Question 1
(a) Given that X follows a binomial model such that 𝑋 ~ 𝐵𝑖(2000,0.025), with the help of software or otherwise, calculate 𝑃(𝑋  60). If you have used a software, state your methodology clearly.
(5 marks)

(b) Compute 𝑃(𝑋  60) using the normal approximation method and comment on your findings.
(10 marks)

Question 2

A statistician with the Singapore Department of Statistics wants to determine the average annual income of people aged between 35 to 45 in Singapore. From last year’s reports, the mean annual income was $50k SGD, with a standard deviation of $5000.
(a) Calculate the required sample size so that the length of the 95% confidence interval of the Mean Annual Income is less than $1000.
(10 marks)

(b) Also determine the required sample size if we want a 90% confidence interval instead.
(5 marks)

(c) Discuss the application of Random Sampling, Systematic Sampling, and Convenience Sampling methods for collection annual income data, including the advantages and disadvantages of each method. Evaluate which method would be most suitable in terms of applicability and effectiveness.