Question

A random sample X1,...,X300 is drawn from a population with a
mean µ = 80 and standard deviation σ = 30 but unknown
distribution.

Let U = (X1 + ...+X100)/100 represent the sample mean of the ﬁrst
100 observations and V = (X100 + ...+X300)/200 represent the sample
mean of the last 200 observations.

a[10 points] What are the approximate distributions of U and V
?

b[10 points] Which probability would you expect to be larger, P(70
<= U <= 90) or P(70 <= U <= 90)?

Answer #1

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A random sample X1,...,X300 is drawn from a population with a
mean µ = 80 and standard deviation σ = 30 but unknown distribution.
Let U = (X1 + ...+X100)/100 represent the sample mean of the ﬁrst
100 observations and V = (X100 + ...+X300)/200 represent the sample
mean of the last 200 observations. a[10 points] What are the
approximate distributions of U and V ? b[10 points] Which
probability would you expect to be larger, P(70 <= U <=...

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