Population and Sample:
(a) Finite Population: A population is said to be finite, if it consists of finite or fixed number of elements (i.e., items, objects, measurements or observations). For example, all the university students in Pakistan, the heights of all the students enrolled in Karachi University, etc. (b) Infinite Population: A population is said to be infinite, if there is no limit to the number elements it can contain. For example, the role of two dice, all the heights between 2 and 3 meters, etc.
(a) Sampling with Replacement: If the sample is taken with replacement from a population finite or infinite, the element drawn is returned to the population before drawing the next element. (b) Sampling without Replacement: If the sample is taken without replacement from a finite population, the element selected is not returned to the population. Probability Samples and Non-Probability Samples:
(a) Simple Random Sampling: refers to a method of selecting a sample of a given size from a given population in such a way that all possible samples of this size which could be formed from this population have equal probabilities of selection. It is a method in which a sample of n is selected from the population of N units such that each one of the NCn distinct samples has an equal chance of being drawn. This method sometimes also refers to ‘lottery method’. (b) Stratified Random Sampling: consists of the following two steps: (i) The material or area to be sampled is divided into groups or classes called ‘strata’. Items within each stratum are homogenous. (ii) From each stratum, a simple random sample is taken and the overall sample is obtained by combining the samples for all strata. (c) Systematic Sampling: is another form of sample design in which the samples are equally spaced throughout the area or population to be sampled. For e.g., in house-to-house sampling every 10th or 20th house may be taken. More specifically a systematic sample is obtained by taking every kth unit in the population after the units in population have been numbered or arranged in some way. (d) Cluster Sampling: One of the main difficulties in large scale surveys is the extensive area that may have to be covered in getting a random or stratified random sample. It may be very expensive and lengthy task to cover the whole population in order to obtain a representative sample. It is not possible to take a simple random or systematic sample of persons from the entire country or from within strata, since there is no such list in which all the individuals are numbered from 1 to N. Even if such a list existed, it would be too expensive to base the enquiry on a simple random sample of persons. Under these circumstances, it is economical to select groups called ‘clusters’ of elements from the population. This is called ‘cluster sampling’. The difference between a cluster and a stratum is that a stratum is expected to be homogenous and a cluster must be heterogeneous as possible. Clusters are also known as the primary sampling units. Cluster sampling may be consisted of: (i) Single-stage Cluster Sampling, (ii) Sub-sampling or Two-stage Sampling, and (iii) Multi-stage Sampling.
(a) Judgement or Purposive Sampling: There are many situations where investigators use judgement samples to gain needed information. For example, it may be convenient to select a random sample from a cart-load of melons. The melons selected may be very large or very small. The observer may use his own judgement. This method is very useful when the sample to be drawn is small. (b) Quota Sampling: is widely used in opinions, market surveys, etc. In such surveys, the interviewers are simply given quotas to be filled in from different strata, with practically no restrictions on how they are to be filled in. Parameters and Statistic:
Sampling and Non-Sampling Errors: (a) Sampling Errors:
E = t – θ
(b) Non-Sampling Errors:
Bias:
B = m – μ Where μ is the true population value and m is the mean of the sample statistics of an infinity of samples.
Precision and Accuracy:
Sampling Distribution:
Example: Take the data of previous example and assume sampling ‘without replacement’, and compute: (a) Population mean, (b) Population standard deviation, (c) Mean of each sample, (d) Sampling distribution table of sample mean w/o replacement, and (e) Mean and standard deviation of sampling distribution. Solution: (a) and (b) Population mean and SD: As calculated above. (c) Mean of each sample: No. of possible samples = NCn = 4C2 = 6 samples Samples (without replacement):
Mean: (d) Sampling Distribution: |