B. Sc. Examination by course unit, 2011 MTH5122 Statistical Methods, Sample Paper Duration: 2 hours Date and time: Apart from this page, you are not permitted to read the contents of this question paper until instructed to do so by an invigilator. You should attempt all questions. Marks awarded are shown next to the questions. Calculators ARE permitted in this examination. The unauthorized use of material stored in pre-programmable memory constitutes an examination offence. Please state on your answer book the name and type of machine used. Statistical functions provided by the calculator may be used provided that you state clearly where you have used them. The New Cambridge Statistical Tables are provided. Complete all rough workings in the answer book and cross through any work which is not to be assessed. Candidates should note that the Examination and Assessment Regulations state that possession of unauthorized materials by any candidate who is under examination conditions is an assessment offence. Please check your pockets now for any notes that you may have forgotten that are in your possession. If you have any, then please raise your hand and give them to an invigilator now. Exam papers must not be removed from the examination room. Examiner(s): B. Bogacka, H. Grossmann c Queen Mary, University of London (2011) TURN OVER Page 2 MTH5122 (2011) Question 1 (15 marks) Let 1 − e−2x (1 + 2x), for x ∈ (0, ∞); FX (x) = 0, otherwise. (a) Write down the conditions for a function F to be a cumulative distribution function. [4] (a) Show that F given above is a cumulative distribution function of a random variable X. [8] (b) Obtain the probability density function of the random variable X. [3] Question 2 (20 marks) Let X be a continuous random variable with the probability density function −(x−α) e , for x > α; fX (x) = 0, otherwise. (a) Show that, for t < 1, the moment generating function for X is MX (t) = eαt /(1 − t). [6] (b) Use the moment generating function to obtain the expectation and the variance of X. [8] (c) Let Y = X − α. Obtain the moment generating function MY (t) and state the distribution of Y . [6] Question 3 (15 marks) Random variables X and Y have joint probability density function given by c(x2 + xy), for 0 < x < 1, 0 < y < 1; fX,Y (x, y) = 0, otherwise. (a) Find the marginal pdf fX (x) and the marginal pdf fY (y) in terms of c. Find c. [6] (b) Are the variables X and Y independent? Explain. [2] (c) Calculate the probability that (0 < X < 1/2 and 0 < Y < 1/2). [4] (d) Find the conditional probability density function of Y |X = x. [3] c Queen Mary, University of London (2011) MTH5122 (2011) Page 3 Question 4 (16 marks) Let Yi ∼ N (µ, σ 2 ), i = 1, . . . , n, be a random sample. iid Consider the following estimators of µ: 1 µ b1 = (Y1 + Y2 ), 2 1 Y2 + . . . + Yn−1 1 µ b2 = Y1 + + Yn , 4 2(n − 2) 4 n µ b3 = 1X Yi n i=1 (a) Show that each of the three estimators is unbiased. [6] (b) Calculate the variance of each estimator and check the estimators’ consistency. [8] (c) Which of the three estimators of the mean would you recommend and why? [2] Question 5 (20 marks) A pharmaceutical company conducted an experiment to compare the mean times (in days) necessary to recover from the effects and complications that follow the onset of the common cold. The experiment compared persons on daily dose of 500 milligrams (mg) of vitamin C to those who were not given a vitamin supplement. For each treatment category 150 adults were randomly selected, and the mean recovery times and standard deviations for the two groups were found to be as shown in the table No supplement 500 mg vitamin C Sample mean 7.2 5.7 Sample variance 12.4 8.3 (a) Do the data indicate that the use of vitamin C reduces the mean time required to recover from complications of the common cold? Use the significance level α = 0.05 to test the relevant hypothesis. [8] (b) State the Central Limit Theorem and explain briefly how it can be applied to obtain the approximate test statistic under the null hypothesis about equality of means of two populations. [12] Question 6 (14 marks) (a) What assumptions about sampling are made for testing the independence of two systems of classification using a contingency table. [4] (b) A study was conducted to determine the effect of early child care on infantmother attachment patterns. In the study 93 infants were classified as either secure or anxious using the Ainsworth strange-situation paradigm. In addition, the infants were classified according to the average number of hours per week that they spent in child care. The data are given in the table below. Attachment Pattern Secure Anxious Low (0–3 h) 24 11 Hours in child care Moderate (4–19 h) 35 10 High (20–54 h) 5 8 Do the data indicate a dependence between attachment patterns and the number of hours spend in child care? Test using α = 0.05. [10] End of Paper c Queen Mary, University of London (2011)