Understanding the Chi-square (χ²) Test

The Chi-square ($\chi^2$) test is a statistical test commonly used to examine the relationship between two categorical variables. It helps determine if there is a significant association between the categories of the variables being studied, or if any observed difference is simply due to random chance.

The test works by comparing the observed frequencies in different categories with the frequencies that would be expected if there were no association between the variables (i.e., under the assumption of independence). The result of the test is a Chi-square statistic and a p-value.

Significance Levels in Statistical Testing

In hypothesis testing, including the Chi-square test, a significance level (denoted by $\alpha$) is chosen before conducting the test. The significance level represents the probability of rejecting the null hypothesis when it is actually true (Type I error).

The null hypothesis for a Chi-square test of association is typically that there is no relationship or no association between the two categorical variables. The alternative hypothesis is that there is a significant relationship or association.

A result is considered statistically significant if the p-value obtained from the test is less than or equal to the chosen significance level ($\alpha$). If the p-value $\le \alpha$, we reject the null hypothesis and conclude that there is a statistically significant relationship between the variables.

Analyzing the Options: Common Alpha Levels

Different significance levels can be chosen depending on the field of study and the consequences of making a Type I error. Common significance levels used in statistics include 0.05 (or 5%) and 0.01 (or 1%). However, other levels like 0.10 (10%), 0.02 (2%), or 0.03 (3%) are also valid choices, although less frequently used as standard thresholds in some disciplines.

Let's look at the levels presented in the options:

• 1% ($\alpha = 0.01$): This is a common, strict significance level. A result significant at the 1% level indicates strong evidence against the null hypothesis.
• 2% ($\alpha = 0.02$): This is a less common but valid significance level. A result significant at the 2% level provides stronger evidence against the null hypothesis than at the 5% level, but less strong than at the 1% level.
• 3% ($\alpha = 0.03$): Similar to 2%, this is a less common but valid significance level. It falls between the 1% and 5% levels in terms of strictness.
• 5% ($\alpha = 0.05$): This is the most commonly used significance level in many fields. A result significant at the 5% level means there is a 5% chance of rejecting the null hypothesis if it were true.
• 10% ($\alpha = 0.10$): This is a less strict significance level than 5%, 3%, 2%, or 1%. While used in some exploratory analyses or fields where higher Type I error risk is acceptable, it is often not considered the standard threshold for declaring a relationship "significant" in many traditional statistical applications compared to the stricter levels.

The question asks at which levels of error a relationship is considered significant according to the $\chi^2$ test. Based on standard statistical practice and the typical options presented in educational contexts, 1%, 2%, 3%, and 5% are all considered valid and commonly used (to varying degrees) significance levels at which a result can be declared significant if the p-value meets the criterion. The 10% level, while usable, is often considered a weaker level of evidence compared to the others in establishing significance.

Conclusion based on Options

Considering the options provided (1%, 2%, 3%, 5%, 10%) and the structure of the final answer choices (which combine these percentages), the levels considered significant in this context are those included in the implied correct group. The correct answer option text "A, B, C and D only" corresponds to the percentages 1%, 2%, 3%, and 5%. This suggests that within the framework of this question, these four levels are considered points at which a Chi-square test result could be deemed significant, unlike the 10% level which is excluded from this specific combination.

Revision Table: Chi-square Test and Significance

Additional Information: Type I and Type II Errors

Understanding significance levels requires knowing about potential errors in hypothesis testing:

• Type I Error ($\alpha$): Rejecting the null hypothesis when it is actually true. The significance level is the maximum probability of making a Type I error. Choosing a lower $\alpha$ (e.g., 1% instead of 5%) reduces the chance of a Type I error but increases the chance of a Type II error.
• Type II Error ($\beta$): Failing to reject the null hypothesis when it is actually false. The probability of a Type II error is denoted by $\beta$.
• Power of a Test ($1 - \beta$): The probability of correctly rejecting a false null hypothesis.

Selecting a significance level involves balancing the risks of Type I and Type II errors based on the context of the study. While 5% is standard, stricter levels like 1% are used when the cost of a Type I error is high. Levels like 2% and 3% are valid intermediate choices, and 10% might be used where detecting any potential relationship is prioritized over minimizing Type I errors, or as a threshold for "marginally significant" results.