5.1 One-tailed test

In a one-tailed test, we focus on either the upper (right) or lower (left) tail of the distribution. For our example, we will consider an upper-tailed test with a significance level (α) of 5% or 0.05. Excel provides a useful function to find z critical values quickly and accurately for this purpose: the NORM.S.INV() function.

Let us consider an upper-tailed test. In this scenario, the z critical value corresponds to the boundary equal to 1 – 5%, or 95% of the z-distribution.

Figure 11 Upper tailed test

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The image depicts the concept of an upper-tailed test in statistics, commonly used in hypothesis testing. In this test, the area under the curve represents the probability distribution, with the majority (95%) of the distribution being under the main portion of the curve.

Figure 11 Upper tailed test

To use NORM.S.INV(), we input this cooresponding propability that we are interested in.

• =NORM.S.INV(0.95)

Figure 12 NORM.S.INV() function

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The image shows a screenshot from Microsoft Excel, where the formula =NORM.S.INV(probability) is being entered. This is a function used to calculate the inverse of the standard normal cumulative distribution, which returns a z-score corresponding to a given probability in a standard normal distribution (mean of 0 and standard deviation of 1). The formula expects a probability value as input and will return the z-score associated with that probability.

Figure 12 NORM.S.INV() function

This formula returns a z critical value of approximately 1.645, which we can round to 1.65 (This value has been used in earlier sections).

For upper tail (e.g., H1: μ > certain value): The critical value would be positive: +1.65

• We would reject the null hypothesis if the calculated z-statistic is greater than 1.65.

For a lower-tailed test, we focus on the left side of the distribution. Here, the z critical value corresponds to the boundary equal to 5% of the z-distribution (Figure 5).

Figure 13 Lower Tailed Test This graph is created by the author.

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The image depicts the concept of an upper-tailed test in statistics, commonly used in hypothesis testing. In this test, the area under the curve represents the probability distribution, with the 5% of the distribution being under the main portion of the curve.

Figure 13 Lower Tailed Test This graph is created by the author.

To find this z critical value using NORM.S.INV(), we input the probability of 5%.

• =NORM.S.INV(0.05)

This returns approximately -1.645, which we can round to -1.65.

Thus, for a lower-tailed test (H1: μ < certain value): The critical value would be negative: -1.65

• We would reject the null hypothesis if the calculated z-statistic is less than -1.65.

This explanation clearly illustrates how the direction of the alternative hypothesis determines the sign of the critical value in one-tailed tests. It also emphasises that the rejection criterion depends on whether we are testing for values significantly greater than (upper tail) or less than (lower tail) a certain value.