1. A marine biologist is interested in the length of a certain fish. For that she has collected a random sample of 46 fishes, registering their lengths in centimeters. The data is the following: (Total marks: 20) c(25.3, 25.6, 15.3, 24.1, 17, 22.8, 27.3, 14.1, 20.6, 27.6, 29, 22.3, 17.1, 19.3, 12.8, 15.8, 14, 28.1, 27.2, 14.1, 25, 27, 21.8, 26.5, 21.9, 24.2, 11.2, 23, 17.1, 20.7, 27.1, 27.9, 21.8, 20.3, 27, 26.4, 31.4, 25.2, 20.2, 10.5, 27.5, 23.9, 17.9, 23.3, 27.6, 20.8) (a) Summarize the data numerically. (5 marks) (b) Interpret the third quantile. (5 marks) (c) Present the data in an appropriate plot. (5 marks) (d) Comment on the distribution of the data. (5 marks) 2. The probability of a student playing football is known to be 0.59; and the probability ot a student playing rugby is known to be 0.5. If the probability of playing both is known to be 0.42, calculate: (Total marks: 25) (a) The probability of not playing rugby. (4 marks) (b) The probability of playing at least one of football and rugby. (4 marks) (c) The probability of a student playing rugby, given that they play tootball. (4 marks) (d) Are playing rugby and football independent? Justify your answer. (5 marks) (e) (harder) A group of 32 randomly selected students attend a special seminar on the health benetits of playing sport. Of these 32, only 5 play neither after the seminar. State a sensible null hypothesis, test it, and interpret. (8 marks) Provide your working for each question. 3. Random variable X is drawn from a normal distribution with mean 9.19 and standard devi- ation 2.88. (Total marks: 25) (a) Calculate the probability of X being less than 6.1 B. (5 marks) (b) What is the probability of X exceeding 8.4? (5 marks) (c) What is the probability of X lying between 9.57 and 11.23? (5 marks) (d) Verity your answers to parts (a) and (b) above using numerical sampling. (5 marks) (e) (Harder) verity your answers to part (c) above using numerical sampling. (5 marks) Express all the probabilities in probability notation. Provide the H code in addition to your answers. 4. Concrete specimens are assessed using a certain standardized test involving stressing the specimen in a rig. The concrete either breaks (fails), or does not break (passes the test; succeeds). It is known that the probability of passing the test is 0.274. A material scientist claims that painting the concrete with a particular paint increases the success rate. She takes 323 samples, paints them, and finds that 90 pass. (Total marks: 30) (a) State a sensible null hypothesis. (5 marks) (b) State the precise definition of the p-value. (5 marks) (c) One-sided or two sided? Justify your answer. (5 marks)