Sample Size and Statistical theory - Sample-Size Question
Sample Size and Statistical theory - Sample-Size Question
Now, we are finally ready to use these concepts to help determine sample size. To proceed, the analyst must specify:
1. Size of the sampling error that is desired.
2. Confidence level, for example, the 95 percent confidence level.
This specification will depend on a trade-off between the value of more accurate information and the cost of an increased sample size. For a given confidence level, a smaller sampling error will "cost" in terms of a larger sample size. Similarly, for a given sampling error, a higher confidence level will "cost" in terms of a larger sample size. These statements will become more tangible in the context of some examples.
Using the general formula for the interval estimate (recall that a and ax are the same)
We know that
Sampling error =
VnT
Dividing through by the sampling error and multiplying by Vn
(sampling error)
and squaring both sides, we get an expression for sample size:
_ z2a2
(sampling error)2
Thus, if we know the required confidence level, and therefore z, and also know the allowed sampling error, then the needed sample size is specified by the formula.
Let us assume that there is a need to be 95 percent sure that our sampling error in estimating the population mean does not exced 0.3. In this case, sampling error = 0.3, and, since the confidence level is 95 percent, z = 2. In our example from Figure 12-1, the population standard deviation is 1.49, so the sample size should be
= 22(1.49)2 =
(3)2 yy
Changing the Confidence Level If the confidence level were changed from 95 percent to 90 percent, the sample size could be reduced because we do not have to be as certain of the resulting estimate. The z term would then be % and the sample size would be
n = (zcr)2 = (|)2(1.49)2 = 65
(sampling error)2 (.3)2
Changing the Allowed Error If the allowed error were increased, the sample size also would decrease, even if a 95 percent confidence level were retained. In our example, if the allowed error were increased to 0.5, then the sample size would be
(zcr)2 _ 4(1.49)2 _ 35 5
(sampling error)2 (0.5)s
Population Size It should be noted that the sample-size calculation is independent of the size of the population. A common misconception is that a "good" sample should have a relatively high percentage of the sampling frame included. Actually, the size of the sample will be determined in the same manner, whether the population is 1000 or 1,000,000. There should be no concern that the sample contain a reasonable percentage of the popu
lation. Of course, if the population is small, the sample size could be reduced.7 Obviously, the sample size should not exceed the population.
Determining the Population Standard Deviation
The procedure just displayed assumes that the population standard deviation is known. In most practical situations it is not known and it must be estimated by using one of several available approaches.
One method is to use a sample standard deviation obtained from a previous comparable survey or from a pilot survey. Another approach is :: estimate cr subjectively. Suppose the task is to estimate the income of a community. It might be possible to say that 95 percent of the people will have an income of between $4000 and $20,000. Assuming a normal distribution, there will be four population standard deviations between the twc figures, so that one population standard deviation will be equal to $4000
Another approach is to take a "worst-case" situation. In our example the largest population variance would occur if half the population would respond with a +2 and the other half with a -2.8 The population variance would then be 4, and the recommended sample size, at a 95 percent confidence level and a 0.3 allowable error, would be 178. Note that the sample size would be larger than desired, and thus the desired accuracy would be exceeded. The logic is that it is all right to err on the side of being too accurate.
Proportions
When proportions are to be estimated (the proportion of people with negative feelings about a change in the starting time of the symphony, for example), the procedure is to use the sample proportion to estimate the unknown population proportion, IT. Because this estimate is based on a sample, it has a population variance, namely,
2 _ TT(1 - IT)
When sampling with relatively small populations the standard error of X is o7Vn
where N is the size of the population. If the sample size is a meaningful percentage of N (such as 30 to 50 percent) then it might be worthwhile to reduce the sample size.
8The population variance would be 0.5(2 - 0)2 + 0.5(-2 - 0)2 = 0.5 x 4 + 0.5 x 4 = 4, since 0.5 of the population responded with a +2, and the population mean, or average, would be zero. See footnote 1 for a calculation formula.
where
Tf = population proportion
P = sample proportion (corresponding to X),
used to estimate the unknown
population proportion cr2, = population variance of P
The formula for sample size is then
Z2TT(1 - Tr)
(sampling error)2
As Figure 12-5 shows, the worst case, where the population variance is at its maximum, occurs when the population proportion is equal to .50:
-rrd - TT) = .25 IT = .50
Because the population proportion is unknown, a common procedure is to assume the worst case. The formula for sample size then simplifies to
z2(.25)
n =
(sampling error)2
Thus, if the population proportion is to be estimated within an error of 0.05 (or 5 percentage points) at a 95 percent confidence level, the needed sample size is
Hit-"
_ .25 I .20 f .15
.10
.05
25 .50 .75 1.0
FIGURE 12-5
A graph of TT(1 - TT).
In general,
Sample size = n = z2cr2 -H (sampling error)2 where
z = 2 for a 95 percent confidence level z = f for a 90 percent confidence level <J = population standard deviation
and
sampling error = allowed sampling error For proportions,
Sample size = n = z2(.25) j- (sampling error)2
FIGURE 12-6
Some useful sample-size formulas.
since z equals 2, corresponding to a 95 percent confidence level, and the allowed sampling error equals 0.05. Figure 12-6 summarizes the two sample size formulas.
Several Questions
A survey instrument or an experiment usually will not be based on just one question. Sometimes hundreds can be involved. It usually will not be worthwhile to go through such a process for all questions. A reasonable approach would be to pick a few representative questions and determine the sample size from them. Included should be the most crucial ones with the highest expected variance.
Now, we are finally ready to use these concepts to help determine sample size. To proceed, the analyst must specify:
1. Size of the sampling error that is desired.
2. Confidence level, for example, the 95 percent confidence level.
This specification will depend on a trade-off between the value of more accurate information and the cost of an increased sample size. For a given confidence level, a smaller sampling error will "cost" in terms of a larger sample size. Similarly, for a given sampling error, a higher confidence level will "cost" in terms of a larger sample size. These statements will become more tangible in the context of some examples.
Using the general formula for the interval estimate (recall that a and ax are the same)
We know that
Sampling error =
VnT
Dividing through by the sampling error and multiplying by Vn
(sampling error)
and squaring both sides, we get an expression for sample size:
_ z2a2
(sampling error)2
Thus, if we know the required confidence level, and therefore z, and also know the allowed sampling error, then the needed sample size is specified by the formula.
Let us assume that there is a need to be 95 percent sure that our sampling error in estimating the population mean does not exced 0.3. In this case, sampling error = 0.3, and, since the confidence level is 95 percent, z = 2. In our example from Figure 12-1, the population standard deviation is 1.49, so the sample size should be
= 22(1.49)2 =
(3)2 yy
Changing the Confidence Level If the confidence level were changed from 95 percent to 90 percent, the sample size could be reduced because we do not have to be as certain of the resulting estimate. The z term would then be % and the sample size would be
n = (zcr)2 = (|)2(1.49)2 = 65
(sampling error)2 (.3)2
Changing the Allowed Error If the allowed error were increased, the sample size also would decrease, even if a 95 percent confidence level were retained. In our example, if the allowed error were increased to 0.5, then the sample size would be
(zcr)2 _ 4(1.49)2 _ 35 5
(sampling error)2 (0.5)s
Population Size It should be noted that the sample-size calculation is independent of the size of the population. A common misconception is that a "good" sample should have a relatively high percentage of the sampling frame included. Actually, the size of the sample will be determined in the same manner, whether the population is 1000 or 1,000,000. There should be no concern that the sample contain a reasonable percentage of the popu
lation. Of course, if the population is small, the sample size could be reduced.7 Obviously, the sample size should not exceed the population.
Determining the Population Standard Deviation
The procedure just displayed assumes that the population standard deviation is known. In most practical situations it is not known and it must be estimated by using one of several available approaches.
One method is to use a sample standard deviation obtained from a previous comparable survey or from a pilot survey. Another approach is :: estimate cr subjectively. Suppose the task is to estimate the income of a community. It might be possible to say that 95 percent of the people will have an income of between $4000 and $20,000. Assuming a normal distribution, there will be four population standard deviations between the twc figures, so that one population standard deviation will be equal to $4000
Another approach is to take a "worst-case" situation. In our example the largest population variance would occur if half the population would respond with a +2 and the other half with a -2.8 The population variance would then be 4, and the recommended sample size, at a 95 percent confidence level and a 0.3 allowable error, would be 178. Note that the sample size would be larger than desired, and thus the desired accuracy would be exceeded. The logic is that it is all right to err on the side of being too accurate.
Proportions
When proportions are to be estimated (the proportion of people with negative feelings about a change in the starting time of the symphony, for example), the procedure is to use the sample proportion to estimate the unknown population proportion, IT. Because this estimate is based on a sample, it has a population variance, namely,
2 _ TT(1 - IT)
When sampling with relatively small populations the standard error of X is o7Vn
where N is the size of the population. If the sample size is a meaningful percentage of N (such as 30 to 50 percent) then it might be worthwhile to reduce the sample size.
8The population variance would be 0.5(2 - 0)2 + 0.5(-2 - 0)2 = 0.5 x 4 + 0.5 x 4 = 4, since 0.5 of the population responded with a +2, and the population mean, or average, would be zero. See footnote 1 for a calculation formula.
where
Tf = population proportion
P = sample proportion (corresponding to X),
used to estimate the unknown
population proportion cr2, = population variance of P
The formula for sample size is then
Z2TT(1 - Tr)
(sampling error)2
As Figure 12-5 shows, the worst case, where the population variance is at its maximum, occurs when the population proportion is equal to .50:
-rrd - TT) = .25 IT = .50
Because the population proportion is unknown, a common procedure is to assume the worst case. The formula for sample size then simplifies to
z2(.25)
n =
(sampling error)2
Thus, if the population proportion is to be estimated within an error of 0.05 (or 5 percentage points) at a 95 percent confidence level, the needed sample size is
Hit-"
_ .25 I .20 f .15
.10
.05
25 .50 .75 1.0
FIGURE 12-5
A graph of TT(1 - TT).
In general,
Sample size = n = z2cr2 -H (sampling error)2 where
z = 2 for a 95 percent confidence level z = f for a 90 percent confidence level <J = population standard deviation
and
sampling error = allowed sampling error For proportions,
Sample size = n = z2(.25) j- (sampling error)2
FIGURE 12-6
Some useful sample-size formulas.
since z equals 2, corresponding to a 95 percent confidence level, and the allowed sampling error equals 0.05. Figure 12-6 summarizes the two sample size formulas.
Several Questions
A survey instrument or an experiment usually will not be based on just one question. Sometimes hundreds can be involved. It usually will not be worthwhile to go through such a process for all questions. A reasonable approach would be to pick a few representative questions and determine the sample size from them. Included should be the most crucial ones with the highest expected variance.
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