Suppose that X1,..., Xm and Y1,...,Ym are independent random samples, with the X; drawn from a normal population with parameters µi and o1, and the Y; drawn from a normal population with parameters 42 and o2. The difference between the sample means X – Ỹ is a linear combination of m+n independent normally distributed random variables and is therefore also normally distributed. (a) Find E(X – Ý) and V(X – Ỹ). (b) Suppose that m = n and of = 2 and o? = 2.5. Find the sample sizes so that X – Ỹ will be within 1 unit of H1 – H2 with probability 0.95.

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M-1 Please solve this problem very clearly and briefly with step by step explanation.

1. Suppose that X1,..., Xm and Y1,...,Ym are independent random samples, with the X; drawn from
a normal population with parameters µi and o1, and the Y; drawn from a normal population with
parameters 42 and o2. The difference between the sample means X – Ỹ is a linear combination of
m+n independent normally distributed random variables and is therefore also normally distributed.
(a) Find E(X – Ỹ) and V(Š – Ỹ).
(b) Suppose that m = n and o? = 2 and of = 2.5. Find the sample sizes so that X – Y will be
within 1 unit of µ1 – H2 with probability 0.95.
Transcribed Image Text:1. Suppose that X1,..., Xm and Y1,...,Ym are independent random samples, with the X; drawn from a normal population with parameters µi and o1, and the Y; drawn from a normal population with parameters 42 and o2. The difference between the sample means X – Ỹ is a linear combination of m+n independent normally distributed random variables and is therefore also normally distributed. (a) Find E(X – Ỹ) and V(Š – Ỹ). (b) Suppose that m = n and o? = 2 and of = 2.5. Find the sample sizes so that X – Y will be within 1 unit of µ1 – H2 with probability 0.95.
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