Critical z-Statistic for Hypothesis Testing | CFA Level 1 Exam

B

Since the z-statistic is larger in magnitude than the critical value, you should reject the null hypothesis. Note that all the other alternatives loosely mean the same thing.

To determine the appropriate decision, let's understand the concept of hypothesis testing and the significance level.

In hypothesis testing, we have a null hypothesis (Ho) and an alternative hypothesis (Ha). The null hypothesis represents the default position or the claim that is initially assumed to be true. The alternative hypothesis is the claim that contradicts the null hypothesis and is typically what we are trying to gather evidence for.

In this case, the null hypothesis is Ho: x = y, which means that x and y are equal. The alternative hypothesis is Ha: x ≠ y, indicating that x and y are not equal.

The significance level, often denoted by α (alpha), is the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. Commonly used significance levels are 0.05 (5%) and 0.01 (1%).

To make a decision in hypothesis testing, we compare the test statistic, which measures the difference between the sample data and the null hypothesis, to the critical value(s) associated with the chosen significance level. If the test statistic falls in the rejection region beyond the critical value(s), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Given that the critical z-statistic for the desired significance level is 1.68 and the calculated z-statistic is -3.2, we need to compare these values.

Since the calculated z-statistic, -3.2, is outside the rejection region defined by the critical value, 1.68, we can conclude that the test statistic strongly indicates a significant difference between x and y. Therefore, we reject the null hypothesis (Ho).

The correct answer is B. Reject the null hypothesis.

Remember, rejecting the null hypothesis does not necessarily imply that the alternative hypothesis is true; it simply suggests that the data provides enough evidence to reject the assumption of equality (x = y) in favor of the alternative hypothesis (x ≠ y).