98年國立清華大學經濟學系統計學線上測驗

3.: The probability that any child in a certain family will have blue eyes is 1/4, and this
feature is inherited independently by different children in the family. If there are five
children in the family and it is known that at least one of these children has blue eyes,
what is the probability that at least three of the children have blue eyes? (10%)

5.: Suppose that a light bulb manufacturing plant produces bulbs with a mean life of 2000
hours and a standard deviation of 200 hours. An inventor claims to have developed an
improved process that produces bulbs with a longer mean life and the same standard
deviation. The plant manager randomly selects 100 bulbs produced by the process. She
says that she will believe the inventor’s claim if the sample mean life of the bulbs is
greater than 2100 hours, otherwise she will conclude that the new process is no better
than the old process. Let μ denote the mean of the new process. Consider the null and
alternative hypothesis H₀ : μ = 2000 vs. H₁ ：μ > 2000 . (10%)
(一) What is the size of the plant manager’s testing procedure?
(二) Suppose that the new process is in fact better and has a mean bulb life of 2150
hours. What is the power of the plant manager’s testing procedure?

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