the one-sided CI, but there is no requirement that both the test and the CI must be calculated the same way. $\endgroup$ – heropup Nov 23 '19 at 17:10

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