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Let X1, X2, . . . , X10 be a sample of size 10 from an...

Let X1, X2, . . . , X10 be a sample of size 10 from an exponential distribution with the density function f(x; λ) = ( λe(−λx), x > 0, 0, otherwise. We reject H0 : λ = 1 in favor of H1 : λ = 2 if the observed value of Y = P10 i=1 Xi is smaller than 6.

(a) Find the probability of type 1 error for this test.

(b) Find the probability of type 2 error for this test.

(c) Let y = 5 be the observed value of Y . Find the p-value for this test. (d) Let y = 5 be the observed value of Y . Construct the exact 95% confidence interval for λ. Hint: λY ∼ Gamma(10, 1).

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