Let $X _ { 1 } , X _ { 2 } , \ldots \ldots , X _ { \mathrm { m } }$ be a random sample from a population with mean $\mu _ { 1 } \text { and variance } \sigma _ { 1 } ^ { 2 } \text {, and let } Y _ { 1 } , Y _ { 2 } , \ldots \ldots , Y _ { n }$ be a random sample from a population with mean $\mu _ { 2 } \text { and variance } \sigma _ { 2 } ^ { 2 } \text {, }$ and that the X and Y samples are independent of one another. Which of the following statements are not true?
A) The natural estimator of $\mu _ { 1 } - \mu _ { 2 } \text { is } \bar { X } - \bar { Y }$
B) The expected value of $\bar { X } - \bar { Y } \text { is } \mu _ { 1 } - \mu _ { 2 }$
C) The expected value of $\mu _ { 1 } - \mu _ { 2 } \text { is } \bar { X } - \bar { Y }$
D) $\bar { X } - \bar { Y }$
Is an unbiased estimator of $\mu _ { 1 } - \mu _ { 2 }$
E) All of the above statements are true.

Question

Quizplus · Accepted Answer

The expected value of a difference in means ($\mu _ { 1 } - \mu _ { 2 }$) is a fixed value based on the population parameters, not a random variable. Therefore, it cannot be expressed as the expected value of the sample means difference ($\bar { X } - \bar { Y }$). Instead, the correct statement is that the expected value of $\bar { X } - \bar { Y }$ is $\mu _ { 1 } - \mu _ { 2 }$, which is what statement B correctly states.

Let X1,X2,……,XmX _ { 1 } , X _ { 2 } , \ldots \ldots , X _ { \mathrm { m } }X1​,X2​,……,Xm​

Let $X _ { 1 } , X _ { 2 } , \ldots \ldots , X _ { \mathrm { m } }$