How to Find a P-Value from a Test Statistic

The evidence in the trial is your data and the statistics that go along with it. All hypothesis tests ultimately use a p-value to weigh the strength of the evidence (what the data are telling you about the population). The p-value is a number between 0 and 1 and interpreted in the following way:

• A small p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, so you reject it.
• A large p-value (> 0.05) indicates weak evidence against the null hypothesis, so you fail to reject it.
• P-values very close to the cutoff (0.05) are considered to be marginal (could go either way). Always report the p-value so your readers can draw their own conclusions.

How to find a p-value from a test statistic

The following figure shows the locations of a test statistic and their corresponding conclusions.

Note that if the alternative hypothesis is the less-than alternative, you reject H₀ only if the test statistic falls in the left tail of the distribution (below –2). Similarly, if H_a is the greater-than alternative, you reject H₀ only if the test statistic falls in the right tail (above 2).

To find the p-value from your test statistic:

1. Look up your test statistic on the appropriate distribution — in this case, on the standard normal (Z-) distribution in the p-value charts (called Z-tables) below.

   
   

2. Find the probability that Z is beyond (more extreme than) your test statistic:

   1. If H_a contains a less-than alternative, find the probability that Z is less than your test statistic (that is, look up your test statistic on the Z-table and find its corresponding probability). This is the p-value. (Note: In this case, your test statistic is usually negative.)

   2. If H_a contains a greater-than alternative, find the probability that Z is greater than your test statistic (look up your test statistic on the Z-table, find its corresponding probability, and subtract it from one). The result is your p-value. (Note: In this case, your test statistic is usually positive.)

   3. If H_a contains a not-equal-to alternative, find the probability that Z is beyond your test statistic and double it. There are two cases:

      If your test statistic is negative, first find the probability that Z is less than your test statistic (look up your test statistic on the Z-table and find its corresponding probability). Then double this probability to get the p-value.

      If your test statistic is positive, first find the probability that Z is greater than your test statistic (look up your test statistic on the Z-table, find its corresponding probability, and subtract it from one). Then double this result to get the p-value.

Similarly, if you’ve got a negative test statistic, its p-value only accounts for the less-than portion of the not-equal-to alternative; double it to also account for the greater-than portion.

For example, when testing H_o: p = 0.25 versus H_a: p < 0.25, the p-value turns out to be 0.1056. This is because the test statistic was –1.25, and when you look this number up on the Z-table (in the appendix) you find a probability of 0.1056 of being less than this value. If you had been testing the two-sided alternative, Ha: p ≠ 0.25, the p-value would be 2 * 0.1056, or 0.2112.

If the results are likely to have occurred under the claim, then you fail to reject H_o (like a jury decides not guilty). If the results are unlikely to have occurred under the claim, then you reject H_o (like a jury decides guilty). The cutoff point between rejecting H_o and failing to reject H_o is another whole can of worms that I dissect in the next section (no pun intended).

Making Conclusions

While 0.05 is a very popular cutoff value for rejecting H_o, cutoff points and resulting decisions can vary — some people use stricter cutoffs, such as 0.01, requiring more evidence before rejecting H_o, and others may have less strict cutoffs, such as 0.10, requiring less evidence.

If H_o is rejected (that is, the p-value is less than or equal to the predetermined significance level), the researcher can say they've found a statistically significant result. A result is statistically significant if it’s too rare to have occurred by chance assuming H_o is true. If you get a statistically significant result, you have enough evidence to reject the claim, H_o, and conclude that something different or new is in effect (that is, Ha).

The significance level can be thought of as the highest possible p-value that would reject H_o and declare the results statistically significant. Following are the general rules for making a decision about H_o based on a p-value:

• If the p-value is less than or equal to your significance level, then it meets your requirements for having enough evidence against H_o; you reject H_o.
• If the p-value is greater than your significance level, your data failed to show evidence beyond a reasonable doubt; you fail to reject H_o.

You may be wondering whether it’s okay to say “Accept H_o” instead of “Fail to reject H_o.” The answer is a big no. In a hypothesis test, you are not trying to show whether or not H_o is true (which accept implies) — indeed, if you knew whether H_o was true, you wouldn’t be doing the hypothesis test in the first place. You’re trying to show whether you have enough evidence to say H_o is false, based on your data. Either you have enough evidence to say it’s false (in which case you reject H_o) or you don’t have enough evidence to say it’s false (in which case you fail to reject H_o).

Setting boundaries for rejecting H_o

• If the p-value is less than 0.01 (very small), the results are considered highly statistically significant — reject H_o.
• If the p-value is between 0.05 and 0.01 (but not super-close to 0.05), the results are considered statistically significant — reject H_o.
• If the p-value is really close to 0.05 (like 0.051 or 0.049), the results should be considered marginally ­significant — the decision could go either way.
• If the p-value is greater than (but not super-close to) 0.05, the results are considered non-significant — you fail to reject H_o.

When you hear a researcher say their results are found to be statistically significant, look for the p-value and make your own decision; the researcher’s predetermined significance level may be different from yours. If the p-value isn’t stated, ask for it.

Testing varicose veins

You didn’t have enough evidence to say the proportion of women with varicose veins is less than 0.25 (your alternative hypothesis). This isn’t declared to be a statistically significant result.

But say your p-value had been something like 0.026. A reader with a personal cutoff point of 0.05 would reject H_o in this case because the p-value (of 0.026) is less than 0.05. The reader's conclusion would be that the proportion of women with varicose veins isn’t equal to 0.25; according to H_a in this case, you conclude it’s less than 0.25, and the results are statistically significant.

However, a reader whose significance level is 0.01 wouldn’t have enough evidence (based on your sample) to reject H_o because the p-value of 0.026 is greater than 0.01. These results wouldn’t be statistically significant.

Finally, if the p-value turned out to be 0.049 and your significance level is 0.05, you can go by the book and say because it’s less than 0.05 you reject H_o, but you really should say your results are marginal, and let the reader decide.