Paper 2 Section B Practice: Sampling and Estimation

Original practice material

These questions total 22 marks. Distinguish population parameters, random estimators and observed estimates.

Answers: Sampling and Estimation Answers.

Question 1: sampling design

A college has four year groups containing 1200, 900, 600 and 300 students respectively. A sample of 100 students is required to investigate weekly independent-study time.

(a) Explain why stratified random sampling by year group may be preferable to a simple random sample. [2]

(b) State how many students should be selected from each year group if proportional allocation is used. [2]

(c) Describe how the students should be selected within each year group. [2]

Question 2: sampling distribution and precision

A population has mean $72$ and standard deviation $15$, but its distribution is not normal. A random sample of size $36$ is taken and $\bar{X}$ denotes its mean.

(a) State an approximate distribution for $\bar{X}$, and identify the result that justifies the approximation. [3]

(b) Find $P(69<\bar{X}<76)$. [3]

(c) Find the smallest sample size $n$ for which

$$P(|\bar{X}-72|<3)>0.95,$$

using the normal approximation. [4]

Question 3: estimates from observed data

A random sample gives the values

$$4,7,9,10,15.$$

(a) Find the sample mean $\bar{x}$. [2]

(b) Find the unbiased estimate $s^2$ of the population variance. [3]

(c) Explain why $\bar{X}$ is called an unbiased estimator of the population mean. [1]