Question 3 (a) Table Q3 is the probability distribution of the lifetime of a lightbulb, X , in months. Table Q3 X 5 8 12 16 24 P(X = x) 0.23 0.10 k 0.28 0.12 Calculate the following values. (1) k (2 marks) (ii) Variance (3 marks) (b) 8% of the light bulbs manufactured on an assembly line are defective. (i) Calculate the probability that the second defective light bulb will be found on the tenth inspection if the light bulbs are inspected one by one. (3 marks) (ii) In a random sample of a light bulbs, the probability to get at least one defective light bulb is greater than 0.9. Calculate the smallest possible value of n. (3 marks) (iii) A random sample of 1800 light bulbs is taken. Calculate the probability that there are at least 152 are defective. (4 marks) (c) In order to examine the effect of temperature on the breaking strength of chains, 9 chains were randomly selected, and each was cut in half. One half, chosen at random, was tested at 15 C and the other at 20"C. The results were shown in the Table Q4. Chain 2 3 4 5 6 7 8 9 15 C 1307 1324 1321 1305 1306 1304 1306 1321 1315 20 C 1312 1320 1318 1303 1298 1297 1312 1310 1306 Test, at the 5% significance level, whether the average breaking strength is less at the higher temperature