A two-tailed test is when we test if a value is different from the assumed mean (µ). Different from means that it can be smaller than or greater than µ. So, we are looking to both sides of µ and thus to both tails of the density curve.
Compared to one-tailed tests, two-tailed tests have critical areas in both tails at the same time, whereas one-tailed tests have critical area either or to the left or to the right, not both at the same time.
Two-tailed test, key points
- Two-tailed tests are applied for testing statistical significance in hypothesis tests.
- Both of the two tails in the density curve have critical areas as opposed to one-tailed tests with a critical area only to either one of the sides (right or left, not both at the same time) .
- If the tested value falls into either of the critical areas, the null hypothesis is rejected.
- The significance level (α) is divided with 2. E.g. a 5% alpha level leaves divided into 2.5% in each tail.
Application of the two-tailed test
Hypothesis testing is applied to calculate the statistical significance of a claim. It can typically be a new finding that is tested to see if it is “significantly” different from the assumed mean (µ).
The two-tailed test is applied to test if a certain value is “significantly” different from µ. Testing if it is ‘different from’ implies to test if it is “significantly” greater than or “significantly” less than µ.
If the value tested for falls beyond the upper limit or lower limit, the null hypothesis is rejected, and it is then inferenced as “statistically significant”. The critical values are set at the α/2 level and determine the upper and lower limits of the rejection areas:
Two-tailed test vs One-tailed test
Two-tailed test example
Say that a production line of tennis balls supplying The International Tennis Federation must produce and deliver all balls with a diameter of 6.58 cm. A simple random sample of 100 balls is planned to be taken and, in the case, that the production has a statistical significance greater than 5%, production will be put on hold and the production line adjusted.
Step 1: Setting the hypotheses
H0: µ = 6.58
HA: µ ≠ 6.58
Step 2: Setting significance level
The production will be put on hold at a statistical significance greater than 5%, so the significance level (α) = 0.05
Step 3: Taking the sample
Now, the company takes a simple random sample of 100 balls and the results are:
- Sample mean (x̄): 6.31
- Sample standard deviation (s): 1.59
Step 4: Calculating test statistics
The Central Limit Theorem says that the greater the sample size the more any probability distribution approximates to the normal distribution and it is usual to see the this applied for n>30. Our n = 100, so our test statistic will be applied with the normal standard table:
The z-score of -1.698 is within the acceptance area as it is does not go below the critical value of -1.96, nor above the 1.96. Looking up in the z-table, a z-score of -1.698 returns a p-value of 0.089.
Step 5: Conclusion
The z-score falls in non-rejection area, so we fail to reject the null hypothesis. Therefore, based on our sample, we do not find evidence that the assumed mean should be different from 6.58. The p-value of 0.089 tells us that there is approximately 8.9% probability of getting a result at least as extreme as ours assuming that the mean is 6.58.
Two-tailed test in MS Excel
The functions =STANDARDIZE; =NORM.S.INV and =NORM.S.DIST can be used to support a hypothesis testing in Excel. Take the following example of quality assurance in a production line:
Say a production line must pack 60 Danish butter cookies per box. The boxes are accepted for shipping if they have a maximum of 63 and a minimum of 57 cookies. The company takes a sample of 100 boxes to test if there should be support for the null hypothesis that the mean value is 60 units.
The sample returns a mean of 56 with a standard deviation of 28:
We fail to reject the null hypothesis considering the following:
- Test statistic (-1.429) > Critical z (-1.96)
- P-value (0.153) > Significance level (0.05)
Some of my preferred material for learnings on two-tailed test:
- Statisticsfun (video 4:34): How to calculate one tail and two tail test
- Jbstatistics (video 9:25): One sided or two-sided test?
- Khan Academy (video 6:34): One tailed and two tailed tests
- Investopedia (text): Two-tailed test definition
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