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Question:
Explain howsample size, confidence level, and confidence interval size are related? What is the role of the project's budget in determining sample size?
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- Answer both parts of the question
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The Statistical Side of Sampling
The sample size for a probability sample depends on the standard error of the mean, the precision desired from the estimate, and the desired degree of confidence associated with the estimate. The standard error of the mean measures sampling errors that arise when estimating a population from a sample instead of including all of the essential information in the population. The size of the standard error is the function of the standard deviation of the population values and the size of the sample.
standard error
σstandard deviation
η = sample size
The precision is the size of the plus-or-minus interval around the population parameter under consideration, and the degree of confidence is the percentage level of certainty (probability) that the true mean is within the plus-or-minus interval around the mean. Precision and confidence are interrelated and, within a given size sample, increasing one may be done only at the expense of the other. In other words, the degree of confidence or the degree of precision may be increased, but not both.
The main factors that have a direct influence on the size of the sample are:
1.The desired degree of confidence associated with the estimate. In other words, how confident does the researcher want to be in the results of the survey? If the researcher wants 100-percent confidence, he or she must take a census. The more confident a researcher wants to be, the larger the sample should be. This confidence is usually expressed in terms of 90, 95, or 99 percent.
2.The size of the error the researcher is willing to accept. This width of the interval relates to the precision desired from the estimate. The greater the precision, or rather the smaller the plus-or-minus fluctuation around the sample mean or proportion, the larger the sample requirement.
The basic formula for calculating sample size for variables (e.g., age, income, weight, height, etc.) is derived from the formula for standard error:
The unknowns in the formula above are (standard error), σ (standard deviation), and η (sample size). In order to calculate the sample size, the researcher must:
1.Select the appropriate level of confidence.
2.Determine the width of the plus-or-minus interval that is acceptable and calculate standard error.
3.Estimate the variability (standard deviation) of the population based on a pilot study or previous experience of the researcher with the population.
4.Calculate sample size (solve for n).
For example, a researcher might choose the 95.5 percent confidence level as appropriate. Using the assumptions of the Central Limit Theorem (that means of samples drawn will be normally distributed around the population means, etc.), the researcher will select a standard normal deviate from the following tables:
Level of ConfidenceZ Value
68.3% 1.00
75.0 1.15
80.0 1.28
85.0 1.44
90.0 1.64
95.0 1.96
95.5 2.00
99.0 2.58
99.7 3.00
This allows the researcher to calculate the standard error (). If, for example, the precision width of the interval is selected at 40, the sampling error on either side of the mean must be 20. At the 95.5 percent level of confidence, Z = 2 and the confidence interval equals .
Then the standard error is equal to 10.
Z = 2 at 95.5% level
CL = Confidence limits
CI = Confidence interval
Having calculated the standard error based on an appropriate level of confidence and desired interval width, we have two unknowns in the sample size formula left, namely sample size (η) and standard deviation (σ). The standard deviation of the sample must now be estimated. This can be done either by taking a small pilot sample and computing the standard deviation or it can be estimated on the knowledge and experience the researcher has of the population. If you estimate the standard deviation as 200, the sample size can be calculated.
The sample size required to give a standard error of 10 at a 95.5 percent level of confidence is computed to be 400. This assumes that assumptions concerning the variability of the population were correct.
Another way of viewing the calculation of sample size required for a given precision of a mean score is to use the following formula:
Z = value from normal distribution table for desired confidence level
η = standard deviation
h = sample size
h = desired precision ±
Using the same information as used in the previous example, the same result is obtained:
η = 400
Tables in most statistical books are provided to allow you, at several given confidence levels, to select the exact sample size given an estimated standard deviation and a desired width of interval.
Determining sample size for a question involving proportions (e.g., those who eat out/don’t eat out, successes/failures, have access to internet/don’t have access, etc.) or attributes is very similar to the procedure followed for variables. The researcher must:
1.Select the appropriate level of confidence.
2.Determine the width of the plus-or-minus interval that is acceptable and calculated the standard error of the proportion .
3.Estimate the population proportion based on a pilot study or previous experience of the researcher with the population.
4. Calculate the sample size (solve for n).
The basic formula for calculating sample size for proportions or attributes is derived from the formula for standard error of the proportion:
= standard error of proportion
ρ = percent of successes
q = percent of nonsuccess (l-p)
Assume that management has specified that there be a 95.5 percent confidence level and that the error in estimating the population not be greater than ± 5 percent (p ± 0.05). In other words, the width of the interval is 10 percent. A pilot study has shown that 40 percent of the population eats out over four times a week.
Substituting in:
η = 384
Another way to view calculating the sample size required for a given precision of a proportion score is to use the following formula:
Z = value from normal distribution table for desired confidence level
ρ = obtained proportion
q = l - ρ
h = desired precision ±
Using the same information as used in the previous example, the same result is obtained:
η = 384
The sample size required to give a 95.5 percent level of confidence that the sample proportion is within ± 5 percent of the population proportion is .384.