BigCarCo, a car manufacturer, receives a shipment of 192 headlights from a supplier. They will accept the shipment unless they are confident that the average brightness of all headlights in the shipment is less than 2000 lumens. To test the shipment, they measure the brightness of a random sample of 18 headlights from the shipment. The results appear in the table below. 1981,6 2012,0 1991,0 1988,5 1994,5 1999,6 1984,0 1982,0 1975,1 1991,6 1981,5 1990,8 1990,7 2001,1 1986,3 1988,2 2004,7 1997,2 a) For the tested headlights, find: I. the mean brightness, and [2 marks] Il. the standard deviation of the brightness. [2 marks] b) Using a significance level of alpha = 1%, test to see if this shipment should be accepted or rejected. Be careful to state any assumption that you need to make about the population distribution. [6 marks] c) Explain briefly how you could check to see if the sample data is consistent with the assumption mentioned in part (b). [1 mark] d) If the mean brightness of the whole shipment really was 2000 lumens, what is the probability of getting an average sample brightness lower than your answer to (a)(i). [3 marks]