Why Use a t Table to Bound a P-Value?
In “Finding a P-Value Using tcdf” and “Using the T-Test Function on a Calculator,” you learned to obtain a p-value from a t distribution using technology. If a calculator is not available, a t table can still tell you how large or small the p-value is. Usually the observed t statistic will fall between two values printed in the table, so you can report a range rather than an exact p-value.
The table does not replace the test setup. You still need the alternative hypothesis to identify the relevant tail, and you need the correct degrees of freedom. For a one-sample t test, calculate \(df=n-1\), as discussed in “Why the t Test Uses n Minus 1 Degrees of Freedom.” Then locate the row for that df and compare the absolute value of the observed t statistic with the positive t values in that row.
Read the Table Before Comparing Values
Many t tables label columns with upper-tail probabilities. In such a table, a positive t value in a row for a given df has the stated probability to its right. For example, if the entry under the upper-tail probability \(0.05\) is \(1.860\), then a t statistic of \(1.860\) has area \(0.05\) to its right for that df.
Here is a small excerpt from a table whose columns are upper-tail probabilities. The critical values are rounded to three decimals, as they commonly are in printed tables.
| df | Upper-tail 0.10 | Upper-tail 0.05 | Upper-tail 0.025 | Upper-tail 0.01 | Upper-tail 0.005 |
|---|---|---|---|---|---|
| 8 | 1.397 | 1.860 | 2.306 | 2.896 | 3.355 |
| 9 | 1.383 | 1.833 | 2.262 | 2.821 | 3.250 |
| 15 | 1.341 | 1.753 | 2.131 | 2.602 | 2.947 |
Always read the headings printed on the table you are given. Some tables show two-tail probabilities instead of upper-tail probabilities, and a table may display only a selection of probability columns. Do not assume a heading means “area in one tail” until you have checked its label.
For an upper-tailed test, compare the positive observed t statistic directly with the row’s positive values. For a lower-tailed test with a negative observed statistic, use symmetry: the area to the left of a negative t equals the area to the right of its positive opposite. For a two-sided test, first find the one-tail area corresponding to the absolute value of t, then double the bounds.
Worked Examples
Worked Example: Bounding a Lower-Tailed P-Value
A fictional greenhouse randomly selects 9 plants from a group of 150. The mean time for these plants to reach a target height is \(\bar{x}=11.2\) days, with \(s=1.5\) days. The greenhouse wants to test whether the true mean time for plants in this group is less than 12 days. Assume the population distribution of times is approximately Normal. Bound the p-value using the table excerpt.
State. Let \(\mu\) be the true mean number of days for plants in this group to reach the target height. The hypotheses are \(H_0:\mu=12\) days and \(H_a:\mu<12\) days.
Plan and check conditions. The plants were randomly selected, supporting the Random condition. For the 10% condition, \(0.10(150)=15\), and \(9\leq15\), so independence is reasonable. Since \(n=9<30\), the large-sample route is not met; the stated approximately Normal population supports the Normal/Large Sample condition. Use a one-sample t test with \(df=9-1=8\).
Do. The standard error is \(s/\sqrt{n}=1.5/\sqrt{9}=0.5\) days. The observed statistic is
This is a lower-tailed test and \(t\) is negative. By symmetry, the p-value is the same as the upper-tail area to the right of \(1.60\) with \(df=8\). In the df 8 row, \(1.397<1.60<1.860\). Those table values correspond to upper-tail areas \(0.10\) and \(0.05\), respectively. A larger t value has a smaller upper-tail area, so
Conclude. Assuming the true mean time is 12 days, the probability of obtaining a t statistic of \(-1.60\) or less is between \(0.05\) and \(0.10\). The table bounds the p-value; it does not give its exact value.
Worked Example: Bounding an Upper-Tailed P-Value
A fictional workshop randomly samples 16 parts from a shipment of 300. Their mean length is 42.3 millimeters and their sample standard deviation is 4 millimeters. The workshop tests whether the true mean length in this shipment exceeds 40 millimeters. Assume the population distribution of lengths is approximately Normal. Use the table excerpt to bound the p-value.
State. Let \(\mu\) be the true mean length, in millimeters, of parts in this shipment. The hypotheses are \(H_0:\mu=40\) millimeters and \(H_a:\mu>40\) millimeters.
Plan and check conditions. Random sampling supports the Random condition. The 10% condition is met because \(0.10(300)=30\) and \(16\leq30\), supporting independence. Since \(n=16<30\), the large-sample route is not met; the stated approximately Normal population supports the Normal/Large Sample condition. The degrees of freedom are \(df=16-1=15\).
Do. The standard error is \(4/\sqrt{16}=1\) millimeter, so the test statistic is
This is an upper-tailed test, so the p-value is the area to the right of \(2.30\). In the df 15 row, \(2.131<2.30<2.602\). The upper-tail headings for these table values are \(0.025\) and \(0.01\). Because \(2.30\) is between the two t values, its upper-tail area is between the two corresponding probabilities:
Conclude. Assuming the true mean length is 40 millimeters, the probability of obtaining a t statistic of \(2.30\) or greater is between \(0.01\) and \(0.025\). That is the p-value range for the test that the mean length exceeds 40 millimeters.
Worked Example: Bounding a Two-Sided P-Value
A fictional recreation group randomly selects 9 participants from a group of 120 and records their weekly training hours. The sample mean is 6 hours and the sample standard deviation is 1.2 hours. The group tests whether the true mean differs from 5 hours. Assume the population distribution is approximately Normal. Use the table excerpt to bound the p-value.
State. Let \(\mu\) be the true mean weekly training time, in hours, for participants in this group. The hypotheses are \(H_0:\mu=5\) hours and \(H_a:\mu\ne5\) hours.
Plan and check conditions. The participants were randomly selected, supporting the Random condition. For the 10% condition, \(0.10(120)=12\), and \(9\leq12\), so independence is reasonable. Since \(n=9<30\), use the stated approximately Normal population to support the Normal/Large Sample condition. The degrees of freedom are \(df=9-1=8\).
Do. The standard error is \(1.2/\sqrt{9}=0.4\) hours. The observed statistic is
For a two-sided test, use \(|t|=2.50\) to find the area in one tail. In the df 8 row, \(2.306<2.50<2.896\), corresponding to upper-tail probabilities \(0.025\) and \(0.01\). Therefore, the one-tail area is between \(0.01\) and \(0.025\). A two-sided p-value includes equally extreme results in both tails, so double both bounds:
Conclude. Assuming the true mean weekly training time is 5 hours, the probability of obtaining a t statistic at least as far from 0 as \(2.50\), in either direction, is between \(0.02\) and \(0.05\). This is the p-value range for testing whether the mean differs from 5 hours.
Common Mistakes and AP Exam Tips
- Using the wrong df row. For a one-sample t test, use \(df=n-1\), not \(n\). Show how you found df so the table lookup is easy to verify.
- Reading the wrong direction. In an upper-tail table, larger positive t values correspond to smaller areas. If t falls between two entries, the probability bounds go in the reverse order from the t values.
- Ignoring the alternative hypothesis. A lower-tailed test with a negative t uses symmetry. A two-sided test requires doubling the one-tail area. Do not double a one-sided p-value.
- Reporting a table entry as the exact p-value. Unless the observed statistic matches a printed t value, the table usually supports a range. State both bounds and use the correct inequality signs.
- Overlooking the table heading. Verify whether the columns show one-tail or two-tail probabilities. If your table gives two-tail areas, do not double them again.
- Forgetting the meaning of the p-value. A full-credit statement identifies the probability, assumes \(H_0\) is true, describes results at least as extreme as observed, and gives the context. The p-value is not the probability that \(H_0\) is true.
When table values are rounded, treat the resulting bounds as approximate bounds. On an exam, a clear comparison such as “\(2.131<2.30<2.602\), so \(0.01<p<0.025\)” shows both the lookup and the reasoning. Do not claim more precision than the table provides.
Check Your Understanding
Use the table excerpt above, and give a p-value range rather than an exact value.
- A lower-tailed test has \(df=9\) and \(t=-1.50\). Between which two upper-tail columns does the p-value fall?
- An upper-tailed test has \(df=15\) and \(t=2.00\). What range can you report for the p-value?
- A two-sided test has \(df=8\) and \(t=-2.50\). What is the one-tail area range, and what is the resulting two-sided p-value range?
- Why must you check whether a table heading gives a one-tail or two-tail probability before using it?
- A one-sample t test has \(n=21\). What degrees of freedom should you use to find the row in the t table?