Assume the rise or the drop of the daily stock price is an independent and
identically distributed (i.i.d.) random variable. Suppose that the
probability of the rise in price for a company’s stock is p , and its daily
stock price will either increase by 1 dollar or drop by 1 dollar. Now an
investor purchases this stock at 20 dollars, what is the average profit (the
expected profit) that she earns by selling the stock after 5 trading days?
What is the variance of the profit? If she holds the stock for 10 trading
days and then sells, how is this different from the case of holding 5 days?(20%)
In the past, a manufacturer on average produces 500 widgets per hour.
Now in order to enhance the production efficiency, an expert is hired to
develop a computerized manufacturing process. 25 hours are
independently drawn from such production process, it is found that the
average amount of 580 widgets is produced per hour, with the standard
deviation of 120. Use the large-sample test to answer the following
questions:
(a) At a 5% significance level, test for the effectiveness of the computerized
manufacturing process. (8%)
(b) Given that the true population mean of production is 540 widgets, at a
5% significance level, calculate the probability of Type II error and the
power of the hypothesis testing performed in part (a). (8%)
(c) If we want to reduce Type II error to about 20%, then how much Type I
error will have to increase? (10%)
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