Chi-square tests #2 - Questions and Answers

Question 1 Distinguish the difference (purpose) between the two different types of chi-square 1.Chi-square test for independence (CTI) 2.Chi-square test for goodness of fit (CGF) 3.What type of variables are we using for a chi-square? Answer Test of independence: A sample of several observations is collected from a population, the researcher wants to determine whether two or more attributes are independent of each other, or associated with each other. Each observation is present at one of the levels of the attributes considered. Step 2 Chi-square goodness of fit: Chi-square goodness of fit test is a non-parametric test used to find how the observed value of a given phenomena is significantly different from the expected value. In Chi- square goodness of fit test the term goodness of fit is used to compare the observed sample distribution with the expected probability distribution. The null hypothesis assumes that there is no significant difference between the observed and expected value and the alternative assumes that there is a significant difference between the observed and expected value. Assumption of goodness of fit test: -The data is obtained from a simple random sampling. -The variable under study is categorical. -The each cell count (for each category) is at least 5 and the observations are independent. The categorical variable are used for a chi-square tests. Question 2 The chi-square test is used when we have interval data. A. True B. False Answer Step 1 The chi-square is a non-parametric test which is used for comparing the categorical variables. Chi-square test is used to test whether there is relation between two-categorical variables or not in the study. It is a statistical method that tests whether at each level of the categorical variable the observed frequencies are similar to the expected frequencies or not. The test involves only two factors that can have any number of levels. Step 2 It is clear that, chi-square test is used only when the data is categorical but not numerical data (interval, ratio). This shows that the statement is false. Result: Option B. False Question 3 Chi-square tests are best used for which type of dependent variable? nominal, ordinal ordinal interval nominal, interval nominal, ratio Answer Step 1 The correct option is: 1. Nominal, ordinal. The Chi-square tests are used to test the categorical variables. Step 2 The categorical variables can be further classified as nominal level and ordinal level. Thus, the Chi-square tests are best used for nominal and ordinal level dependent variable. Question 4 Explain the Chi - Square test. Answer Step 1 Introduction: The chi-square test can be used to test the in the following scenarios: To test the Goodness of fit of the variables when their expected and observed frequencies are given. To test the independence of the categorical variables by making it into a contingency table. To test the significance of the single variance with the given variance. 1). Goodness of fit: Goodness of fit test is applied to check how well the sample data obtained fits the distribution of the selected population. It can also be viewed as whether the frequency distribution fits the given pattern. Most commonly used test to check the goodness of fit is the chi-square test. There are two values involved are observed and the expected values. The observed value represents the frequency of particular category in the sample and the expected value is obtained from the given distribution. Moreover, it summarizes the difference between the expected and observed values of the given data. The hypotheses are stated as given below: Null hypothesis: Data comes from the specified distribution. Alternative hypothesis: Data does not come from the specified distribution. The chi-square test statistic is calculated using the formula given below: x² = Σ(Oᵢ - Eᵢ)² / Eᵢ where, Oᵢ-Represents the observed values Eᵢ-Represents the expected values Step 2 2). Test for independence: In test for independence, we test whether there is an association between the categorical variables. Null hypothesis: There is no association between the two categorical variables. Alternative hypothesis: There is no association between the two categorical variables. The chi-square test statistic is calculated using the formula given below: x² = ΣΣ(Oᵢⱼ - Eᵢⱼ)² / Eᵢⱼ where, Oᵢⱼ-Represents the observed values in iᵗʰ column and jᵗʰ row Eᵢⱼ-Represents the expected values in iᵗʰ column and jᵗʰ row Step 3 3). Test for single variance: Here, the chi-square test is used to compare the single sample variance with the population variance. Null hypothesis: The given sample variance is equal to the population variance. Alternative hypothesis: The given sample variance is not equal (less or greater) to the population variance. The chi-square test statistic is calculated using the formula given below: x² = (n-1)s² / σ² where, n-Sample size s²-Sample variance σ²-Population variance Step 4 Examples for some research questions based on chi square tests: 1.Is there significant association between blood pressure and exercise level? 2.Whether the class has less variation in statistics marks than the other past statistics classes marks? 3.To check whether health insurance benefits vary by the size of the company. 4.To test the claim that the subject distribution of books in the library fits the distribution of books checked out by students. 5.The level of education and the amount of proceeded foods in an individual's diet are independent or not.