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Data analysis: hypothesis testing

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# 5.1 Acceptance and rejection regions

The z-score table you created in Activity 5 represents the area under the normal distribution bell curve left of z (as shown in Figure 17).

Figure 17 Area under the curve left of z

The entries in this table can be used to determine whether to accept or reject the null hypothesis.

Suppose a marketing team at a company wishes to test the hypothesis that a new ad campaign will lead to a significant increase in sales. The team could use a one-tailed test with the reject region in the upper (right) tail and an alpha level of 1%.

Using the table created in Activity 5, the team can identify the range of z-scores that correspond to this test. They can then calculate the test statistic based on the data collected from the sales during and after the ad campaign. If the calculated test statistic falls within the rejection region identified by the table, the team can reject the null hypothesis and conclude that the ad campaign has had a significant impact on sales. This information can be used by the marketing team to justify the investment in the ad campaign and to make informed decisions about future marketing strategies.

In the context of the marketing team's hypothesis testing, the reject region for the one-tailed test with an alpha level of 1% corresponds to the range of z-scores that fall within the top 1% of the normal distribution. Conversely, the acceptable range refers to the range of z-scores that corresponds to the remaining 99% of the distribution to the left of z. Using the table created in Activity 5, the marketing team can identify the specific range of z-scores that correspond to the acceptable range and the reject region. Based on this table, the z-score of 2.33 corresponds to the upper limit of the acceptable range, as the area to the left of z = 2.33 represents approximately 99% of the area under the curve.

Therefore, if the team obtains a calculated z-score that is greater than 2.33, they can reject the null hypothesis and conclude that the new ad campaign has had a significant impact on sales. This information can help the marketing team make data-driven decisions about future campaigns and allocate resources effectively to maximise sales and profits. Figure 18 below illustrates this.

Figure 18 One-tailed test with Alpha level of 1%

Other than creating a z-score table, you calculate the region to the left of z by using the Excel formula NORM.S.DIST(z, cumulative). For example, you can calculate the region left of z when z = 2.33 by simply entering 2.33 as a z-score and setting the cumulative to be ‘TRUE’ in this Excel formula.

Figure 19 Calculate the region left of z without using z-score table 1
Figure 20 Calculate the region left of z without using z-score table 2

You should get a value reading of 0.9901, which is exactly what you found in the z-score table in row 2.3 and column 0.03.

Here is another question. If you want to test hypotheses using the two-tailed test with the alpha level equal to 0.5%, how can you determine the z-scores region to reject the null hypothesis?

The two-tailed test requires you to divide the levels of alpha by 2.

Therefore, α for the two-tailed test = 0.05/2 = 0.0250

As the z-score table shows the area to the left of the value of z, a two-tailed test requires you to identify two entries. The area of one entry covers 0.975 (97.5%) of the area (where 0.025 of the area is outside the value of z on the right tail), and the area of another entry covers 0.025 of the area on the left tail.

Using the z-score table, you can determine the z-score = 1.96 or -1.96. Therefore, you will reject the null hypothesis for obtained z-score > 1.96 or z-score