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1. TESTS RAVI KUMAR M,038

2. CONTENTS Statistical Test Introduction Chi-Square Test Introduction Purpose Types Assumptions t-Test Introduction Purpose Types Assumptions z-Test Introduction Purpose Types Assumptions F-Test Introduction Purpose Types Assumptions Conclusion References

3. STATISTICAL TESTING Statistical testing is a way to analyze data to draw conclusions about a population based on a sample. It helps us make informed decisions and test hypotheses. For example, in the context of libraries, we may want to know whether certain user demographics affect library usage, or if two different library services have the same impact on user satisfaction. To do this, we use hypothesis testing, which involves comparing sample data to a hypothesis or expected outcome.

4. Chi-Square Test A Chi-Square test is a statistical test used to determine if there is a significant association between two categorical variables. It's particularly useful when dealing with nominal or ordinal data. Purpose : Tests the relationship between categorical variables, such as user preferences or library service usage.

5. Types Of Chi-Square Test Chi-Square Test of Independence : Tests if two categorical variables are independent of each other. Chi-Square Goodness-of-Fit Test : Tests if observed categorical data matches an expected distribution. Assumptions : The data must be categorical (nominal or ordinal). Observations should be independent. Expected frequency for each category should generally be at least 5.

6. Example : Testing if the frequency of library visits is independent of user demographics (e.g., age groups, gender). Formula : χ2=∑ (Oi​−Ei​) 2 ​ Ei​ Oi​: Observed frequency Ei​: Expected frequency

7. Chi-Square Test Example Example : A library wants to see if the distribution of book genres (Fiction, Non-fiction, Sci-Fi, History) is independent of user age groups (Under 18, 18-35, 36+). Null Hypothesis (H₀) : The distribution of book genres is independent of user age. Alternative Hypothesis (H₁) : The distribution of book genres is dependent on user age. Interpretation : A significant result would suggest that the age group influences genre preferences.

8. T-TEST A t-test is a statistical test used to determine if there is a significant difference between the means of two groups. In library science, t-tests can be a valuable tool to analyze various aspects of library services and user behavior. Purpose : Compares the means of two groups to see if there’s a significant difference between them. Example : Comparing the average number of books borrowed by male and female users or by students vs. faculty members.

9. Types OF T- test One-sample t-test : Comparing a sample mean with a known population mean (e.g., average library checkout rate). Independent two-sample t-test : Comparing the means of two independent groups (e.g., male vs female usage). Paired t-test : Comparing two related groups (e.g., number of books borrowed before and after a new library service). Assumptions : Data should be approximately normally distributed. The samples should be independent (for the independent t-test) and the differences should be normally distributed (for paired t-tests). The population standard deviation is unknown.

10. Formula Formula for two-sample t-test:

11. t-Test Example Example : A library wants to compare the average number of books borrowed per month by two groups of users: students and faculty. Null Hypothesis (H₀) : The average number of books borrowed by students and faculty is the same. Alternative Hypothesis (H₁) : The average number of books borrowed by students and faculty is different. Interpretation : A significant t-test result will indicate a difference in borrowing behavior between students and faculty.

12. z-Test A z-test is a statistical test used to determine if there is a significant difference between a sample mean and a population mean or between the means of two large samples. It's particularly useful when the population standard deviation is known or when the sample size is large enough (typically, n > 30). Purpose :Tests the difference between a sample mean and a known population mean when the sample size is large or the population variance is known.  

13. Assumptions :The population is normally distributed or the sample size is large enough to apply the Central Limit Theorem (n > 30). The population variance is known (for one-sample or two-sample z-tests). Formula : Xˉ: Sample mean μ: Population mean σ : sigma σ: Population standard deviation n: Sample size

14. Example : A library director claims that the average number of books borrowed per student is 10. A sample of 50 students shows an average of 12 books borrowed. Null Hypothesis (H₀) : The average number of books borrowed is 10. Alternative Hypothesis (H₁) : The average number of books borrowed is not 10. Interpretation : A significant z-test result would indicate the average borrowed books differ from the claim.

15. F-test An F-test is a statistical test used to compare the variances of two populations. It's often used in conjunction with other statistical tests, such as ANOVA, to determine if the variances of different groups are equal. Purpose : Compares the variances of two or more groups to determine if they are significantly different. Often used in ANOVA.

16. Types of F-Tests: One-Way ANOVA: Compares the means of more than two groups. The F-test is used to determine if there are significant differences between the means of the groups. Two-Way ANOVA: Compares the means of multiple groups across two factors. The F-test is used to determine the main effects of each factor and the interaction effect between the factors. Formula :

17. F-Test Example Example : Comparing the variance in the number of books borrowed by male and female students. Null Hypothesis (H₀) : The variances in borrowing behavior are the same for both genders. Alternative Hypothesis (H₁) : The variances in borrowing behavior are different for male and female students. Interpretation : A significant result indicates that one group’s borrowing behavior varies more than the other.

18. CONCLUSION Chi-Square Test is used for categorical data to examine relationships between variables or to assess how well observed data matches an expected distribution. It's ideal for understanding associations or deviations from expectations in categorical datasets. t-Test is employed to compare the means of two groups, typically when the sample size is small or the population variance is unknown. It is best used when testing for differences in means, particularly with continuous data. z-Test is similar to the t-test but is used when the population variance is known or the sample size is large. It is commonly used in large sample hypothesis testing for means or proportions, where normality is assumed. F-Test is primarily used for comparing variances or testing the overall significance of a regression model, such as in Analysis of Variance (ANOVA). It helps determine whether the variability within groups is significantly different or whether models can explain data variance.

19. References SPSS Statistical Tests: Understanding When to Use Them . ( n.d .). Encyclopedia Hub. https://www.thegioiso309.com/showinfo-1-24873-0.html Ash7540 - Overview . ( n.d .). GitHub . https://github.com/Ash7540/ Biswal , A. (2024, October 22). Chi-squared Test: Key Insights for Statistical Analysis . Simplilearn.com. https://www.simplilearn.com/tutorials/statistics-tutorial/chi-square-test