108年 - 108 國立臺北大學_日間學士班轉學生招生考試試題_統計學系日間學士班2、3年級：統計法#147796

(a) What is the probability that a randomly chosen entering student is a freshman who will eventually graduate?

(b) Find the probability that a randomly chosen entering student will eventually graduate.

(c) What is the probability that a randomly chosen entering student is either a freshmen or will eventually graduate, or both?

(d) Are the events "Eventually graduates" and "Enters as junior college transfer" statistically independent?

(a) Find the mean daily total revenues from the sale of hamburgers.

(b) Find the standard deviation of total revenues from the sale of hamburgers.

(c) Daily costs (in dollars) are given byㄒ $$C = 100 + 0.95X$$ where X is the number of hamburgers sold. Find the mean and standard deviation of daily profits from sales. (15%)

(a) What is the probability that the lowest of the other bids will be less than the consultant's cost estimate of $10,000?

(b) If the consultant submits a bid of $12,000, what is the probability that she will secure the contract?

(c) The consultant decides to submit a bid of $12,000. What is her expected profit from this strategy?

(d) If the consultant wants to submit a bid so that her expected profit is as high as possible, discuss how she should go about making this choice.

(a) Find the probability that the sample standard deviation is more than $3,000.

(b) Find the probability that the sample standard deviation is less than $1,500.

(10%)5. Independent random samples were taken of male and female clients of University Entrepreneurship Centers. These clients were considering starting a business. Of 94 male clients, 53 actually started a business venture, as did 47 of 68 female clients. Find and interpret the p-value of a test of equality of the population proportions against the alternative that the proportion of female clients actually starting a business is higher than the proportion of male clients.

(a) What is the probability \( \alpha \) of a Type I error using the decision rule if the population mean is in fact 800 kilowatts per day?

(b) What is the probability \( \beta \) of a Type II error using this decision rule if the population mean is in fact 740 kilowatts per day?

(i)Would the value of \( \alpha \) be larger than, smaller than, or the same as that found in (a)?

(ii)Would the value of \( \beta \) be larger than, smaller than, or the same as that found in (b)?

(i)Would the value of \( \alpha \) be larger than, smaller than, or the same as that found in (a)?

(ii)Would the value of \( \beta \) be larger than, smaller than, or the same as that found in (b)?

(a) Find and interpret the coefficient of determination.

(b) Test at the \( 10\% \) significance level against a two-sided alternative the null hypothesis that \( \beta_{1} \) is 0.

(c) Find