SCHOOL OF MATHEMATICAL SCIENCES BACHELOR OF SCIENCE (HONS) IN ACTUARIAL STUDIES BACHELOR OF SCIENCE (HONS) IN FINANCIAL ANALYSIS BACHELOR OF SCIENCE (HONS) IN FINANCIAL ECONOMICS BACHELOR OF SCIENCE (HONS) IN INDUSTRIAL STATISTICS ACADEMIC SESSION: MARCH 2020 SEMESTER MAT 204 4 INTRODUCTION TO STATISTICS / MST 2014 MATHEMATICAL STATISTICS I ASSIGNMENT: INDIVIDUAL PROJECT DUE DATE: 3 July 2020 @5 pm INSTRUCTIONS TO CANDIDATES 1. This assignment will contribute s 40% to your final grade . 2. This is an individual project . Answer ALL quesitons. The assignment must be typewritten with double line spacin g 3. Marks will be allocated for correctness and clarity of the work . Use R language for the computational part. Include the R script, solutions and conclusion to you report for each question. Provide your solution carefully and neatly, preferably in complete and grammatically correct sentences. Comment your code with the # sign. Every function should have a comment. 4. Make sure you begin a new page for each question and state the question number clearly. 5. Submit the completed softcopy of your assignment to ELearn , with the following file name: student_ID _#.doc (for example: 17005577. doc). IMPORTANT Assignments must be submitted on their due dates. If an assignment is submitted after its due date, the following penalty will be imposed: ? One to two days late : 20% deducted from the total marks awarded. ? Three to five days late : 40% deducted from the total marks awarded. ? More than five days late : Assignment will not be marked. Question 1 This question is designed to discover the Central Limit Theorem (CLT) and learn to describe the predictable pattern of the generating empirical distributions of sample means. Consider a random sample from the assigned distribution based on your Student ID ’s number (refer to Appendix ). (a) Describe the parameters and the probability density function (pdf) of the random variable. Plot th e density graph of the distirbution. (b) Generate random sample s of size n = 5, n = 50 and n = 100 from the assigned dist ribution . (c) Calculate the sample mean, sample standard deviation from the sample s generated in part (b).

(d) Perfom 500 repetitions for part (b) and (c) . Plot the relative fre quency histogram for the values of the sample means for 500 samples , each with n = 5, n = 50 and n = 100 . (e) Compare graphically the histogram in part (d) to the density of a suitable Normal distribution. You might find the following R command useful: Curve(dnorm(x,mean=, sd= , add=TRUE, lwd=2, col=”red”)) (f) Commen t on your findings in the context of the CLT . 12, , ..., n X X X Question 2 A new study indicates that babies ma y choose not to learn from someone they do not trust. A group of 60 babies, aged 13 to 16 months, were randomly divided into two groups. Eac h baby watched an adult express great excitement while looking into a box. The babies were then checked the box and it either had a toy in it (the adult could be trusted) or it was empty (the adult was not reliable). The same adult then turned on a push -on li ght with her/his forehead, and the number of babies who imitated the adult’s behaviour by doing the same thing was counted. The results are in the table below. Imitated Did not imitate Reliable Unr eliable 18 10 12 20 Test whether there is evidence that babies are more likely to imitate those they consider reliable at (a) 1% significance level? (b) 5% significance level? Justify your findings. Notes for Question 2: (I) You may consider using function prop.test to perform the above hypothesis test. (II) Present complete procedures of hypothesis testing for the above problem such as null hypothesis, alternative hypothesis, significance level, test statistics value, p-value etc..in your findings. (III) State your conclusion clearly. Appendix Last digit of your student ID Eg: Student ID: 1703557 7 Distribution 0 Continuous Uniform for 1 Geometric with probability of success of 0. 08 2 Poisson with 3 Continuous Uniform for 4 Exponential for 5 Poisson with 6 Normal with 7 Gamma for 8 Geometric with probability of success of 0.18 9 Exponential for - END OF PAPE R - ? ? 5, 20 4.2 ? = ? ? 0,10 2 ? = 2.5 ? = 2 25, 2 ?? == 3, 2 ?? == 2.5 ? = Sample R Scripts: Question 1 data< -replicate (100,{ mm< -rnorm(9,80,5) a< -mean(mm) }) hist(a,nclass=20,col="red",xlab="values for 100 Subjects",main="Plot") Question 2: #data< - c(490, 400) out of c(500, 500) x=c(490,400) n=c(500,500) alpha=0.05 res=prop.test(x,n, alternative = "greater") res