# Manually calculate a p-value

## Introduction to calculating a p-value

The p-value is calculated using the test statistic calculated from the samples, the assumed distribution, and the type of test being done (lower-tailed test, upper-tailed test, or two-tailed test).

The p-value for:
• 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)
• a two-sided test is specified by: p-value = 2 * P(TS > |ts| | H0 is true) = 2 * (1 - cdf(|ts|))
Where:
P
Probability of a random variable having a range of values.
TS
Random variable associated with the assumed distribution.
ts
The test statistic calculated from your sample.
cdf()
Cumulative density function of the assumed distribution.

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:

1. Choose Calc > Probability Distributions > Choose the appropriate distribution.
2. Choose Cumulative probability.
3. Provide the parameters if necessary.
4. Choose Input constant and enter the test statistic.
5. Click OK.
The result is the probability of observing a random variable less than the test statistic, cdf(ts).
• 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 an upper-tailed test, the p-value is equal to two times the p-value for the lower-tailed test if the test statistic is negative, and for the upper-tailed test if the test statistic is positive; p-value = 2 * (1 - cdf(|ts|)).

## Example of calculating a lower-tailed p-value

Suppose you do a one-sample lower-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.

1. Choose Calc > Probability Distributions > Normal.
2. Choose Cumulative probability.
3. If necessary, in Mean, enter 0 and, in Standard deviation, enter 1.
4. Choose Input constant and enter –1.785.
5. Click OK.

This value is the probability of observing a random variable less than the test statistic, 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 test statistic is 1.785 (ts= 1.785). You want to calculate a p-value for the z test.

1. Choose Calc > Probability Distributions > Normal.
2. Choose Cumulative probability.
3. If necessary, in Mean, enter 0 and, in Standard deviation, enter 1.
4. Choose Input constant and enter 1.785.
5. In Optional storage, enter K1. Click OK. K1 contains the probability of observing a random variable less than your test statistic, P(TS < 1.785) = 0.9629. For an upper-tailed test, you need to subtract this probability from 1.
6. Choose Calc > Calculator.
7. In Store result in variable, enter K2.
8. In Expression, enter 1-K1. Click OK.
9. Choose Data > Display Data.
10. Choose K2. Click OK.

This value is the probability of observing a random variable greater than the test statistic, 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.

1. Since the test statistic is positive, calculate an upper-tailed p-value. When the test statistic is negative, calculate a lower-tailed p-value and in step 5 enter K2 in Optional storage. Click OK.
2. 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.
3. Choose Calc > Calculator.
4. In Store result in variable, enter K3.
5. In Expression, enter 2*K2. Click OK.
6. Choose Data > Display Data.
7. Choose K3. Click OK.

This value is 2 times the probability of observing a random variable greater than the absolute value of the test statistic, 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.

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