Hypothesis Testing: 2 Sided Test (aka 2 Tailed Test)

Introduction

The 2 sided hypothesis test is used to examine both sides of the data. In other words, one must use this test to test both areas under the left and right tails of the normal distribution. Hence, the p-value must be multiplied by 2 in order to account for both tail areas. 

This type of test is usually used to determine whether a claim is true or false. It is not used for "greater than" or "less than" scenarios; rather, a two-sided hypothesis test is used when your alternative hypothesis employs the " \(\neq\) " symbol. 

Example

The manager of the FAO Schwarz factory states that 0.08 of its produced toys are defective. In order to test this, an employee of a FAO Schwarz store takes a random sample of 739 toys delivered from the factory and inspects them. He finds that 52 of them are defective. At a significance level of 0.05, is there evidence to support the claim that the manager of the FAO Schwarz factory is lying?

Step 1: Name Test: 2 Sided Hypothesis Test

Step 2: Define Test: Since this is a 2 sided test:

H_o = p_o = 0.08

H_A = p_o ≠ 0.08

Step 3: Check Conditions:

1. Data from random sample

2. N ≥ 10n

3. np_o ≥ 10 and nq_o ≥ 10

Step 4: Calculate test statistic and p-value

\(z = pˆ - p_o \over \sqrt{p_oq_o \over n}\)  Note: pˆ is calculated by doing 52/739

p-value: For a 2 sided test, the p-value is equal to 2 times the p-value for the lower-tailed p-value, if the value of the test statistic from your sample is negative. The p-value is equal to 2 times the p-value for the upper-tailed p-value if the value of the test statistic from your sample is positive.

So in this example one must calculate the z-score for the lower end (H_A: p_o < 0.08) and multiply its p-value by 2.

Step 5: Conclusion:

Once everything is solved, you will discover that the p-value is 0.16 So the conclusion is as follows:

I calculated a p-value of 0.16 which is greater than the significance level of 0.05. Therefore, we fail to reject the null hypothesis and the data fails to support the claim that the manager is lying.