Sampling

1Random sampling

Non Random sampling

Biased/ non biased issues

2Stratified Sampling

  • Choose a sample based on certain criteria like age group, level of education, state of origin etcetc
  • 25 – 29 years old
  • Diploma
  • Sabah
  • Curly hair
  • Foreign working husband
  • Wake up after 11 am on Sunday
  • Why stratified sampling?
  • So that mean and variance of samples are well managed.
  • Small value in variance is said to represent ‘..a well representative sample’ (Crawshaw and Chambers, 1994).
  • Example: a population consists of the numbers 3/4/4/5/7/9/10/14
  • (a)Calculate the mean and variance of this population
  • A random sample of size 4 is taken with replacement from this population.
  • (b) calculate the variance of the sample mean

Note: when sampling without replacement from a population ofsize N, the variance of the sample mean is

The numbers are then split into 2 groups as follows:

Group 1: 3, 4, 4, 5

Goup 2: 7, 9, 10, 14

(d) calculate the variance of the numbers in each group.

A sample of size 4 is obtained by independently selecting 2 numbers, at random. Without replacement , from each of the 2 groups. The population mean is to be estimated by

Y= , where and are the sample means of the 2 groups.

Find the variance of Y.

State , giving a reason, which of the 3 samples is likely to give the best estimate of t he population mean.

Solution:

Population 3/4//4/5/7/9/10/14

By calculator mean = 7; population variance 12.5

For a random sample size 4 with replacement , variance of sample mean is given by

Variance where we get 3.125 as the variance.

For a random sample size 4 taken without replacement , variance of the sample mean is given by

Variance = = 1.786

d)group 1 / 3,4,4,5 variance 0.5 =

group 2 / 7,9,10, 14 variance 6.5 =

e) we now take a proportionate stratified sample of size of 2 from each stratum (group 1 and group 2) . note that the numbers in each group are similar to each other and form 2 distinguishable strata. Between them , they cover the complete population.

The estimate of the population mean is

Y =

Now Var (Y) =

Also

Var (= = 0.16

Var (

Therefore variance of stratified estimate is

Var (Y) =

The variance of the stratified estimate is much smaller than the randm sample variances and so provides the most precise estimate of the population mean.

The next best estimate is the random sample , size 4, taken without replacement , then last of all, the random sample taken with replacement.

NON RANDOM SAMPLING

  • Cluster sampling
  • Quota sampling

(find the notes yourself please).

Reference: Crawshaw, J. and Chambers, J. ( 1994) A Level Statistics. 3rd Ed. London: Stanley Thornes Ltd