Let $X \square \operatorname { Bin } \left( m , p _ { 1 } \right) \text { and } Y \square \operatorname { Bin } \left( n , p _ { 2 } \right)$ with X and Y independent variables, and let $\hat { p } _ { 1 } = X / m \text { and } \hat { p } _ { 2 } = Y/ n$ Which of the following statements are not correct?
A) $E \left( \hat { p } _ { 1 } - \hat { p } _ { 2 } \right) = p _ { 1 } - p _ { 2 } \text {, so } \hat { p } _ { 1 } - \hat { p } _ { 2 }$
Is an unbiased estimator of $p _ { 1 } - p _ { 2 }$
B) When both m and n are large, the estimator $\hat { p } _ { 1 } - \hat { p } _ { \mathbf { 2 } }$
Individually has approximately normal distributions.
C) When both m and n are large, the estimator $\hat { p } _ { 1 } - \hat { p } _ { 2 }$
Has approximately a normal distribution.
D) $V \left( \hat { p } _ { 1 } - \hat { p } _ { 2 } \right) = p _ { 1 } q _ { 1 } / m \quad p _ { 2 } q _ { 2 } / n , \text { where } q _ { l } = 1 - p _ { l } \text { for } i = 1,2$
E) All of the above statements are correct.

Question

Quizplus · Accepted Answer

The variance of the difference between two independent binomial proportions $\hat { p } _ { 1 } - \hat { p } _ { 2 }$ is given by $V \left( \hat { p } _ { 1 } - \hat { p } _ { 2 } \right) = \frac{p _ { 1 } q _ { 1 }}{m} + \frac{p _ { 2 } q _ { 2 }}{n}$, not $p _ { 1 } q _ { 1 } / m \quad p _ { 2 } q _ { 2 } / n$ as stated. The correct formula adds the variances of $\hat { p } _ { 1 }$ and $\hat { p } _ { 2 }$, reflecting the properties of variance for independent variables.

Let X□Bin⁡(m,p1) and Y□Bin⁡(n,p2)X \square \operatorname { Bin } \left( m , p _ { 1 } \right) \text { and } Y \square \operatorname { Bin } \left( n , p _ { 2 } \right)X□Bin(m,p1​) and Y□Bin(n,p2​)

Let $X \square \operatorname { Bin } \left( m , p _ { 1 } \right) \text { and } Y \square \operatorname { Bin } \left( n , p _ { 2 } \right)$