To determine the appropriate hypothesis-testing test for comparing the average performance across more than two sample groups drawn from a normally distributed population, we need to consider the type of analysis involved. When comparing means of more than two groups, the most suitable test is the F-test, commonly known as ANOVA (Analysis of Variance).

Let's break down why each option is or isn't suitable:

1. Chi-square test: This test is generally used for categorical data to test the independence of variables or goodness of fit, not for comparing means of continuous data. Hence, it is not suitable here.
2. F-test: The F-test is designed to compare the variances across multiple groups to determine if there are any statistically significant differences between the means of these groups. Since we are dealing with more than two groups, the F-test (ANOVA) is appropriate for this scenario.
3. z-test: This test is typically used when comparing the means of large samples (n > 30) or when the population variance is known. It is limited to comparing two sample means rather than more than two.
4. t-test: The t-test is used for comparing the means of two groups. It is not appropriate for comparing more than two groups without modification (like repeated tests that increase error rates).

Therefore, the correct answer is the F-test because it allows for the comparison of the means across more than two sample groups effectively and is suitable for normally distributed populations with potentially unequal variances.