P-value Formula


The P-value is short for probability value. It defines the probability of getting a result that is either the same or more extreme than the other actual observations. The P-value represents the probability of occurrence of the given event. The P-value is used as an alternative to the rejection point to provide the least significance for which the null hypothesis would be rejected. The smaller the P-value, the stronger is the evidence in favor of the alternative hypothesis given observed frequency and expected frequency.

P-value Description Hypothesis Interpretation
P-value ≤ 0.05
It indicates the null hypothesis is very unlikely.
Rejected
P-value > 0.05
  It indicates the null hypothesis is very likely.
Accepted or it “fails to reject”.
P-value > 0.05 The P-value is near the cut-off. It is considered as marginal The hypothesis needs more attention.

What Is the P-value Formula?

P-value is an important statistical measure, that helps to determine whether the hypothesis is correct or not. P-value always only lies between 0 and 1. The level of significance(α) is a predefined threshold that should be set by the researcher. It is generally fixed as 0.05. The formula for the calculation for P-value is:

Step 1: Find out the test static Z is

\(Z = \frac{\hat{p}-p 0}{\sqrt{\frac{p 0(1-p 0)}{n}}}\)
Where,

\(\hat{p}=\) Sample Proportion
\(\mathrm{P0}=\) assumed population proportion in the null hypothesis

N = sample size

Step 2: Look at the Z-table to find the corresponding level of P from the z value obtained.

 

Solved Example Using P-value Formula

Example 1: 

A statistician is testing the hypothesis H0: μ = 120 using the approach of alternative hypothesis Hα: μ > 120 and assuming that α = 0.05. The sample values that he took are as 

n =40, σ = 32.17 and x̄ = 105.37. What is the conclusion for this hypothesis?

Solution:

We know that,
\(\sigma_{\bar{x}}=\dfrac{\sigma}{\sqrt{n}}\)
Now substitute the given values
\(\sigma_{\bar{x}}=\dfrac{32.17}{\sqrt{40}}=5.0865\)

As per the test static formula, we get

t = (105.37 – 120) / 5.0865

Therefore, t = -2.8762

Using the Z-Score table, finding the value of P(t > -2.8762)

we get,

P (t < -2.8762) = P(t > 2.8762) = 0.003

Therefore,

If P(t > -2.8762) =1 - 0.003 =0.997

P- value =0.997 > 0.05

As the value of  p > 0.05, the null hypothesis is accepted.

Answer: Null hypothesis is accepted.

 

Example 1: 

What conclusion should be made with respect to an experiment when the significance level is 0.05 (α = 0.05)?

Solution:

Since the p-value of 0.068 is greater than α = 0.05, it would fail to reject the null hypothesis.

Answer: As the value of p < 0.05, the null hypothesis is rejected.