7.5 Calculating the Sample Size for a Confidence Interval

Usually we have no control over the sample size of a data set.  However, if we are able to set the sample size, as in cases where we are taking a survey, it is very helpful to know just how large it should be to provide the most information.  Sampling can be very costly, in both time and product.  Simple telephone surveys will cost approximately $30.00 each, for example, and some sampling requires the destruction of the product.  Selecting a sample that is too large is expensive and time consuming.  But selecting a sample that is too small can lead to inaccurate conclusions.  We want to find the minimum sample size required to achieve the desired level of accuracy in the confidence interval.

Calculating the Sample Size for a Population Mean

The margin of error [latex]E[/latex] for a confidence interval for a population mean is

[latex]\displaystyle{E=\frac{z \times \sigma}{\sqrt{n}}}[/latex]

where [latex]z[/latex] is the [latex]z[/latex]-score so that the area under the standard normal distribution in between [latex]-z[/latex] and [latex]z[/latex] is the confidence level [latex]C[/latex].

Rearranging this formula for [latex]n[/latex] we get a formula for the sample size [latex]n[/latex]:

[latex]\displaystyle{n=\left(\frac{z \times \sigma}{E}\right)^2}[/latex]

In order to use this formula, we need values for [latex]z[/latex], [latex]E[/latex] and [latex]\sigma[/latex]:

• The value for [latex]z[/latex] is determined by the confidence level of the interval, calculated the same way we calculate the [latex]z[/latex]-score for a confidence interval.
• The value for the margin of error [latex]E[/latex] is set as the predetermined acceptable error, or tolerance, for the difference between the sample mean [latex]\overline{x}[/latex] and the population mean [latex]\mu[/latex].  In other words, [latex]E[/latex] is set to the maximum allowable width of the confidence interval.
• An estimate for the population standard deviation [latex]\sigma[/latex] can be found by one of the following methods:
  • Conduct a small pilot study and use the sample standard deviation from the pilot study.
  • Use the sample standard deviation from previously collected data.  Although crude, this method of estimating the standard deviation may help reduce costs significantly.
  • Use [latex]\displaystyle{\frac{\mbox{Range}}{4}}[/latex] where [latex]\mbox{Range}[/latex] is the difference between the maximum and minimum values of the population under study.

Calculating the Sample Size for a Population Proportion

The margin of error [latex]E[/latex] for a confidence interval for a population proportion is

[latex]\displaystyle{E=z \times \sqrt{\frac{p \times (1-p)}{n}}}[/latex]

where [latex]z[/latex] is the [latex]z[/latex]-score so that the area under the standard normal distribution in between [latex]-z[/latex] and [latex]z[/latex] is the confidence level [latex]C[/latex].

Rearranging this formula for [latex]n[/latex] we get a formula for the sample size [latex]n[/latex]:

[latex]\displaystyle{n=p \times (1-p) \times \left(\frac{z}{E}\right)^2}[/latex]

In order to use this formula, we need values for [latex]z[/latex], [latex]E[/latex] and [latex]p[/latex]:

• The value for [latex]z[/latex] is determined by the confidence level of the interval, calculated the same way we calculate the [latex]z[/latex]-score for a confidence interval.
• The value for the margin of error [latex]E[/latex] is set as the predetermined acceptable error, or tolerance, for the difference between the sample proportion [latex]\hat{p}[/latex] and the population proportion [latex]p[/latex].  In other words, [latex]E[/latex] is set to the maximum allowable width of the confidence interval.
• An estimate for the population proportion [latex]p[/latex].  If no estimate for the population proportion is provided, we use [latex]p=0.5[/latex].

There is an interesting trade-off between the level of confidence and the sample size that shows up here when considering the cost of sampling.  The table below shows the appropriate sample size at different levels of confidence and different margins of error, assuming [latex]p=0.5[/latex].  Looking at each row, we can see that for the same margin of error, a higher level of confidence requires a larger sample size.  Similarly, looking at each column, we can see that for the same confidence level, a smaller margin of error requires a larger sample size.

_{Watch this video: Sample Size for Confidence Intervals by ExcelIsFun [7:54]}

Concept Review

In order to construct a confidence interval, a sample is taken from the population under study.  But collecting sample information is time consuming and expensive.  The minimum sample size required to achieve the desired level of accuracy is determined before collecting the sample data.

• Sample size for population means:  [latex]\displaystyle{n=\left(\frac{z \times \sigma}{E}\right)^2}[/latex]
• Sample size for population proportions:  [latex]\displaystyle{n=p \times (1-p) \times \left(\frac{z}{E}\right)^2}[/latex]

After calculating the value of [latex]n[/latex] from the formula, round the value of [latex]n[/latex] up to the next integer.

Attribution

“7.2 The Central Limit Theorem for Sums“ in Introductory Statistics by OpenStax is licensed under a Creative Commons Attribution 4.0 International License.

“8.4 Calculating the Sample Size n: Continuous and Binary Random Variables“ in Introductory Business Statistics by OpenStax is licensed under a Creative Commons Attribution 4.0 International License.