One-tailed Hypothesis test

What is a one-tailed Hypothesis test?

In a directional hypothesis test or a one-tailed test, the statistical hypotheses (H₀ and H₁) specify either an increase or a decrease in the population mean. That is, they make a statement about the direction of the effect.

Table of Contents.

For example:

- A special training program has been expected to increase student performance (Marks obtained) in the class 12^th CBSE board exam.
- Now we are going to test that the average mark of students may increase from the previous year’s average marks.

One-tailed Hypothesis test

- This situation incorporates the directional prediction into the statement of H₀ and H₁.
- The result is a directional test, or what commonly is called a one-tailed test.

One-tailed Hypothesis test

Hypothesis for a Directional Test

In the first step of the hypothesis test, the directional prediction is incorporated into the statement of the hypotheses. Thus, the two hypotheses would state:

H₀ : Average Marks are NOT increased.

(Before and after training -average marks are the same. The training program does not work.)

H₁ : Average Marks are increased.

(The training program works as predicted.)

In the second step of the process, the critical region is located entirely in one tail of the distribution.

Measuring Effect Size

A measure of effect size is intended to provide a measurement of the absolute magnitude of a treatment effect, independent of the size of the sample(s) being used.
One concern with hypothesis testing is that a hypothesis test does not really evaluate the absolute size of a treatment effect.

Cohen’s d – Measuring and reporting effect size

Cohen’s d. Cohen (1988): Effect size can be standardized by measuring the mean difference in terms of the standard deviation. The resulting measure of effect size is computed as

The mean difference is determined by the difference between the population mean before treatment and the population mean after treatment.
The population mean after treatment is unknown. Therefore use the sample mean µt of the treated population.
The sample mean is expected to be representative of the population mean and provides the best measure of the treatment effect.

Evaluating the size of a treatment effect

Cohen’s d measures the degree of separation between two distributions, and a separation of one standard deviation (d = 1.00) represents a large difference.

Cohen’s d simply describes the size of the treatment effect and is not influenced by the number of scores in the sample.

Comparison of One-Tailed vs. Two-Tailed Tests

A two-tailed test is more rigorous and, therefore, more convincing than a one-tailed test.
The two-tailed test demands more evidence to reject H₀ and thus provides a stronger demonstration that a treatment effect has occurred.
One-tailed tests should be used only in situations when the directional prediction is made before conducting analysis and there is a strong justification for making the directional prediction.
There is the argument that one-tailed tests are more precise because they test hypotheses about a specific directional effect instead of an indefinite hypothesis about a general effect.
Precaution: If a two-tailed test fails to reach significance, never follow up with a one-tailed test as a second attempt to salvage a significant result for the same data.

Concerns about Hypothesis Testing: Measuring Effect Size

There are two serious limitations to using a hypothesis test to establish the significance of a treatment effect.

The first concern is ⇒

The focus of a hypothesis test is on the data rather than the hypothesis.
When the null hypothesis is rejected, a strong probability statement is made about the sample data, not about the null hypothesis.
A significant result permits the conclusion: “This specific sample mean is very unlikely (p < .05) if the null hypothesis is true.” This conclusion does not make any definite statement about the probability of the null hypothesis being true or false.
The fact that the data are very unlikely suggests that the null hypothesis is also very unlikely, but we do not have any solid grounds for making a probability statement about the null hypothesis. It cannot be concluded that the probability of the null hypothesis being true is less than 5% simply by rejecting the null hypothesis with α = .05.

The second concern is ⇒

A significant treatment effect does not necessarily indicate a substantial treatment effect. In particular, statistical significance does not provide any real information about the absolute size of a treatment effect.
The test simply establishes that the results obtained in the analysis are very unlikely to have occurred if there is no treatment effect.
The hypothesis test reaches on conclusion by –

calculating the standard error
Difference between means
Demonstrates that the obtained mean difference is substantially bigger than the standard error.

What is a one-tailed Hypothesis test?

Hypothesis for a Directional Test

Measuring Effect Size

Cohen’s d – Measuring and reporting effect size

Evaluating the size of a treatment effect

Comparison of One-Tailed vs. Two-Tailed Tests

Concerns about Hypothesis Testing: Measuring Effect Size

The first concern is ⇒

The second concern is ⇒

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