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One-tailed test

A one-tailed test is when we are testing a value in only one of the tales of the density curve. A right tailed test is when we test for a value greater than the assumed mean (µ). A left tailed test is when we test for a value less than µ. One-tailed tests are also known as ‘directional hypothesis tests’,

When a value is tested for whether it is different from the assumed mean it is a two-tailed test, as we are looking both at the left end tail for the ‘less than’ values and at the right end tail for the ‘greater than’ values.

 

 

One-tailed test, key points

  • One-tailed tests are applied to answer for the questions: Is our finding significantly greater than our assumed value? Or: Is our finding significantly less than our assumed value?
  • If we are analyzing for the question that the findings are different from the assumed mean, we are talking two-tailed tests
  • Alpha (α) is not divided by two when looking up the z-score or t-score, as it is only looked for in one side.

 

Right-tailed test (greater than), left-tailed test (less than)

An example of a one-tailed test can be that an analyst run a sample and find a sample mean that seems “relatively much” larger than the assumed mean. The analyst wishes to test whether this is finding can be concluded as “significantly” greater and thereby if the null hypothesis, that claims a smaller value, can be rejected.

This “greater than” example is also called a right-tailed test as it looks to the right for values greater than the mean. The same method is applied for less than examples where we will look to the left for values less than the mean estimate:

One-tail test

 

 As with hypothesis test in general the rejection area is in the smaller area under the tails. If the new finding falls into the rejection area, the analyst will reject the null hypothesis (H0) as there is sufficient evidence to support the alternative hypothesis.

 

One-tailed test vs two tailed test

When an analyst wishes to test whether a sample mean is different from the assumed mean, it is a two-tailed test. ‘Different from’ means that it can be both ‘greater than’ or ‘less than’, and the analyst therefore will look to both sides of the mean. To the right tail for the positive values and to the left tail for the negative values: 

One-tailed vs two-tailed test

 

 

Formulas for one-tailed test statistics

First step in any hypothesis testing, whether it’s a one-tailed test or a two-tailed test, the statistician will set the hypotheses and significant level (α). The formulas for a one-tailed test for one mean:

Test statistic formulas

 

As shown, for small samples with unknown sigma, the Student’s t-distribution is applied. The formula is the same as for when using the z-statistics, but the t-table returns greater values than the z-tables, as the it works with a greater degree uncertainty.

For one-tailed tests the statistician will apply the entire alpha-level when looking up in the respective z or t-table. For two-tailed tests the statistician will apply the alpha-level divided by two (one half of the alpha-level on each side of the mean estimate in the density curve).

  

One-tailed test, example:

Say that Melinda is a portfolio manager in an investment firm and is supposed to score the average of 145.00 points on some point scale for ROI for a given product. She scores 150.51 and wishes to prove that her score is ‘significantly’ greater than the set goal of the 145. She asks her statistician colleague to test her results.

The statistician sets up a hypothesis test through a one-sided test whereas the result that she is testing for is greater than assumed mean of 145:

H0: μ ≤ 145.00

Ha: μ > 145.00

Alpha (α) = 0.05

The score is an average calculation of the total of 15 sales of the product, so the sample size (n) is 15 and the sample standard deviation (s) is in this case calculated to be 8.8.

Melinda hopes that the statistician will prove that she’s done “significantly” better than the 145 points and therefore that the null hypothesis will be rejected. This will occur if her result falls into the rejection area which will be calculated through the test statistics:

test statistic calculation

The p-value is 0.015, below the significance level (α) of 0.05, so it falls into the rejection region and can thereby be qualified as “significant” compared to the mean of the 145:

One-tailed test example

 

So, Melinda can state that she has outperformed her goal with “statistical significance”. She can say, based on the sample statistics, there is only a 1.47% chance that a score, as extreme as her score, would be observed when the assumed mean is 145.

Had the significance level (α) been set to 1%, the result would have fallen into the non-rejection area and the null hypothesis could not have been rejected. In this case, Melinda’s score would not have been qualified as “statistically significant”, although there would still be only 1.47% chance of observing that an extreme score.

 

One-tailed test with Excel

One-tailed t-test in Excel can be done by applying the =T.TEST formula to calculate the p-value:

T.TEST(array1,array2,tails,type) with the arguments:

  • Array1: The first dataset (in our Melinda-example, the 15 observed scores)
  • Array2: the second dataset (the value in the hypotheses)
  • Tails: one-tailed or two-tailed
  • Type: The kind of t-test that we conduct (in the Melinda-example it is the 3rd type which is a ‘Two-sample unequal variance’).

If the p-value, calculated with the =T.TEST function, falls below the significance level (α) and thereby into the rejection area, the H0 is rejected:

One-tailed test in Excel

 

The calculations can also be done by the running the descriptive statistics function: Data >> Data Analysis >> Descriptive statistics:

One-tailed test in Excel

Learning statistics

Some of my preferred learning materials on one-tailed test:

 

 

 

Carsten Grube

Carsten Grube

Freelance Data Analyst

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