1. Sampling and Sampling Distribution
Welcome To
A Session on
Sampling and Sampling Distribution

2. What are various approaches to sample size determination?

3. There are two alternative approaches for determining the size of the sample.
The first approach is “to specify the precision of estimation desired and then to determine the sample size necessary to insure it.”
The second approach “uses Bayesian statistics to weigh the cost of additional information against the expected value of the additional information.”

4. What are the strengths and weaknesses of each of the approaches?

5. The first approach is capable of giving a mathematical solution, and as such is a frequently used technique of determining “n’.
The limitation of this technique is that it does not analyze the cost of gathering information vis-à-vis the expected value of information .

6. The second approach is theoretically optimal, but it is seldom used because of the difficulty involved in measuring the value of information. This might have led researchers to use the first approach.

7. Sample Size and its Determination
In sampling analysis the most ticklish question is: what should be size of the sample ?
How large or small should be ‘n’?
If the sample size (‘n’) is too small, it may not serve to achieve the objectives
If it is too large, this may involve huge cost and waste of resources.

8. General rule
As a general rule, the sample must be of an optimum size.
Technically, the sample size should be large enough to give a confidence interval of desired width.
What are the points one should keep in mind in determining the size of samples?

9. The question is: what should be the size of samples ?
Nature of universe: Universe may be either homogenous or heterogeneous in nature. If the items of the universe are homogenous, a small sample can serve the purpose. But if the items are heterogeneous, a large sample would be required. Technically, this can be termed as the dispersion factor.
Number of classes proposed: If many class-groups (groups and sub-groups ) are to be formed, a large sample would be required because a small sample might not be able to give a reasonable number of items in each class- group.

10. The question is: what should be the size of samples ?
Standard of accuracy and acceptable confidence level: If the standard of accuracy or the level of precision is to be kept high, the sample size has to be larger. For doubling the accuracy for a fixed significance level, the sample size has to be increased fourfold.
Availability of finance: In practice, the size of the sample depends upon the amount of money available for the accuracy for study purposes. This factor should be kept in view while determining the size of sample, for large samples resulting in increasing the cost of sampling estimates.

11. The question is: what should be the size of samples ?
Other considerations: Nature of units, size of the population, size of questionnaire, availability of trained investigators, the conditions under which the sample survey is being conducted, and the time available for completion of the study are a few other considerations to which a researcher must pay attention while selecting the size of the sample.

12. Key Questions
What are reasonable estimates of key proportions to be measured in the study ? ( If one cannot guess what will be the key proportions, the safest procedure is to assume the same to be 0.50, which maximizes the expected variance and therefore indicates a sample size that is sure to be large enough their ideas.)
What degree of accuracy is desired in the study ?

13. Key Questions
How far can we allow the sample estimates of key proportions to deviate from the true proportions in the population as a whole?
What confidence level do we want to use ?
How confident do we want to be that the sample
estimate is as accurate as we wish ?

14. Key Questions What is the size of the population that the
sample is supposed to represent ?
If it is desired to measure the difference between
the two subgroups with regard to a proportion,
what is the minimum difference one expects to
find statistically significant ?

15. What are the specific pieces of information needed to estimate the size of samples ?
Population size
Precision level (acceptable error )
Standard deviation of the population
The value of the standard variate at a given
confidence level (it is 1.96 for a 95%
confidence level )

16. The formula for computing the standard errors concerning various measures based on samples is as under:

17. Sample size determination
The sample size must be large enough:
To allow for reliable analysis of cross-
tabulation;
To provide for desired levels of accuracy in
estimates of proportions; and
To test for the significance of differences
between proportions .

18. Points to be kept in mind in cross- tabulations:
Each category of an independent variable
included in a cross-tabulations should contain
at least 50 cases;
The expected number of cases in each cell of a
table should be at least 5.
Other things being equal, the sample size depends on expected precision level…….
Continued…….

19. Precision and sample size
Precision (Interval width)
Approximate sample size
± 10%
± 7%
± 5%
± 3%
± 2%
± 1%

20. Example 1
Determine the size of the sample for estimating the true weight of the cereal containers for the universe with N = 5000 on the basis of the following information:
The variance of weight = 4 ounces on the basis
of past records.
Estimate should be within 0.8 ounces of the true
average weight with 99% probability.

21. Solution
In the given problem, the following data/statistics are given:
N = 5000;
p = 2 ounces (since the variance of weight =
4 ounces);
e= 0.8 ounces (since the estimate should be
within 0.8 ounces of the true average
weight):
z= 2.57 (as per the table of area under normal
curve for the given confidence level of
99%.)
Continued…….

22. Solution
In case of finite population, the confidence interval for  is given by:
Continued…….

23. where
Z = the value of the standard variate at a given confidence level ( to be read from the table giving the areas under normal curve) and it is 1.96 for a 95% confidence level
n = Size of the sample
Standard deviation of the population (to be estimated from past experience or on the basis of a trial sample)

24. Putting the values in the above – mentioned formula, we get
If the precision (acceptable error) is taken as equal to e, then we have
Putting the values in the above – mentioned formula, we get

25. Hence, the sample size is estimated at 41with finite population.
Continued…….

26. Will there be a change in the size of the sample if infinite population in the given case is assumed? If so, by how much change?
Continued…….

27. The size of the sample in the event of population being infinite may be estimated as under:
In the given case the sample size remains the same even if the population is assumed to be infinite.

28. Example 2
What should be the size of the sample if a simple random sample from a population of 4000 items is to be drawn to estimate the per cent defective within 2 per cent of the true value with per cent probability?
What would be the size of the sample if the population is assumed to be infinite in the given case?
Continued………

29. Solution:
Given:
N=4000;
e= .02 (since the estimate should be within 2% of true
value);
z= (as per table of area under normal curve for
the given confidence level of 95.5%).
As we have not been given the value being the proportion of defectives in the universe, let us assume it to be p = .02 (This may be on the basis of experience or on the basis of past data or may be the result of a pilot study).
Continued…….

30. Solution:
If the population happens to be finite as in this case, then the sample size may be estimated as under:
Continued…….

31. Solution:
If the population happens to be infinite, then the sample size may be estimated as under:

32. Example 3
If the proportion of a target population with a certain characteristic is 0.50, the z statistics is 1.96, and the desired accuracy is level, what will be the sample size?
Continued……

33. n – the desired sample size ( when population is> 10,000) .
Based on the information, the sample size may be computed by using the following formula:
n=(z2pq)/d2
Where,
n – the desired sample size ( when population is> 10,000) .
z – the standard normal deviate, usually set at 1.96 ( or more
simply at 2.0), which corresponds to the 95% confidence
level .
p -- the proportion in the direct population estimated to have a
particular characteristic. If there is no reasonable
estimate, then use 50% ( 0.50) .
q – 1.0 – p.
d -- degree of accuracy desired, usually set at 0.05 or
occasionally at 0.02
Continued……

34. Solution Putting the values in the question, we get=
If we use the more convenient 2.0 for the z statistic, then the sample size is :
Putting the values in the question, we get
n = {( 2)2 ( 0.50) ( 0.50)}/ ( 0.05)2 =
Continued……….

35. Note that the numerator in this case is 1
Note that the numerator in this case is 1.0. This means that when the proportion is assumed to be .50 and 95 percent confidence level is set by using z equal to 2.0, the formula for sample size is simply:
N= 1.0/d2

36. Example 4
If the entire population is less than 10,000, the sample size may be determined as under:
nf=n/{1+(n/N)}
Where,
nf – the desired sample size when population
is less than 10,000
n – desired sample size when the population is more
than 10,000
N – the estimate of the population size

37. Example 5.
In the event of n being 400 and the population size being 1000, what will be the sample size?
Putting the values in the formula, we get,
nf= 400/{ 1+ (400/1000) }
= 286

38. Thank You
For
Attending the Session