How to Find p Value from Test Statistic

P-value (Probability Value) is the probability of obtaining the observed result, or a more extreme result, assuming that the null hypothesis (H₀) is true.

It helps determine whether the observed result is statistically significant.
A smaller p-value provides stronger evidence against the null hypothesis.

The following formula helps calculate the p-value in statistical hypothesis testing:

Test Statistic

A test statistic measures how closely our data matches the distribution predicted by the null hypothesis of the statistical test you’re using.

Steps to Find p-value from Test Statistic

Step 1: State the Hypotheses

First, define the null hypothesis (H₀) and the alternative hypothesis (H₁) based on the problem.

Example:

H₀: μ = 10 (The population mean is 10.)
H₁: μ ≠ 10 (The population mean is not 10.)

Where,

H₀ = Null hypothesis
H₁ = Alternative hypothesis
μ = Population mean

Step 2: Calculate the Test Statistic

Use the sample data to calculate the test statistic (t) using the following formula:

t = (x̄ − μ) / (s / √n)
Where,
t = Test statistic
x̄ = Sample mean
μ = Population mean under the null hypothesis
s = Sample standard deviation
n = Sample size

Step 3: Find the P-value

After calculating the t-value, use a t-distribution table or statistical software to find the p-value.

Calculate the degrees of freedom (df = n − 1).
Find the corresponding p-value using the calculated t-value and df.
For a one-tailed test, use the p-value directly.
For a two-tailed test, multiply the one-tailed p-value by 2.

Step 4: Make the Decision

Compare the p-value with the significance level (α), which is usually 0.05.

If p-value ≤ 0.05, reject the null hypothesis (H₀).
If p-value > 0.05, fail to reject the null hypothesis (H₀).

This decision helps determine whether the result is statistically significant.

P-value Table

The table below summarizes the interpretation of p-values in hypothesis testing.

P-value	Description	Decision / Interpretation
P-value < 0.05	Strong evidence against the null hypothesis.	Reject the null hypothesis (H₀).
P-value = 0.05	The result is at the significance level.	Decision depends on the chosen significance level (α).
P-value > 0.05	Weak evidence against the null hypothesis.	Fail to reject the null hypothesis (H₀).
P-value is close to 0.05	The result is considered marginal.	More data or further analysis may be required.

Practical Applications of p-value

Hypothesis Testing: Helps determine whether the observed results are statistically significant and whether to reject the null hypothesis.
Medical Research: Used to evaluate the effectiveness of new medicines, treatments, and clinical trials.
Engineering: Helps test the reliability of systems, evaluate new designs, and validate experimental results.
Business and Marketing: Used to compare strategies, analyze customer behavior, and make data-driven decisions.
Quality Control: Helps monitor product quality, identify defects, and improve manufacturing processes.

Example 1: A researcher wants to test whether the population mean is greater than 80. The hypotheses are:

H₀: μ = 80
H₁: μ > 80

The significance level is α = 0.05. A random sample of 36 observations is selected from a population with a known standard deviation of 18. The sample mean is found to be 86.

Using the p-value approach, determine whether the null hypothesis should be rejected or not rejected.

Solution:

Given,
n = 36
σ = 18
x̄ = 86
μ = 80
Step 1: Calculate the Standard Error
Standard Error = σ / √n
= 18 / √36
= 18 / 6
= 3
Step 2: Calculate the Test Statistic (Z)
Z = (x̄ − μ) / (σ / √n)
= (86 − 80) / 3
= 6 / 3
= 2.00
Step 3: Find the P-value
From the Z-table, P(Z > 2.00) = 0.0228
So, P-value = 0.0228
Step 4: Compare with the Significance Level
P-value = 0.0228
α = 0.05
Since 0.0228 < 0.05, we reject the null hypothesis (H₀).

Example 2: A statistical test gives a p-value of 0.12. If the significance level is 5% (α = 0.05), determine whether the null hypothesis should be rejected.