(problem 1 to 3, if the statement is true, prove it. otherwise, disprove. 1. there exists a rational number and an irrational number y such that y is rational. [no need to show a number is rational or irrational in your proof

Thompson and thompson is a steel bolts manufacturing company. their current steel bolts have a mean diameter of 127 millimeters, and a variance of 36. if a random sample of 35 steel bolts is selected, what is the probability that the sample mean would differ from the population mean by greater than 0.5 millimeters? round your answer to four decimal places.