A Directional Test of Hypothesis, also known as a One-tailed Test, is a type of hypothesis test used to determine whether there is a statistically significant effect in a specific direction. In contrast to a two-tailed test, which tests for deviations in both directions (greater than or less than), a one-tailed test is focused only on one direction of interest—either greater than (right-tailed) or less than (left-tailed) the hypothesized value.

Key Features:

1. One-tailed Test: It looks for evidence of the effect in only one direction.

   • Right-tailed test: Tests if the parameter is greater than a certain value.
   • Left-tailed test: Tests if the parameter is less than a certain value.

2. Null Hypothesis $H_0$: The null hypothesis states that there is no effect or the effect is equal to the hypothesized value (e.g., $H_0: \mu = \mu_0$).

3. Alternative Hypothesis $H_1$: The alternative hypothesis is directional.

   • For a right-tailed test: $H_1: \mu > \mu_0$
   • For a left-tailed test: $H_1: \mu < \mu_0$

4. Critical Region: In a one-tailed test, the critical region is placed entirely in the tail of the distribution that corresponds to the direction of the test.

5. P-value Interpretation: The p-value is calculated only for the specified tail of the distribution. If the p-value is less than the significance level (e.g., 0.05), the null hypothesis is rejected in favor of the directional alternative.

Example:

If you want to test whether a new drug increases recovery time faster than the current standard, you would set up a right-tailed test:

• $H_0: \mu = \mu_0$ (no improvement in recovery time)
• $H_1: \mu > \mu_0$ (drug improves recovery time)

If the calculated statistic is significantly higher than expected under the null hypothesis, you reject $H_0$ and conclude that the drug improves recovery time.

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Do you want further details or have any questions?

5 Related Questions:

1. What are the advantages and disadvantages of using a one-tailed test over a two-tailed test?
2. How do you choose between a right-tailed and a left-tailed test in practice?
3. What is the impact of sample size on the power of a one-tailed test?
4. Can you explain how critical values are determined in a one-tailed test?
5. How do you calculate the p-value for a one-tailed test using normal or t-distribution?

Tip:

Always ensure that the use of a one-tailed test is justified, as it limits your ability to detect an effect in the opposite direction.