Question 1 [15 marks]. Suppose that you are asked to conduct a sample survey for a large primary school, to estimate average number of hours in a typical week during the terms (eg, Week 6 of Term 1) that a student spends on school homework among students in Year 3 to Year 6 of this school. It is known that this school has 4 classes in each Grade (Year level) and the amount school homework given to different classes in the same grade is similar (or the same) but varies from one grade to another.

a) Describe carefully what the target population is. [1 marks]

b) State the sampling unit. [1 marks]

c) Indicate carefully the sampling frame (or frames) that you would use for this survey. Assume that you can access the list of students' IDs for each class of this school, and information about the class and grade (Year level) for each student. [1 marks]

d) Indicate the most essential quantity/information for this survey, which you must record on each observation unit. [1 marks]

e) If you are asked to use stratified random sampling for this survey, suggest one vari- able that you would like to consider for stratification. Explain a main reason of using this variable for the stratification. [2 marks]

f) If you are asked to take a "1 in 25" systematic sample from the students of Year 3 to Year 6 in the school. For this part only, assume that each class has 30 students. As there are 4 classes in each grade within this school, there should be a total of 480 students in Year 3 to Year 6.

i) What is the sample size(s) that you will or is likely to have? [1 marks]

ii) Carefully describe how you will draw this systematic sample to ensure the (sample) mean of this systematic sample is unbiased. [2 marks]

g) Now you are asked to take a (one stage) cluster sample of size 3 from the 16 Year 3 to Year 6 classes, to estimate average number of hours per week spent on school homework. This time you are told that those 16 classes do NOT have the same class size. (Note that the amount school homework given to different classes in the same grade is similar but varies from one grade to another.).

i) Briefly describe how you will draw the sample (list the steps). [2 marks]

ii) Explain and show (write out relevant expression) how you will estimate the pa- rameter of interest here. For this part only, assume that class size varies across the 16 classes in the study. [2 marks]