Section B (41 marks) 7. Five metal ingots were chosen at random and measurements were made of their breaking strength s and their hardness h . The results are shown in the table below. 5 5.5 6 Breaking strength (s tonnes per cm) 6.5 T 60 68 71 Hardness (h) 70 73 The relationship between the variables can be modelled by the regression equation h = as +b. (a) Write down the value of a and of b. [2] (b) Use the regression equation to estimate the hardness of a metal ingot when its breaking strength is 6.3 tonnes per cm. [2] There are 120 metal ingots in total. The cumulative frequency below shows the distribution of the hardness of all metal ingots. Hardness less than Cumulative Frequency 55 0 60 30 65 56 70 75 108 80 120 (c) A metal ingot is randomly selected. Find the probability that the hardness of the selected ingot is (i) less than 75; (ii) at least 65; (ii) at least 75, given that its hardness is at least 65. Two metal ingots are randomly selected. The probability that the hardness of [6] the two selected ingots are both at least 70 is 476 (d) Find