Forever 21 sells women’s flip-flops in (oddly enough) 21 different colors. Despite this vast available color selection, 50% of all flip-flop purchases are in white. Suppose 30 buyers are selected at random.
a. Find the mean, variance, and standard deviation of the number of buyers who purchase white flip-flops.
b. Find the probability that the number of white flip-flops purchased will be within two standard deviations of the mean. Compare this with the predicted result from Chebyshev’s rule.
c. Suppose two groups of 30 customers are independently selected. What is the probability of at least one group having exactly 15 people who buy white flip-flops?
In early 2013 there was a worldwide glut of solar panels. This forced prices down and made this form of energy production affordable to many more people. This oversupply of solar panels may have caused some manufacturers to cut corners. Sainty Solar is a leading producer and claims the proportion of its solar panels that are defective is 0.02. Solar Solutions, a leading installer, receives a shipment of 50,000 solar panels. Before accepting the entire lot, Solar Solutions selects a random sample of 25 panels and thoroughly tests each one. If four or more panels are found to be defective, the entire shipment will be sent back. Otherwise the shipment will be accepted.
a. Suppose the claim is true: The actual proportion of defectives is p = 0.02. What is the probability that the shipment will be rejected? (This is one type of error probability. The company would be making a mistake if this event occurred. It would be rejecting the shipment when the proportion of defectives is as claimed.)
b. Suppose the actual proportion of defectives is p = 0.05. What is the probability that the shipment will be accepted? (This is another type of error probability. In this case, the company would also be making a mistake. It would be accepting the shipment when the proportion of defectives is too high.)
c. Suppose the actual proportion of defectives is p = 0.07. What is the probability that the shipment will be accepted?