A company that manufactures smartphones developed a new battery that has a longer life span than that of a traditional battery. From the date of purchase of a smartphone, the distribution of the life span of the new battery is approximately normal with mean 22 months and standard deviation 3 months. For the price of $20, the company offers a two-year warranty on the new battery for customers who purchase a smartphone. The warranty guarantees that the smartphone will be replaced at no cost to the customer if the battery no longer works within 18 months from the date of purchase. 1. in how many months from the date of purchase is it expected that 20 percent of the batteries will no longer work? Justify your answer.
2. Suppose one customer who purchases the warranty is selected at random. What is the probability that the customer selected will require a replacement within 18 months from the date of purchase because the battery no longer works?