Suppose that λ% ( 0 <λ <1 ) of the population never buy lotteries. Among the
remaining (1−λ )% population, the probability of buying lotteries is 0.8 when the
lottery stake(彩票獎金)accumulates more than ten million dollars; the probability of
buying lotteries is 0.3 when the stake is smaller than ten million dollars. The probability
for the lottery to have more than ten million dollars stake is 0.9; the probability for the
lottery to have less than ten million dollar stake is 0.1.
(一) When is the probability of buying lotteries for a given lottery stake in the whole
population? (Remark: the given stake could be larger or smaller than ten million
dollars.) (Hint: The answer is a function of λ .) (7%)
(二) In a random sample of 100 residents, only 15 of them bought lotteries for a specific
lottery stake. Use the Maximum Likelihood Estimation (MLE) method to estimate
the unknown parameter λ . (8%)
Answer the following questions.
(一) What is the difference between Type I and Type II errors in hypothesis testing?
How do α and β relate to Type I and Type II errors? (4%)
(二) If the average monthly salary of workers with an undergraduate degree of
economics is NT$30,000. You collect a sample (large sample, n ) and use a 95%
confidence interval to estimate the population mean. What is meant by the phrase
“95% confidence interval”? (4%)
(三) You estimate a simple regression and obtain the determinant of coefficient
R2 = 0.81. Please interpret the result. (4%)
(四) Show the formula of Laspeyres’ Price Index and indicate its shortcoming. (4%)
Advertisements by California Fitness Center claim that completing their course will
result in losing weight. A random sample of nine participants reported their weights
before and after completing the course (See the table below). At the 0.05 significance
level, can you conclude that the participants lose weight?(計算至小數點第一位即可)
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