Q.Distinguish between Stratified Sampling and Cluster Sampling. Why might a researcher prefer stratified sampling over cluster sampling when the population's sub-groups differ greatly from each other?

A value computed from sample data is a statistic; a value describing the whole population is a parameter.

In statistical inference we distinguish two ideas:

• A parameter is a numerical measure (mean $\mu$, standard deviation $\sigma$, proportion $P$) describing the entire population — usually unknown and fixed.
• A statistic is a numerical measure ($\bar{x}$, $s$, sample proportion $p$) computed from a sample; it is used to estimate the population parameter.

Since the question asks about a measure computed from sample data, the answer is statistic.

Equal, known probability of selection for each unit of the population is the defining feature of a random sample; option (d).

Reasoning. In a random sample every item in the universe (population) has an equal and known chance of being selected, which removes selection bias.

• A statistic is a value computed from a sample.

• A parameter is a value describing the whole population.

• Entire data refers to a census, not a sample.

The quoted definition of a random sample is given by Harper.

A random (simple random) sample gives every unit of the population an equal and independent chance of being included. In the Tamil Nadu HSC Business Maths & Statistics syllabus, this particular wording is attributed to Harper.

A finite subset of individuals selected from a population is, by definition, a sample.

In sampling theory the entire collection of items under study is the population (or universe); a finite subset of statistical individuals chosen from that population is a sample. A study of the whole population is a census. Hence the finite subset described is a sample.

A sample is a finite subset of the population.

In the Tamil Nadu HSC Statistics syllabus, the population (or universe) is the entire collection of statistical individuals, while a sample is only a portion of it selected for study. Since a sample is a manageable part rather than the whole, it is described as a finite subset of the statistical individuals of the population.

This defines a random sample.

A random sample is one selected so that each item in the universe (population) has an equal and known chance of being included. This equal-opportunity property removes selection bias and lets the results be generalised to the population.

(By contrast, a statistic is a value computed from a sample, and a parameter is a value describing the population — neither is a method of selection.)

Simple random sampling gives every unit an equal chance. With $N$ units, one is chosen at the first draw, so each unit's probability is $\frac{1}{N}$.

Reasoning. In simple random sampling (without replacement) from a population of $N$ units, all units are equally likely to be selected. At the first draw there are $N$ equally likely units and exactly one is picked, so the probability that any specified unit is drawn is

$\frac{\text{favourable}}{\text{total}} = \frac{1}{N}.$