Critical Value Method and p-value Method to Hypothesis Testing | Two-Tailed Test

In this article, we will learn how to apply Hypothesis Testing to the two-tailed test using the critical value method and p-value method step by step. Also, we will learn how to look at the z-table. It is highly recommended to go through the basics of Hypothesis Testing first.

6 Steps of Hypothesis Testing

Below is the 6-step process to solve any problem using Hypothesis Testing.

Step 1: Formulate Null Hypothesis (H0)

Step 2: Formulate Alternative Hypothesis (H1)

Step 3: Set Significance Level (α)

Step 4: Find the Right Test

Step 5: Execute the Test and calculate the test statistics value

Step 6: Make a Decision

We will solve a problem to understand the above steps.

Example of Hypothesis Testing

A random sample of 10 individuals was drawn from the population of interest which has a mean of 27. Assuming that the population is approximately normally distributed with a variance of 20. Perform a Hypothesis Test with a significance level of 0.05 to conclude that the mean is different from 30 years.

Solution

• sample size, n=10
• sample mean, =27 
• population variance, =20
• significance level, α=0.05 

Step 1: Formulate Null Hypothesis (H0)

• H0 =  μ=30
• where μ is the population mean

Step 2: Formulate Alternative Hypothesis (H1)

•  H1 = μ 30

Step 3: Set Significance Level (α)

• α=0.05, given in the question

Step 4: Find the Right Test

• Since population variance is known and data is drawn from a single population, the z-test can be used.

Step 5: Execute the Test and calculate the test statistics value

• z-statistics =  
• where
  = sample mean
  μ = population mean
  σ = population standard deviation
  n = sample size
• z-statistics = = -2.12

Step 6: Make a Decision

• Decision:  There can be two final decisions.
  1. Accept Null Hypothesis
  2. Reject Null Hypothesis
• Using the value of test statistics (calculated in the previous step), there are two ways of making a decision.
  1. Critical Value Method
  2. p-value Method

1. Critical Value Method       

In the critical value method, we find out if the test statistics value lies in the rejection region or not.

Rejection Region

• The rejection region is present at the extreme end of population distribution.
• The area of the rejection region depends on the level of significance α.
• For the two-tailed test, the rejection region is present on both ends.
• If the value of test statistics lies in the rejection region, we reject the Null Hypothesis.

Critical Value

• It is the cutoff using which Rejection Region can be separated from the non-rejection Region.
• It can be calculated using the level of significance α from the z-table.
• Since the distribution is symmetrical, α/2 can be used to find critical values on both ends in the case of a two-tailed test.
• For a two-tailed test, there are two critical values. The absolute value for both the critical values are same, they only differ by sign.

The value of test statistics is compared with the critical value to make the decision.

Null Hypothesis is Rejected if

• test statistics < negative critical value
• test statistics > positive critical value

Null Hypothesis is Accepted if

• negative critical value < test statistics < positive critical value

Let’s continue solving the above example where α was 0.05. Since it’s a two-tailed test, the rejection region is present on both ends and there exist two critical values.

• α/2 = 0.025
• The x-axis of distribution starts from 0 and ends at 1.
• negative critical value can be found using 0+0.025 =0.025
• positive critical value can be found using 1-0.025 =0.975
• Find the 0.025 value in the z table
• 
• To calculate the critical value corresponding to 0.025, add the corresponding left (-1.9) and top (0.06)  from the z-table, hence negative critical value = -1.9+0.06 = -1.96.
• Find the 0.975 value in the z table.
  
• To calculate the critical value corresponding to 0.975, add the corresponding left (1.9) and top (0.06)  from the z-table, hence positive critical value = 1.9+0.06 = 1.96.
• test statistics value is -2.12 from step 5.

  

• since -2.12<-1.96, -2.12 is present in the rejection region hence we reject the null hypothesis.

2. p-value Method     

p-value (probability value)

• Assuming the Null Hypothesis is True, the p-value is the probability of obtaining test results at least as extreme as the result actually observed.
• the p-value is the area of test statistics value.
• In the left-tailed test, the p-value is the area to the left of the test statistics value.
• In the right-tailed test, the p-value is the area to the right of the test statistics value.
• In the two-tailed test, the p-value is twice the area to the left or to the right of the test statistics value.
• if the p-value<α, reject the Null Hypothesis

p-value for a Two-Tailed Test

You can see from the above figure, that the p-value is the area of the test statistics value. Normally Distributed data is symmetrical on both sides, hence the rejection region area is α/2 on both sides.

Null Hypothesis is Rejected if

• p-value < α/2 
• 2*p-value < α 

Let’s continue solving the above example where α was 0.05 and the z-statistics value is -2.12.

• Find the area of -2.12 in the z-table.
• 
• To calculate the p-value from the z-table, find such values from left and top which add up to the test statistics value and their corresponding value returns you the area of test statistics, which is nothing but the p-value.
• 2*0.0170 = 0.0340, hence p-value is 0.0340.
• since 0.0340 < 0.05, we reject the null hypothesis.

We Reject the Null Hypothesis using both the critical value method and the p-value method. Hence conclude that the mean is different from 30.

Critical value vs p-value

Both critical value and p-value methods are used to perform the same operation i.e. either accept or reject the null hypothesis. But there is a difference in the way they are executed.

In the P-value method, you just need to compute the p-value to perform the test whereas, in the critical value method you need to compute the value of test statistics (z or t statistic) and find the critical value corresponding to the given confidence or significance level.

Because of this reason, most statistical software prefers using the p-value approach over the critical value approach.

End Notes

Thank you for reading this article. By the end of this article, we have learned how to perform the two-tailed test using the critical value method and p-value method.

I hope this article was informative. Feel free to ask any query or give your feedback in the comment box below.