A random sample of five pairs of observations were selected, one of each
pair from a population with mean μ1 , the other from a population with μ2.
The data are shown as below (Note: The conditions that are required for
small-sample inferences hold.) (18%)
A random variable X is distributed uniformly between 1 and a , where
a is an unknown parameter.
(a) Write down the probability density function of X. (5%)
(b) Compute E(X) and Var(X) as functions of a . (10%)
(c) With one sample observation of 2.5, test H0 : a = 11 against
H1 : a < 11 at the 10% significance level (i.e. α = 0.1). (5%)
(d) What is the P-value in (c)? (Hint: P-value is the smallest level of type I
error for which the null hypothesis is rejected.) (5%)
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