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Let X1, X2, . . . , Xn be a random sample from a population following...

Let X1, X2, . . . , Xn be a random sample from a population following a uniform(0,2θ)

(a) Show that if n is large, the distribution of Z =( X − θ) / ( θ/√ 3n) is approximately N(0, 1).

(b) We can estimate θ by X(sample mean) and define W = (X − θ ) / (X/√ 3n) which is also approximately N(0, 1) for large n. Derive a (1 − α)100% confidence interval for θ based on W.

(c) Find a (1 − α)100% confidence interval for θ directly from Z defined in (a).

(d) Using the result in (b), find a (1 − α)100% confidence interval for the variance of X which follows a uniform(0,2θ) distribution.

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