Question 3 (a) (b) A certain type of carrot has amass which is normally distributed wii mean 300 gm and standard deviation of 30 gm. 6) 20 of these carrots are chosen at random. Find the probability that the mean mass of these carrots falls between 290 gm and 305 gm. (5 marks) A new culture method thought to be able to improve the mass of these carrot weight. Following this, a sample 20 carrots of the new culture method's been sampled to test whether t1 has increased. Assuming the variance remains the same, and the sample mean ofthese 30 carrots is 315 gm. (ii) Explain why a one-tail test is appropriate. (1 mark) (iii) Carry out the test at the 5% signicance level. (7 marks) (iv) Explain whether it was necessary to use the Central Limit theorem in part (iii) (1 mark) It is known that 70% of the residents of a city like to watch a particular TV program. In a survey, 30 residents in that city were randomly sampled, and asked if they like the particular TV program (i) State the Binomial distribution applicable to this problem. (3 marks) (ii) Find the probability that exactly 25 residents will like the TV program. (2 marks) (iii) Find the probability that more 28 residents will like the TV program. (5 marks) (iv) Find the expected number of 30 residents like the TV program. (1 mark)