Introduction to Sample Size formula:
Sample size is nothing but the number of observations that constitute the experiment / research. It is generally denoted by n.
Sample size determination is closely related to estimation. Sometimes, we may need to know how large a sample is necessary in order to make an accurate estimate. The answer depends on
The margin of error.
The degree of confidence.
In considering large sample confidence intervals for the mean, since the error of estimate is given by E = E = Zα/2 (), we can solve to find n, the sample size.
Solving, we have
n = [(Zα/2 x σ) / E]2
Example for Sample Size Formula:
Q 1: What sample size should be selected to estimate the mean age of workers in the large factory to with in ±1 year at a 95 percent confidence level if the standard deviation for the ages is 3.5 years?
Sol: We are given that α=0.05, Zα/2 = 1.96 {the value is obtained from the normal distribution chart}, and E=1.
Substituting into the formula, we get the sample size as
n = [(Zα/2 x σ) / E]2 = [(1.96 x 3.5)/1]2 = 47.0596 ≈ 48
That is, in order to be 95 percent certain that the estimate is with in 1 year of the true mean age, a sample of at least 48 is selected. Having problem with Convert Fractions to Decimals keep reading my upcoming posts, i will try to help you.
Note: For a large enough sample size (n ≥ 30), when the population standard deviation σ is unknown, we can replace σ with s in the above equations
Example on Error in the Estimate:
Q 1: The president of a large community college wishes to estimate the average distance commuting students travel to the campus. A sample of 64 students was randomly selected and the standard deviation of the sample is worked out to be 5 mi. Find the error in the estimate of the distance commuting students travel to the campus at 95 percent confidence level..
Sol: given α=0.05, Zα/2 = 1.96, s=5, n=64 and σx ≈ s / √n = 0.625
Using the above mentioned formula, the error in the estimate, E would be
E = ± (1.96 x 0.625) = ± 1.225
Sample size is nothing but the number of observations that constitute the experiment / research. It is generally denoted by n.
Sample size determination is closely related to estimation. Sometimes, we may need to know how large a sample is necessary in order to make an accurate estimate. The answer depends on
The margin of error.
The degree of confidence.
In considering large sample confidence intervals for the mean, since the error of estimate is given by E = E = Zα/2 (), we can solve to find n, the sample size.
Solving, we have
n = [(Zα/2 x σ) / E]2
Example for Sample Size Formula:
Q 1: What sample size should be selected to estimate the mean age of workers in the large factory to with in ±1 year at a 95 percent confidence level if the standard deviation for the ages is 3.5 years?
Sol: We are given that α=0.05, Zα/2 = 1.96 {the value is obtained from the normal distribution chart}, and E=1.
Substituting into the formula, we get the sample size as
n = [(Zα/2 x σ) / E]2 = [(1.96 x 3.5)/1]2 = 47.0596 ≈ 48
That is, in order to be 95 percent certain that the estimate is with in 1 year of the true mean age, a sample of at least 48 is selected. Having problem with Convert Fractions to Decimals keep reading my upcoming posts, i will try to help you.
Note: For a large enough sample size (n ≥ 30), when the population standard deviation σ is unknown, we can replace σ with s in the above equations
Example on Error in the Estimate:
Q 1: The president of a large community college wishes to estimate the average distance commuting students travel to the campus. A sample of 64 students was randomly selected and the standard deviation of the sample is worked out to be 5 mi. Find the error in the estimate of the distance commuting students travel to the campus at 95 percent confidence level..
Sol: given α=0.05, Zα/2 = 1.96, s=5, n=64 and σx ≈ s / √n = 0.625
Using the above mentioned formula, the error in the estimate, E would be
E = ± (1.96 x 0.625) = ± 1.225
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