Introduction to calculating a p-value
The p-value is calculated using the sampling distribution of the test statistic under the null hypothesis, the sample data, and the type of test being done (lower-tailed test, upper-tailed test, or two-sided test).
- a lower-tailed test is specified by: p-value = P(TS
ts | H0 is true) = cdf(ts) - an upper-tailed test is specified by: p-value = P(TS
ts | H0 is true) = 1 - cdf(ts) - assuming that the distribution of the test statistic under H0 is symmetric about 0, a two-sided test is specified by: p-value = 2 * P(TS
|ts| | H0 is true) = 2 * (1 - cdf(|ts|))
- P
- Probability of an event
- TS
- Test statistic
- ts
- observed value of the test statistic calculated from your sample
- cdf()
- Cumulative distribution function of the distribution of the test statistic (TS) under the null hypothesis
Minitab automatically displays p-values for most hypothesis tests. But you can also use Minitab to “manually” calculate p-values. To manually calculate a p-value in Minitab:
- Choose .
- Choose Cumulative probability.
- Provide the parameters if necessary.
- Choose Input constant and enter the test statistic.
- Click OK.
- For a lower-tailed test, the p-value is equal to this probability; p-value = cdf(ts).
- For an upper-tailed test, the p-value is equal to one minus this probability; p-value = 1 - cdf(ts).
- For a two-sided test, the p-value is equal to two times the p-value for the lower-tailed p-value if the value of the test statistic from your sample is negative. However, the p-value is equal to two times the p-value for the upper-tailed p-value if the value of the test statistic from your sample is positive.
Example of calculating a lower-tailed p-value
Suppose you do a one-sample lower-tailed z test and the resulting value of the statistic calculated from the data is −1.785 (ts= −1.785). You want to calculate a p-value for the z-test.
- Choose .
- Choose Cumulative probability.
- If necessary, in Mean, enter 0 and, in Standard deviation, enter 1.
- Choose Input constant and enter –1.785.
- Click OK.
This value is the probability that the test statistic assumes a value equal to or less than that value actually observed based on your sample (under H0). P(TS < −1.785) = 0.0371. Therefore, the p-value = 0.0371.
Example of calculating an upper-tailed p-value
Now suppose you do a one-sample upper-tailed z test and the resulting value of the statistic calculated from the data is 1.785 (ts= 1.785). You want to calculate a p-value for the z test.
- Choose .
- Choose Cumulative probability.
- If necessary, in Mean, enter 0 and, in Standard deviation, enter 1.
- Choose Input constant and enter 1.785.
- In Optional storage, enter K1. Click OK. K1 contains the probability that the test statistic assumes a value equal to or greater than that value actually observed based on your sample (under Ho). P(TS < 1.785) = 0.9629. For an upper-tailed test, you need to subtract this probability from 1.
- Choose .
- In Store result in variable, enter K2.
- In Expression, enter 1-K1. Click OK.
- Choose .
- Choose K2. Click OK.
This value is the probability that the test statistic assumes a value equal to or greater than that value actually observed based on your sample (under H0). P(TS > 1.785) = 0.0371. Therefore, the p-value = 0.0371.
Note
Because the normal distribution is a symmetric distribution, you could enter −1.785 as the input constant (in step 4) and then you do not have to subtract the value from 1.
Example of calculating a two-tailed p-value
Suppose you perform a one-sample two-tailed z test and the resulting test statistic is 1.785 (ts= 1.785). You want to calculate a p-value for the z test.
- Since the calculated value of the test statistic from the sample is positive, calculate an upper-tailed p-value. When the calculated value of the test statistic from the sample is negative, calculate a lower-tailed p-value and in step 5 enter K2 in Optional storage. Click OK.
- This value is the p-value for a one-tailed test. For a two-tailed test, you need to multiply by this value by 2.
- Choose .
- In Store result in variable, enter K3.
- In Expression, enter 2*K2. Click OK.
- Choose .
- Choose K3. Click OK.
This value is 2 times the probability of that the test statistic assumes a value equal to or greater than the absolute value of that value actually observed based on your sample (under H0). 2* P(TS > |1.785|) = 2 * 0.0371 = 0.0742. Therefore, the p-value = 0.0742.
Note
Depending on the test or type of data, the calculations do change, but the p-value is usually interpreted the same way.
