A retailer purchases a batch of 1000 fluorescent light bulbs from a wholesaler at a cost of $2 per bulb. The wholesaler agrees to replace each defective bulb with one that is guaranteed to function properly for a charge of $0.20 per bulb. The retailer sells the bulbs at a price of $2.50 per bulb and gives his customers free replacements if they bring defective bulbs back to the store. Let X be the number of defective bulbs in a typical batch, and assume that the mean and standard deviation of X are 50 and 10, respectively.
a. Find the mean and standard deviation of the profit (in dollars) the retailer makes from selling a batch of light bulbs.
b. Find an interval with the property that the retailer can be approximately 95% sure that his profit will be in this interval.