P-value

In Statistics, the researcher checks the significance of the observed result, which is known as test static. For this test, a hypothesis test is also utilized. The P-value or probability value concept is used everywhere in the statistical analysis. It determines the statistical significance and the measure of significance testing. In this article, let us discuss its definition, formula, table, interpretation and how to use P-value to find the significance level etc. in detail.

P-value Definition

The P-value is known as the probability value. It is defined as the probability of getting a result that is either the same or more extreme than the actual observations. The P-value is known as the level of marginal significance within the hypothesis testing that 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 at which the null hypothesis would be rejected. If the P-value is small, then there is stronger evidence in favour of the alternative hypothesis.

P-value Table

The P-value table shows the hypothesis interpretations:

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.

P-value Formula

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

Step 1: Find out the test static Z is

\(z = \frac{\hat{p}-p0}{\sqrt{\frac{po(1-p0)}{n}}}\)

Where,

\(\hat{p}\) = Sample Proportion

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.

P-Value Example

An example to find the P-value is given here.

Question: A statistician wants to test the hypothesis H0: μ = 120 using the alternative hypothesis Hα: μ > 120 and assuming that α = 0.05. For that, he took the sample values as

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

Solution:

We know that,

\(\sigma _{\bar{x}}=\frac{\sigma }{\sqrt{n}}\)

Now substitute the given values

\(\sigma _{\bar{x}}=\frac{32.17 }{\sqrt{40}}\) = 5.0865

Now, using the test static formula, we get

t = (105.37 – 120) / 5.0865

Therefore, t = -2.8762

Using the Z-Score table, we can find the value of P(t>-2.8762)

From the table, 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

Therefore, from the conclusion, if p>0.05, the null hypothesis is accepted or fails to reject.

Hence, the conclusion is “fails to reject H0.

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