Using a T Table to Bound a P-Value

Why Bound a P-Value?

A one-sample t table often gives selected critical values rather than a p-value for every possible test statistic. It can still tell you how small or large the p-value is: compare the observed statistic with the critical values listed for the test’s degrees of freedom. The result is usually an interval, such as \(0.025 < p < 0.05\), rather than an exact p-value.

The interval is useful because it places the evidence on a meaningful scale and lets you compare the result with a significance level. The key is to use the correct degrees of freedom and match the table’s tail convention to the alternative hypothesis. A correct comparison with the wrong tail can produce a misleading answer.

The One-Sample t Statistic and Its Reference Distribution

For a one-sample test of a population mean, let \(\mu_0\) be the mean specified by the null hypothesis, \(\bar{x}\) the sample mean, \(s\) the sample standard deviation, and \(n\) the sample size. The test statistic measures how many estimated standard errors separate the observed sample mean from \(\mu_0\).

Definition: For a one-sample t test of \(H_0:\mu=\mu_0\), the test statistic is \(t=(\bar{x}-\mu_0)/(s/\sqrt{n})\), and its degrees of freedom are \(df=n-1\). Under the null model and the test’s conditions, the statistic follows a t distribution with \(n-1\) degrees of freedom.

The t reference model is appropriate when observations are independent and arise from a random sample or a suitable randomized design. For a small sample, the population distribution should be approximately normal, without severe skewness or outliers. For a larger sample, the sample mean is generally more robust to moderate departures from normality, although severe skewness and influential outliers still deserve attention. These are conditions for the test, not consequences of looking up a value in a table.

For a one-sample test, the degrees of freedom are always \(n-1\). They do not depend on the value of the sample mean or the direction of the alternative hypothesis. A table row for the wrong degrees of freedom changes the critical values and therefore can change the p-value bounds.

How to Read an Upper-Tail t Table

A common t table lists positive critical values \(t^*\) by degrees of freedom and upper-tail area. In that format, the entry in the row for \(df\) and the column for \(\alpha\) is defined by

P(T_{df}>t^*)=\alpha

Here \(T_{df}\) is a random variable with a t distribution having \(df\) degrees of freedom. For example, the table entry \(t^*=1.833\) at \(df=9\) and upper-tail area \(0.05\) means that a t value greater than 1.833 has probability 0.05 under that distribution. It does not mean that the probability of falling between zero and 1.833 is 0.05.

The t distribution is symmetric around zero. Thus an upper-tail table can also be used for a left-tailed test: the lower-tail critical value for area \(\alpha\) is the negative of the positive upper-tail critical value for area \(\alpha\). For a two-sided test, the p-value includes extreme values in both tails. If the observed statistic is \(t_{\text{obs}}\), then

p_{\text{two-sided}}=2P(T_{df}\geq |t_{\text{obs}}|)

The factor of 2 applies because the two tails have equal probability at equally distant positive and negative values. Check the table heading carefully: some tables show one-tail areas, while others show two-tail areas. A column labelled “two-tail probability” already refers to both tails; do not double that entry again.

Alternative hypothesis	Observed statistic to compare	Tail probability from an upper-tail table
\(H_a:\mu>\mu_0\)	\(t_{\text{obs}}\)	\(P(T_{df}\geq t_{\text{obs}})\)
\(H_a:\mu<\mu_0\)	\(-t_{\text{obs}}\), using symmetry, or compare with negative critical values	Lower-tail probability for \(t_{\text{obs}}\)
\(H_a:\mu\neq\mu_0\)	\(\|t_{\text{obs}}\|\)	\(2P(T_{df}\geq \|t_{\text{obs}}\|)\)

Why Comparing Critical Values Bounds the P-Value

For a right-tailed test, the p-value is the area to the right of the observed statistic. As that statistic moves farther to the right, the area beyond it gets smaller. This decreasing-tail property is what makes a table of a few critical values enough to bound the p-value.

Suppose a table gives critical values \(c_{\text{large}}\) and \(c_{\text{small}}\) for upper-tail areas \(\alpha_{\text{small}}\) and \(\alpha_{\text{large}}\), respectively, where \(\alpha_{\text{small}}<\alpha_{\text{large}}\). The critical value for the smaller area is farther to the right, so \(c_{\text{large}}>c_{\text{small}}\). If the observed statistic lies between them, then the p-value lies between the corresponding areas.

\alpha_{\text{small}}<\alpha_{\text{large}},\qquad c_{\text{small}}<t_{\text{obs}}<c_{\text{large}} \quad\Longrightarrow\quad \alpha_{\text{small}}<P(T_{df}\geq t_{\text{obs}})<\alpha_{\text{large}}

To justify the implication, define the upper-tail probability \(S(t)=P(T_{df}\geq t)\). If \(a<b\), the event \(T_{df}\geq b\) is contained within the event \(T_{df}\geq a\), so \(S(b)\leq S(a)\). Since a t distribution is continuous, the tail area strictly decreases as the cutoff increases. Given \(c_{\text{small}}<t_{\text{obs}}<c_{\text{large}}\), it follows that \(S(c_{\text{large}})<S(t_{\text{obs}})<S(c_{\text{small}})\). Substituting the table-defined areas gives the stated p-value bounds. For a two-sided test, apply this argument to \(|t_{\text{obs}}|\) and then double both one-tail bounds.

Worked Examples

Worked Example: Bounding a Right-Tailed P-Value

In a hypothetical sample of 10 independently selected packages, the mean fill amount is 53.2 grams and the sample standard deviation is 4.9 grams. The test is \(H_0:\mu=50\) grams against \(H_a:\mu>50\) grams. Assume the sampling and distribution conditions are reasonable: the packages were selected independently, the sample is small, and the measurements show no severe skewness or outliers. Bound the p-value using an upper-tail t table.

First calculate the standard error and test statistic:

SE=\frac{s}{\sqrt{n}}=\frac{4.9}{\sqrt{10}} =\frac{4.9}{3.1623}\approx 1.549\text{ grams}

t_{\text{obs}}=\frac{\bar{x}-\mu_0}{SE} =\frac{53.2-50}{1.549} =\frac{3.2}{1.549}\approx 2.066

The degrees of freedom are \(df=10-1=9\). For a right-tailed test, use the positive statistic as the upper-tail cutoff. At \(df=9\), the table gives a critical value of 1.833 for upper-tail area 0.05 and 2.262 for area 0.025. Because \(1.833<2.066<2.262\), the p-value is between those areas:

$$ 0.025<p<0.05 $$

The data are inconsistent with the null model at a level of 0.05 because the p-value is below 0.05. The table does not establish an exact p-value; it establishes this range.

Worked Example: Bounding a Two-Sided P-Value

A hypothetical random sample of 15 independent measurements has \(\bar{x}=18.45\) units and \(s=2.4\) units. Test \(H_0:\mu=20\) units against \(H_a:\mu\neq20\) units. Suppose the sample values are reasonably symmetric with no severe outliers, supporting the small-sample t procedure. Use the table to bound the two-sided p-value.

Calculate the standard error and test statistic:

SE=\frac{2.4}{\sqrt{15}} =\frac{2.4}{3.8730}\approx 0.6197\text{ units}

t_{\text{obs}}=\frac{18.45-20}{0.6197} =\frac{-1.55}{0.6197}\approx -2.501

The degrees of freedom are \(df=15-1=14\). Since the alternative is two-sided, use the magnitude \(|t_{\text{obs}}|=2.501\). At \(df=14\), the critical value for upper-tail area 0.025 is 2.145, and the critical value for upper-tail area 0.01 is 2.624. Therefore, the one-tail area beyond 2.501 is between 0.01 and 0.025. Doubling these bounds gives the two-sided p-value range:

0.01<P(T_{14}\geq 2.501)<0.025

$$ 0.02<p<0.05 $$

The two-sided p-value is between 0.02 and 0.05. This range accounts for results at least as far from 20 units as the observed sample mean, in either direction. Reporting the one-tail range as the final two-sided p-value would understate the p-value by a factor of two.

Worked Example: Bounding a Left-Tailed P-Value

In a hypothetical study, 20 independent devices have a mean operating time of 98.7 hours and a sample standard deviation of 3.8 hours. The test is \(H_0:\mu=100\) hours against \(H_a:\mu<100\) hours. Assume the devices were randomly selected, the measurements are independent, and their distribution has no severe skewness or outliers. Find a table-based p-value bound.

The standard error and test statistic are

SE=\frac{3.8}{\sqrt{20}} =\frac{3.8}{4.4721}\approx 0.8497\text{ hours}

t_{\text{obs}}=\frac{98.7-100}{0.8497} =\frac{-1.3}{0.8497}\approx -1.530

The degrees of freedom are \(df=20-1=19\). For a left-tailed test, the relevant area is to the left of \(-1.530\). At \(df=19\), the positive critical values for upper-tail areas 0.10 and 0.05 are 1.328 and 1.729. By symmetry, the corresponding lower-tail critical values are \(-1.328\) and \(-1.729\). Since \(-1.729<-1.530<-1.328\), the area to the left of the observed statistic is between 0.05 and 0.10:

$$ 0.05<p<0.10 $$

The negative sign matters: the observed statistic lies in the direction specified by the alternative, so this is a small-to-moderate left-tail area. Using the positive value 1.530 directly as though this were a right-tailed test would instead calculate the opposite tail.

Common Mistakes and Reporting

A p-value bound is a statement about the tail probability under the null model. For example, \(0.025<p<0.05\) says that, if the null hypothesis and the test conditions hold, the probability of a test statistic at least as extreme in the specified direction or directions is between 0.025 and 0.05. It is not the probability that the null hypothesis is true, nor is it the probability that chance alone caused the data.

Using the wrong row. For a one-sample test, calculate \(df=n-1\), then use that row. The sample size itself is not the degrees of freedom.
Using the wrong tail. A right-tailed test uses the right tail, a left-tailed test uses the left tail, and a two-sided test includes both. Follow the alternative hypothesis, not just the sign of the statistic.
Doubling twice or not at all. If the table gives one-tail areas, double the relevant area for a two-sided p-value. If its column is already labelled two-tail probability, do not double it again.
Calling a bound an exact p-value. If the statistic lies between two critical values, report the corresponding interval. Do not report one of the table’s column headings as though it were the exact p-value.
Ignoring table precision. Critical values are usually rounded. If a statistic is extremely close to a listed value, recognize that rounding limits the precision of the bound rather than claiming a sharper result.

A clear report names the direction of the test, the degrees of freedom, the statistic, and the p-value range. For example: “For the right-tailed one-sample t test, \(t=2.066\) with \(df=9\), and the table gives \(0.025<p<0.05\).” Add the conclusion in context and compare the range with the chosen significance level, without implying that the p-value is an effect size or a measure of practical importance.

Key takeaway: Find \(df=n-1\), identify the tail from \(H_a\), and compare the appropriate statistic (or its absolute value) with critical values in that row. The tail areas attached to the critical values then give a justified p-value interval.

Check Your Understanding

Use the table format and reasoning from this tutorial to answer the following.

A one-sample test has \(n=12\). What degrees of freedom should be used, and why?
For a right-tailed test, the observed statistic falls between the critical values for upper-tail areas 0.10 and 0.05. What does that imply about the p-value?
A two-sided test has \(t_{\text{obs}}=-2.3\). Which statistic should be compared with the positive critical values in an upper-tail table, and why?
A left-tailed test has a negative observed statistic. Explain how symmetry lets you use a table that lists only positive critical values.
A table’s heading says “two-tail probability.” What should you check before deciding whether to double a listed probability?