(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?

(15%)2. A hamburger stand sells burgers for $1.45 each. Daily sales have a distribution with mean 530 and standard deviation 69. (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%)3. A consultant knows that it will cost her $10,000 to fulfil a particular contract. The contract is to be put out for bids, and she believes that the lowest bid, excluding her own, can be represented by a distribution that is uniform between $8,000 and $20,000. Therefore, if the random variable X denotes the lowest of all other bids (in thousands of dollars), its probability density function is $$f_x(x) = \begin{cases} \frac{1}{12}, & \text{for } 8 < x < 20 \\ 0, & \text{for all other values of } x \end{cases}$$ (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.

(10%)4. It is believed that first-year salaries for newly qualified accountants follow a normal distribution with standard deviation $2,500. A random sample of 16 observations was taken. (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.

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