Common Core Standards: Math
4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
Students should understand how the mean, proportion, and margin of error can all be gleaned from a sample survey. In order to do that, they have to know what these wonderful things are.
The population mean is the average value of the parameter for the whole population, and is expressed as μ. Students should know that it's nearly impossible to find the true value of μ, so we use the mean of our sample to estimate it. The larger of a sample we have, the closer we'll get to μ. We mean it.
Rather than finding μ, it may just be better to find the population's proportion. Population proportions can be given as point estimates (single values) or intervals (a range of values). That is, we can say that 80% of residents prefer candidate X, or that 75 - 85% prefer candidate X. Students should know when it's appropriate to use each one, depending on the accuracy of the study and the variability of the numbers.
Our sample estimate, randomly chosen, is nothing more than an estimate of our population mean or proportion. That means there will always be some margin of error involved. The only way we won't have a margin of error is if we had all the money and time in the world (yes, please!) to sample everyone in the population.
Before calculating this margin of error we must choose how confident we want to be about our results. For example, if we choose a 95% confidence interval (very standard) this means that out of 100 randomly chosen sample surveys, our result would be within our confidence interval 95 out of 100 times. Not too shabby.
For a 95% confidence interval our equation for the margin of error (where p is equal to the proportion in decimal form and N is our sample size) becomes:
But what if our results aren't in proportion form? How do we calculate margin of error for a mean score instead? That's when we use this equation (where N is our sample size and s is our standard deviation).
Armed with all this information, students should be able to emerge victorious in battles with population means, proportions, and margins of error.