A hospital claims that 75% of people who come to its emergency room are seen by a doctor within 30 minutes of checking in. To verify this claim, an auditor inspects the medical records of 55 randomly selected patients who checked into the emergency room during the last year. Only 32 (58.2%) of these patients were seen by a doctor within 30 minutes of checking in.

a. If the wait time is less than 30 minutes for 75% of all patients in the emergency room, what is the probability that the proportion of patients who wait less than 30 minutes is 0.582 or less in a random sample of 55 patients?

b. Based on your answer to part (a), is there convincing evidence that less than 75% of all patients in the emergency room wait less than 30 minutes? Explain your reasoning.