STA4322 Exam3 (closed book, 9:30 - 10:45, 8/6/2009) Name
1. In hypothesis testing, we need two hypotheses H0and Ha, a rejection region RR, and
two types of errors αand β. State the two main relations between αand βand the
other terms. (10%)
Sol: α=P{RR|H0}and 1 −β=P{RR|Ha}
2. (i) When we say that ˆ
θis a MVUE (minimum variance unbiased estimator) for an
unknown parameter θ, what do we means. Explain it using commonly used statistical
notation. (10%)
Sol: (i) E[ˆ
θ] = θand for and other unbiased estimator ˜
θ,V ar(˜
θ)≥V ar(ˆ
θ)
(ii) What we say that a test is the uniformly most powerful test in hypothesis testing,
what do we mean? Use an example to explain this. (10%)
Sol: Let H0:p=p0and Ha:p > p0. Then the uniformly most powerful test is the
most powerful for test for all the simple hypothesis Ha:p=pawith pa> p0.
3. The minimum requirement for the hardness index of an alloy is 65. Fifty specimen of
the alloy was collected and the sample mean is 68 with sample standard deviation 8.
Can the company claim that their alloy meets the hardness requirement? Make you
conclusion by p-value. (15%)
Sol: To guarantee the product meets the requirement, we need to put Haas µ > 65.
Thus H0is µ≤65.
p=value =P( ¯y≥68)
=P(Z≥68 −65
8/√50 )
=P(Z≥2.65) = 0.004
4. Let y1,y2,..., ynbe a random sample from a Poisson distribution with unknown mean
λ. Find the most powerful test for H0:λ= 1 against Ha:λ= 2. You are allowed to
have one threshold to be determined by the type I error probability. (10%)
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