1
$\begingroup$

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?

$\endgroup$

1 Answer 1

1
$\begingroup$

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.

$\endgroup$
4
  • $\begingroup$ 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, 2019 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, 2019 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, 2019 at 16:35
  • 1
    $\begingroup$ @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, 2019 at 17:10

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .