2. Let X denote the amount of time it takes for a student to complete a quiz a ECU11022. X is normally distributed with a standard deviation of 12 minutes, i.e., X-N(,90). The lecturer assumes that the mean for the entire class is 45 minutes, i.e., =45. You would like to investigate whether the assumption is true. Based on what you have learned in class, you take a random sample of 30 students, ask them the time it took them to prepare for the exam, and compute the mean X. a) Provide an expression for the distribution of the sample mean. b) Based on your experience, you think that the sample average time to prepare for the midterm is above 60 minutes. Compute the probability that the average preparation time is more than 60 minutes. Discuss the result. c) What do you conclude about the lecturer's assumption about the average time it takes to complete a quiz? d) What would happen if you reduced the sample size to 10 students?