Statistics and probalbility

{a} A quality assurance engineer realized that the diameter of the tubes produced by a machine is normally distributed with a standard deviation of 8.025 cm. Show clearlyF all the steps in answering the following questions and give your answers to four decimal places. (i) The engineer rejected the tubes with the diameter smaller than 59.3 cm. he discovered that the proportion of the tubes he rejected is 5% of the population, as shown in the diagram. Prove that the mean of diameter is 73 cm. (4 marks} 3' {ii} Find the range of the diameter that is two standard deviations from the mean. Hence. state the proportion of the tubes with the diameters in this range. (5 marks) {iii} If a tube is selected randome from the population, nd the probabilityF that the tube diameter is in between T0 cm to 30 cm. (4 marks) {iv} State the mode of the population for the diameter of the tubes. Briefly explain your answer. (2 marks}