Drawing conclusions based on data from a random sample

A good description of how this section might play out in the classroom can be found on pp. 8–10 of Progressions for the Common Core State Standards in Mathematics: High School Statistics and Probability.  

The focus of this section is on standards S-IC.B.3 and S-IC.B.4. This section will require a substantial time commitment. Students build on what they have learned about distributions and sampling variability as they use data from a random sample to learn about the value of a population proportion or a population mean.

It is probably easiest to begin by focusing on estimating a population proportion. Simulation can be used to approximate the sampling distribution of a sample proportion. This can be done using physical simulations (for example from a bag of beads that contains 40% blue beads) or using one of the many technology applets (for example, rossmanchance.com/applets/Reeses3/ReesesPieces.html which builds up a sampling distribution of the proportion of orange candies in a random sample after the user specifies the population proportion and the sample size). It is recommended that at least one physical simulation be carried out before turning to technology so that students understand the process that the technology is implementing.

Simulated sampling distributions become the basis for the important discussion of how the sampling distribution provides information about the anticipated accuracy of estimates based on random samples.

Margin of error associated with an estimate of a population proportion can be motivated based on the simulated sampling distribution in one of two ways:

1. In the context of the sampling distribution of a sample proportion, general properties can be described. These include:

• The sampling distribution is approximately normal if the sample size is large enough and the population proportion is not too close to 0 or 1.

• The sampling distribution is centered at the value of the population proportion, meaning that sample proportions tend to cluster around the actual value of the population proportion.

• The standard deviation of the sampling distribution (the standard deviation of the sample proportion) is approximately equal to $\sqrt{p(1-p)/n}$.

Based on these properties, the margin of error for estimating a population proportion is approximately $2\sqrt{p(1-p)/n}$.

2. A less formal approach introduces $1/\sqrt{n}$ as a conservative estimate of the margin of error (this is the maximum value of $2\sqrt{p(1-p)/n}$, which occurs when $p = ½$. The simulated sampling distributions can then be used to convince students that this value is reasonable (and at times a bit large—this is the “conservative” part) as a way to describe error.

Once margin of error has been developed, the formula and/or simulation results can be used to explore the relationship between sample size and margin of error.

Students should practice interpreting margin of error in context. This is a good place to bring in statements from the media where a margin of error is reported and to have students explain what is meant by the margin of error.

The sampling distribution also provides information that allows testing of claims about a population proportion. By comparing an observed sample proportion to what would be expected under a specified model (for example, a model that specifies that the population proportion is 0.6), a decision can be reached about whether the observed data are consistent with the model or whether it provides evidence that the model is not a believable description of the population.

The final part of this section should consider using sample data to estimate a population mean. Simulating to obtain a sampling distribution is not as easy here as in the case of proportions, but there are a number of good applets that can be used to carry out such simulations, e.g., rossmanchance.com/applets/SampleMeans/SampleMeans.html).