Probability and Statistics Problems and Solutions, Exams of Mathematics

Download Probability and Statistics Problems and Solutions and more Exams Mathematics in PDF only on Docsity! SECTION A 1. A panel of experts rated the attractiveness of a sample of 20 catwalk models on a scale from 0 to 100, with 100 representing perfect attractiveness. The resulting data, ordered from smallest to largest, were as follows. 65, 71, 72, 73, 75, 75, 76, 77, 77, 77, 77, 79, 80, 80, 83, 83, 83, 87, 87, 99. (i) Compute the median and upper and lower quartiles of the given data. [3 marks] (ii) Represent the data by a boxplot. Comment on your plot. [5 marks] 2. (i) For two independent events A and B, express P (A∪B) in terms of P (A) and P (B). [2 marks] (ii) For three events C,D,E which are mutually exclusive and exhaustive, express P (C) in terms of P (D) and P (E). [2 marks] (iii) For two events F,G with P (F ) 6= 0 and P (G) 6= 0, write down the definition of the conditional probability P (G | F ). [2 marks] Express P (G | F ) in terms of P (F ), P (G) and P (F | G). [2 marks] 3. The diagram below shows a network of unreliable components. For each component X, denote by P (X) the probability that X is working. Given that P (A) = P (B) = P (E) = 0.9, P (C) = P (D) = 0.85, and that whether any component X is working or not is independent of whether any other component is working or not, calculate the network reliability (that is, the probability that a signal will pass successfully from START to FINISH). START   A   B     C Q Q Q Q Q Q  D     E FINISH 1 [8 marks] 4. In a fibre-optic cable communications system, the number of transmission errors which occur in any 1 minute period is a Poisson random variable with mean 4.5. Denoting by X the number of errors which occur during a particular one minute period, find the values of (i) Var[X] [1 mark] (ii) P (X ≤ 2) [2 marks] (iii) P (X ≥ 1) [2 marks] (iv) P (X = 0 | X ≤ 2) [3 marks] 5. A random variable X is normally distributed with mean 20 and variance 9. Calculate (i) P (X ≤ 26) [2 marks] (ii) P (15 < X < 21) [3 marks] (iii) The value a such that P (X < a) = 0.6554. [3 marks] 2 8. A public policy group investigating the political preferences of people living in the same household selected 100 homes, each containing exactly 4 voters. The residents were asked separately for their opinion (in favour or against) on a proposed new law. The resulting data were as follows. Number in favour 0 1 2 3 4 Frequency 17 46 30 6 1 (i) Use the above data to calculate the proportion of people in the population who are in favour of the new law. [2 marks] (ii) It is suggested that the above data might be expected to follow a binomial distri- bution. Discuss reasons why this might or might not be a reasonable assumption. [4 marks] (iii) Calculate the binomial probabilities P (X = x) for x = 0, 1, 2, 3, 4, where X is a ran- dom variable following the binomial distribution with n = 4 and success probability p equal to the proportion computed in part (i) above. [4 marks] (iv) Use a goodness of fit test at the 5% level to assess whether the binomial distribution is appropriate for the given data. [6 marks] (v) Suppose now that you are asked to test whether the above data are well described by a binomial distribution with success probability p = 0.5. Without carrying out further detailed calculations, explain how the test procedure would differ from that in part (iv) above. [4 marks] 5 9. It is estimated that three-quarters of adults in the UK are regular internet users. Denote by X the number of regular internet users in a random sample of 60 UK adults. (i) Assuming independence between individuals in the sample, what is the exact distri- bution of X? (Give the values of any parameters of the distribution.) [3 marks] (ii) Use an appropriate approximation (either normal or Poisson) to calculate approx- imately the probability that the number of regular internet users in the sample is betwen 45 and 50, i.e. P (45 ≤ X ≤ 50). [6 marks] Justify your choice of approximating distribution, explaining both why the distribu- tion you chose (normal or Poisson) is appropriate, and also why the alternative would not have been appropriate. [3 marks] (iii) It is estimated that 5% of UK adults over the age of 65 are regular internet users. Use an appropriate approximation (either normal or Poisson) to calculate approximately the probability that in a sample of 60 UK adults over the age of 65 the number of regular internet users is less than 5. (You may assume independence between individuals in the sample,) [5 marks] Again, justify your choice of approximating distribution. [3 marks] 6 10. (a) In a study of treatments for scoliosis, a condition involving curvature of the spine, two different treatments, A and B, were compared. For each of a sample of 240 patients, the researchers recorded which treatment was used and whether the treatment was a success or failure (treatment being deemed to have failed if the curvature of the spine increased by 6o on two successive examinations). The resulting data were as follows. Treatment A Treatment B Success 94 71 Failure 17 58 Test at the 5% level the hypothesis that the success or failure of treatment is inde- pendent of which particular treatment is used. What advice would you give to medical practitioners in the light of the result of your statistical test? [8 marks] (b) An experiment was conducted to test the ability of trained dogs to detect bladder cancer. A large random sample was taken from a population in which 17% of indi- viduals actually had bladder cancer. Of those who actually had bladder cancer, 40% were successfully identified. Of those who did not have bladder cancer, 10% were incorrectly identified as having cancer. (i) For an individual chosen at random from the population (who may or may not actually have bladder cancer), what is the probability that the dog will identify them as having bladder cancer? [2 marks] (ii) Given that an individual is identified by the dog as having bladder cancer, what is the probability that they actually do have bladder cancer? [4 marks] (iii) Given that an individual is identified by the dog as not having bladder cancer, what is the probability that they actually do have bladder cancer? [4 marks] (iv) Do your results suggest that trained dogs are able to detect bladder cancer, or not? Explain your answer. What further analysis of the data can you suggest to determine whether trained dogs are able to detect bladder cancer? [2 marks] 7