Research Methodology - Dr. Krishan K. Pandey
Research Methodology - Dr. Krishan K. Pandey Research Methodology - Dr. Krishan K. Pandey
Sampling Fundamentals 179 Since $p is actually what we are trying to estimate, then what value we should assign to it ? One method may be to take the value of p = 0.5 in which case ‘n’ will be the maximum and the sample will yield at least the desired precision. This will be the most conservative sample size. The other method may be to take an initial estimate of p which may either be based on personal judgement or may be the result of a pilot study. In this context it has been suggested that a pilot study of something like 225 or more items may result in a reasonable approximation of p value. Then with the given precision rate, the acceptable error, ‘e’, can be expressed as under: e = z ⋅ or e z pq 2 2 = n 2 z ⋅ p ⋅q or n = 2 e The formula gives the size of the sample in case of infinite population when we are to estimate the proportion in the universe. But in case of finite population the above stated formula will be changed as under: 2 pq n z ⋅ p ⋅ q ⋅ N n = 2 2 e N − 1 + z ⋅ p ⋅ q b g Illustration 6 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 95.5 per cent probability? What would be the size of the sample if the population is assumed to be infinite in the given case? Solution: In the given question we have the following: N = 4000; e = .02 (since the estimate should be within 2% of true value); z = 2.005 (as per table of area under normal curve for the given confidence level of 95.5%). As we have not been given the p value being the proportion of defectives in the universe, let us assume it to be p = .02 (This may be on the basis of our experience or on the basis of past data or may be the result of a pilot study). Now we can determine the size of the sample using all this information for the given question as follows: 2 z ⋅ p ⋅ q ⋅ N n = 2 2 e N − 1 + z ⋅ p ⋅ q b g
180 Research Methodology = 2 b2. 005g b. 02gb1−. 02gb4000g 2 2 b. 02g b4000 − 1g + b2. 005g b. 02gb1− . 02g 3151699 . 3151699 . = = = 187. 78 ~ 188 15996 . + . 0788 16784 . But if the population happens to be infinite, then our sample size will be as under: 2 z ⋅ p ⋅q n = 2 e = 2 b2. 005g ⋅b. 02gb1− . 02g 2 . 02 b g . 0788 = = 196. 98 ~ 197 . 0004 Illustration 7 Suppose a certain hotel management is interested in determining the percentage of the hotel’s guests who stay for more than 3 days. The reservation manager wants to be 95 per cent confident that the percentage has been estimated to be within ± 3% of the true value. What is the most conservative sample size needed for this problem? Solution: We have been given the following: Population is infinite; e = .03 (since the estimate should be within 3% of the true value); z = 1.96 (as per table of area under normal curve for the given confidence level of 95%). As we want the most conservative sample size we shall take the value of p = .5 and q = .5. Using all this information, we can determine the sample size for the given problem as under: 2 z p q n = 2 e = 2 b196 . g ⋅b. 5gb1− . 5g . 9604 = = 1067. 11 ~ 1067 2 . 03 . 0009 b g Thus, the most conservative sample size needed for the problem is = 1067. DETERMINATION OF SAMPLE SIZE THROUGH THE APPROACH BASED ON BAYESIAN STATISTICS This approach of determining ‘n’utilises Bayesian statistics and as such is known as Bayesian approach. The procedure for finding the optimal value of ‘n’ or the size of sample under this approach is as under:
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Sampling Fundamentals 179<br />
Since $p is actually what we are trying to estimate, then what value we should assign to it ? One<br />
method may be to take the value of p = 0.5 in which case ‘n’ will be the maximum and the sample<br />
will yield at least the desired precision. This will be the most conservative sample size. The other<br />
method may be to take an initial estimate of p which may either be based on personal judgement or<br />
may be the result of a pilot study. In this context it has been suggested that a pilot study of something<br />
like 225 or more items may result in a reasonable approximation of p value.<br />
Then with the given precision rate, the acceptable error, ‘e’, can be expressed as under:<br />
e = z ⋅<br />
or e z pq<br />
2 2<br />
=<br />
n<br />
2<br />
z ⋅ p ⋅q<br />
or n =<br />
2<br />
e<br />
The formula gives the size of the sample in case of infinite population when we are to estimate<br />
the proportion in the universe. But in case of finite population the above stated formula will be<br />
changed as under:<br />
2<br />
pq<br />
n<br />
z ⋅ p ⋅ q ⋅ N<br />
n =<br />
2 2<br />
e N − 1 + z ⋅ p ⋅ q<br />
b g<br />
Illustration 6<br />
What should be the size of the sample if a simple random sample from a population of 4000 items is<br />
to be drawn to estimate the per cent defective within 2 per cent of the true value with 95.5 per cent<br />
probability? What would be the size of the sample if the population is assumed to be infinite in the<br />
given case?<br />
Solution: In the given question we have the following:<br />
N = 4000;<br />
e = .02 (since the estimate should be within 2% of true value);<br />
z = 2.005 (as per table of area under normal curve for the given confidence level of 95.5%).<br />
As we have not been given the p value being the proportion of defectives in the universe, let us<br />
assume it to be p = .02 (This may be on the basis of our experience or on the basis of past data or<br />
may be the result of a pilot study).<br />
Now we can determine the size of the sample using all this information for the given question as<br />
follows:<br />
2<br />
z ⋅ p ⋅ q ⋅ N<br />
n =<br />
2 2<br />
e N − 1 + z ⋅ p ⋅ q<br />
b g