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

question 45

Multiple Choice

Let $X _ { 1 } , X _ { 2 } , \ldots \ldots , X _ { \mathrm { m } }$ be a random sample from a normal population with mean $\mu _ { 1 } \text { and known variance } \sigma _ { 1 } ^ { 2 } \text {, }$ and let $Y _ { 1 } , Y _ { 2 } , \ldots \ldots , Y _ { n }$ be a random sample from a normal 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 true?

• A) \(\bar { X }\)
• Is normally distributed with expected value \(\mu _ { 1 } \text { and variance }σ _ { 1 } ^ { 2 } / m\)
• B) \(\bar { Y }\)
• Is normally distributed with expected value \(\mu_{2} \text { and variance } σ_{2}^{2} / n\)
• C) \(\bar { X } - \bar { Y }\)
• Is normally distributed with expected value \(\mu _ { 1 } - \mu _ { 2 } \text { and variance } \left(σ _ { 1 } ^ { 2 } / m + \sigma _ { 2 } ^ { 2 } / n \right) \text {. }\)
• D) \(\bar { X } - \bar { Y }\)
• Is an unbiased estimator of \(\mu _ { 1 } - \mu _ { 2 }\)
• )
• E) All of the above statements are true.

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