This question requires us to match different statistical tests with their appropriate applications in hypothesis testing. Let's analyze each test and application provided.

Understanding the Statistical Tests

Here's a brief overview of the statistical tests mentioned in List-II:

• Z-test: This test is used to compare the means of two groups or to compare a sample mean to a known population mean. It is typically used when the sample size is large (usually n > 30) or when the population standard deviation is known.
• Chi-square test (χ² test): This test is used for categorical data. Common applications include testing the goodness of fit (comparing observed frequencies to expected frequencies from a theoretical distribution) and testing independence between two categorical variables.
• F-test: This test is primarily used to compare variances between two populations. However, it is also the basis for Analysis of Variance (ANOVA), which is used to compare the means of three or more groups.

Analyzing the Applications

Now let's look at the applications described in List-I and determine which test is appropriate for each:

• (a) Testing the goodness of fit of a distribution: This involves comparing observed data frequencies to the frequencies expected under a specific theoretical distribution (like normal, uniform, etc.). The standard test for this purpose is the Chi-square test.
• (b) Testing the significance of the differences among the average performance of more than two sample groups: When you want to compare the means of three or more independent groups to see if there's a statistically significant difference among them, you use Analysis of Variance (ANOVA). The test statistic calculated in ANOVA follows an F-distribution, and the test is often referred to as an F-test in this context.
• (c) Testing the significance of the difference between the average performance of two sample groups (Large-sized): When comparing the means of two groups and the sample size is large, the Z-test is the appropriate test to use. If the sample size were small and the population standard deviation unknown, a t-test would be used, but the question specifically mentions 'Large-sized'.

Matching the Items

Based on the analysis, we can match the items:

• (a) Testing the goodness of fit of a distribution → (ii) Chi-square test
• (b) Testing the significance of the differences among the average performance of more than two sample groups → (iii) F-test (ANOVA)
• (c) Testing the significance of the difference between the average performance of two sample groups (Large-sized) → (i) Z-test

This gives us the matching code: (a) - (ii), (b) - (iii), (c) - (i).

Conclusion

The correct matching is (a) - (ii), (b) - (iii), (c) - (i).

Revision Table: Statistical Tests & Applications

Additional Information on Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population based on sample data. It involves formulating a null hypothesis ($\text{H}_0$) and an alternative hypothesis ($\text{H}_1$), choosing an appropriate statistical test, setting a significance level ($\alpha$), calculating a test statistic, and making a decision based on the p-value or critical region.

The choice of the statistical test depends on several factors:

• The type of data (e.g., continuous, categorical).
• The number of groups being compared.
• Whether the data meets assumptions of the test (e.g., normality, equal variances).
• Sample size.

Understanding these factors is crucial for applying the correct statistical test in research and data analysis.