(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.

(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.

(20%)6. Supporters claim that a new windmill can generate an average of at least 800 kilowatts of power per day. Daily power generation for the windmill is assumed to be normally distributed with a standard deviation of 120 kilowatts. A random sample of 100 days is taken to test this claim against the alternative hypothesis that the true mean is less than 800 kilowatts. The claim will be accepted if the sample mean is 776 kilowatts or more and rejected otherwise. (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?

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