2 Alpha (α) levels

The process of hypothesis testing is a fundamental aspect of scientific research and statistical analysis. It provides a structured approach to evaluate claims and make decisions based on empirical evidence. Let us explore the two possible outcomes of hypothesis testing in more detail:

1. Rejecting the Null Hypothesis (H₀)

When decision-makers reject the null hypothesis, it implies that there is sufficient evidence to support the alternative hypothesis (H₁). This outcome occurs when the observed data is statistically significant and unlikely to have occurred by chance alone.

2. Failing to Reject the Null Hypothesis (H₀)

When decision-makers fail to reject the null hypothesis, it means that there is insufficient evidence to support the alternative hypothesis (H₁). This outcome occurs when the observed data is not statistically significant and could have occurred by chance.

To illustrate these concepts, let us return to our coffee example. Suppose we want to test the null hypothesis that the average foam height in a caffè latte is 1 cm (H₀: µ = 1 cm foam). We might randomly sample 60 cups of caffè latte throughout the day, measure the foam height, and calculate the average and test statistic.

Figure 4 Caffè latte

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The image shows a glass cup filled with a layered coffee drink, likely a latte or cappuccino. The drink has three distinct layers: a creamy base at the bottom, a darker coffee layer in the middle, and a thick layer of foam on top. The foam appears light and fluffy, creating a soft peak above the rim of the cup.

Figure 4 Caffè latte

Consider a study where three decision-makers each sample 60 cups of caffè latte:

• Decision-maker 1 finds an average foam height of 1.1 cm.

• Decision-maker 2 finds an average foam height of 1.5 cm.

• Decision-maker 3 finds an average foam height of 2.6 cm.

Decision-maker 1’s result (1.1 cm) is close to the null hypothesis value. In this case, we might fail to reject H₀, as the observed difference could be due to random variation. However, this does not prove that the true average foam height is exactly 1 cm. It only indicates that we do not have enough evidence to conclude otherwise.

Decision-maker 3’s result (2.6 cm) is far from 1 cm. Here, we would likely reject H₀. This suggests that the observed difference is statistically significant and unlikely to have occurred by chance.

Decision-maker 2’s result (1.5 cm) is less clear-cut. Although the average foam height of 1.5 cm is not far away from 1 cm, does it go far enough to be considered sufficient?

In order to answer this question, you would need to introduce the concept of ‘statistical significance’, which you will look at in more detail in the next section.