4.5 Directional hypotheses

As you may recall from an earlier discussion, directional hypotheses are refered to as one-tailed hypotheses. These hypotheses use iequality signs in their statements. To test a directional hypothesis, you would be expected to perform a one-tailed test, which aims to verify whether the data from the sample supports the anticipated direction of the relationship or difference.

To conduct a one-tailed test, decision-makers establish a critical value to determine whether to reject or retain the null hypothesis. This process typically involves setting a significance level (α). For instance, when α = 0.05, the z critical value for a one-tailed test in a normal distribution is 1.65.

You may notice that the z critical value of 1.65 for a one-tailed test with α = 0.05 differs from the 1.96 used in two-tailed tests. This difference arises from the nature of these tests and how they distribute the significance level. In a one-tailed test, we focus on only one direction of the distribution (either upper or lower tail). Consequently, we allocate the entire α (0.05) to one tail of the distribution. To understand this better, consider that if one tail is associated with 0.05, then both tails would be associated with 2 x 0.05 = 0.10, which is equivalent to a 90% confidence level. At a 90% confidence level, the z-score equals 1.65. This relationship explains why we use 1.65 as the z critical value in one-tailed tests with α = 0.05, rather than the 1.96 used in two-tailed tests at the same significance level.

In this case:

• The null hypothesis would be rejected if the z-statistic exceeds 1.65.
• Only the upper tail region of the distribution is considered for rejection in a one-tailed test.
• The area in the tail above z = +1.65 represents 0.05 of the distribution.

Unlike in a two-tailed test, the alpha level does not need to be divided by two for a one-tailed test. This approach allows decision-makers to focus on detecting effects in a specific direction, making it particularly useful when there’s a strong theoretical or practical reason to expect a particular outcome.

Figure 9 One-tailed test – right (upper) tail.

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a symmetrical graph reminiscent of a bell. The graph points out z-score axis and hours of study axis, as well as circle the rejection regions of Null Hypothesis on right hand side of the bell curve.

Figure 9 One-tailed test – right (upper) tail.

In summary, decision-makers use one-tailed tests to evaluate directional hypotheses, which predict the direction of a difference or association between two variables. The critical value for these tests depends on the chosen significance level (α), and the test aims to determine if the data supports the predicted direction.

One-tailed tests are versatile and can be used for both "greater than" and "less than" scenarios. Let us consider an example to illustrate this flexibility:

Imagine a department store where management believes the average customer spend per visit is £65. However, the service manager suspects customers are spending less. We can formulate the following hypotheses:

• H₀: µ ≥ £6

• H₁: µ < £65

In this case, the null hypothesis (H₀) states that the average spend is greater than or equal to £65, while the alternative hypothesis (H₁) suggests it is less than £65.

To test this directional hypothesis, we conduct a one-tailed test. The alternative hypothesis predicts that the true value of µ will be lower than £65, so we focus on the lower tail of the normal distribution for our rejection region.

At an alpha level of 0.05, the z critical value for the lower tail is -1.65. This means we would reject the null hypothesis if our calculated z-statstic falls below -1.65. Graphically, this rejection region appears in the lower tail of the normal distribution curve.

Figure 10 One-tailed test – left (lower) tail.

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a symmetrical graph reminiscent of a bell. The graph points out z-score axis and customer spending axis, as well as circle the rejection regions of Null Hypothesis on left hand side of the bell curve.

Figure 10 One-tailed test – left (lower) tail.

In general, the one-tailed test is not restricted to a specific direction and can be used in either direction, depending on the research question and the hypothesis being tested. The test is utilised to determine if the data supports a directional hypothesis, and a critical value is established based on the significance level chosen for the test.