The manufacturer of a certain type of new cell phone battery claims that the average life span of the batteries is 500 charges; that is, the battery can be charged at least 500 times before failing. To investigate the claim, a consumer group will select a random sample of cell phones with the new battery and use the phones through 500 charges of the battery. a. If the claim is true, what is P( X ≤ 36.7) ?b. Based on the answer to part (a), if the claim is true, is a sample mean lifetime of 36.7 hours unusually short?c. If the sample mean lifetime of the 100 batteries were 36.7 hours, would you find the manufacturer's claim to be plausible? Explain. d. If the claim is true, what is P( X ≤ 39.8)?e. Based on the answer to part (d), if the claim is true, is a sample mean lifetime of 39.8 hours unusually short?f. If the sample mean lifetime of the 100 batteries were 39.8 hours, would you find the manufacturer's claim to be plausible? Explain.