# p-Value in Hypothesis Testing

The p-value is the probability, computed using the test statistic, that measures the support (or lack of support) provided by the sample for the null hypothesis.

Once we have the test statistic, we look into the z-table to calculate a value greater than or less than that. This will depend on whether it’s a two-tailed or a one-tailed test.

### Lower-tailed Test

In a left-tailed hypothesis test, the decision rule will be as follows:

Assume that the level of significance is 10% (a = 0.10)

Under **critical value approach**, the decision rule will be as follows:

*Reject the null hypothesis if test-statistic < -1.28*

If the test statistic is -1.46, we will reject the null hypothesis.

We can calculate the p-value as follows: We will look for the value of F(Z) where Z=-1.46. This gives us p-value = 0.072.

Under **p-value approach**, the decision rule will be as follows:

Since p-value (0.072) is less than a (0.10), we reject the null hypothesis. This is illustrated below:

### Upper-tailed Test

In a right-tailed hypothesis test, the decision rule will be as follows:

Assume that the level of significance is 4% (a = 0.04)

Under **critical value approach**, the decision rule will be as follows:

*Reject the null hypothesis if test-statistic > 1.75*

*1.75 is the value of z in z-table where F (Z) is (1 – 0.04) = 0.96*

If the test statistic is 2.29, we will reject the null hypothesis.

We can calculate the p-value as follows: We will look for the value of F(Z) where Z=2.29. This gives us p-value = 0.011. *Note: At z=2.29, P(Z)=0.989. p-value = 1 – 0.989 = 0.011.*

Under **p-value approach**, the decision rule will be as follows:

Since p-value (0.011) is less than a (0.04), we reject the null hypothesis. This is illustrated below:

### Two-tailed Test

In a two-tailed hypothesis test, the decision rule will be as follows:

Assume that the level of significance is 3% (a = 0.03)

Under **critical value approach**, the decision rule will be as follows:

*Reject the null hypothesis if test-statistic <= -2.17 or test-statistic >=2.17*

*2.17 is the value of z in z-table where F (Z) is (1 – 0.015) = 0.985*

If the test statistic is -2.74 or 2.74, we will reject the null hypothesis.

We can calculate the p-value as follows: We will look for the value of F(Z) where Z = 2.74. *At z=2.74, P(Z)=0.9690. Probability of getting a value greater than 2.74 = 1-0.9690 = 0.0031. Since it’s a two tailed test, p-value = 2*0.0031 = 0.0062.*

Under **p-value approach**, the decision rule will be as follows:

Since p-value (0.0062) is less than a (0.03), we reject the null hypothesis. This is illustrated below:

- What is Hypothesis Testing
- Test Statistic, Type I and type II Errors, and Significance Level
- Decision Rule in Hypothesis Testing
- p-Value in Hypothesis Testing
- Selecting the Appropriate Test Statistic
- Hypothesis Testing with t-statistic
- Hypothesis Testing with z-statistic
- Tests Concerning Differences in Means
- Paired Comparision Tests - Mean Differences When Populations are Not Independent
- Hypothesis Tests Concerning Variances
- Chi-square Test – Test for value of a single population variance
- F-test - Test for the Differences Between Two Population Variances
- Non-parametric Tests

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