Question

Let \( X \) and \( Y \) be two random variables such that \( p_{11} + p_{10} + p_{01} + p_{00} = 1 \), where \( p_{ij} = P(X = i, Y = j) \), \( i, j = 0, 1 \). Suppose that a realization of a random sample of size 60 from the joint distribution of \( (X,Y) \) gives \( n_{11} = 10 \), \( n_{10} = 20 \), \( n_{01} = 20 \), \( n_{00} = 10 \), where \( n_{ij} \) denotes the frequency of \( (i,j) \) for \( i,j = 0,1 \). If the chi-square test of independence is used to test \[ H_0: p_{ij} = p_i p_j \text{ for } i,j = 0,1 \quad \text{against} \quad H_1: p_{ij} \neq p_i p_j \text{ for at least one pair } (i,j), \] where \( p_i = p_{i0} + p_{i1} \) and \( p_j = p_{0j} + p_{1j} \), then which one of the following statements is true?

• A. Under \( H_0 \), the test statistic follows central chi-square distribution with one degree of freedom and the observed value of the test statistic is \( \frac{20}{3} \)
• B. Under \( H_0 \), the test statistic follows central chi-square distribution with three degrees of freedom and the observed value of the test statistic is \( \frac{20}{3} \)
• C. Under \( H_0 \), the test statistic follows central chi-square distribution with one degree of freedom and the observed value of the test statistic is \( \frac{16}{3} \)
• D. Under \( H_0 \), the test statistic follows central chi-square distribution with three degrees of freedom and the observed value of the test statistic is \( \frac{16}{3} \)

Hint:

In a chi-square test of independence, the degrees of freedom are calculated as \( (r-1)(c-1) \), where \( r \) and \( c \) are the number of rows and columns in the contingency table.

Answer 1

The Correct Option is A

The given joint distribution is for two random variables \( X \) and \( Y \), and the chi-square test of independence is being used. The chi-square statistic for the test is given by: \[ \chi^2 = \sum_{i,j} \frac{(n_{ij} - E_{ij})^2}{E_{ij}}, \] where \( E_{ij} \) is the expected frequency for each cell, computed under the assumption of independence, \( E_{ij} = n \cdot p_i p_j \), where \( n \) is the sample size, and \( p_i, p_j \) are the marginal probabilities. For the given question, we have 2 rows and 2 columns, which gives 1 degree of freedom. Thus, the test statistic follows a central chi-square distribution with one degree of freedom. The observed value of the test statistic is calculated to be \( \frac{20}{3} \). Final Answer: \[ \boxed{\text{(A) Under \( H_0 \), the test statistic follows central chi-square distribution with one degree of freedom and the observed value of the test statistic is } \frac{20}{3}}. \]