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Math

Unless otherwise stated, the material in this guide is from The Learning Centre at Centennial College.  Content has been adapted for the NWP Learning Commons in March 2022. This work is licensed under a Creative Commons BY 4.0 International License.

Sample Size

We can calculate the required sample size to reach a certain confidence level. 

For the confidence level, the value we add/subtract to the mean is called the sampling error, E. Thus, the confidence intervals could be expressed as 

\[\mu \approx \bar{x} \pm E\]

where

\[E=z_{\alpha/2}\left(\frac{\sigma}{\sqrt{n}}\right)\]

We can rearrange E to solve for the sample size required to reach a specific confidence level.

Determining the Sample Size for a Confidence Interval for \(\mu\).

\[n=\frac{(z_{\alpha/2})^2\sigma^2}{E^2}\]

when you do not have \(\sigma\) and the sample size is expected to be small.

\[n=\frac{(t_{\alpha/2})^2\sigma^2}{E^2}\]

Determining the Sample Size for a Confidence Interval for Proportions, \(p\).

\[E=z_{\alpha/2}\sqrt{\frac{pq}{n}}\]

\[n=\frac{(z_{\alpha/2})^2 pq}{E^2}\]

Example: Suppose you wish to estimate a population mean correct to within 0.15 with a confidence level of 0.90. \(\sigma^2\) is approximately equal to 5.4. Find the sample size required.

Solution:

The population variance is 5.4, so we know we can use the z-statistic a 90% confidence level. \(z_{0.10/2}=1.645, E=0.15\)

\[n=\frac{(1.645)^2(5.4)}{(0.15)^2}=649.446\]

You need more than 649 data points, so we round up to a sample size 650 to obtain a 90% confidence interval with a sampling error within 0.15.

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