Evaluate statistical claims and conclusions

Detailed Explanation: Evaluate statistical claims and conclusions

To evaluate a statistical claim, check whether the data and the method support the conclusion—not just whether the sample result seems large or small.

Use this process:

1. Identify the population and the sample.
2. Check how the sample was selected. A random, representative sample is more trustworthy than a convenience sample.
3. Consider sample size and variability. Larger samples usually give more precise results.
4. Use the margin of error or confidence interval. Ask whether the claimed value is consistent with the interval.
5. Decide whether the conclusion is justified in context.

Worked example

A news article states:

“A majority of city residents support building a new library.”

The article is based on a random sample of $(800)$ city residents. In the sample, $(54\%)$ supported the proposal. The poll reports a margin of error of $(4\%)$ at the $(95\%)$ confidence level.

Step 1: Identify the population and sample

• Population: All residents of the city
• Sample: The $(800)$ residents who answered the poll

Because the sample was randomly selected, it is likely to represent the city’s residents reasonably well. However, the result could still vary from sample to sample.

Step 2: Find the confidence interval

The sample percentage is $(54\%)$, with a margin of error of $(4\%)$.

So the approximate $(95\%)$ confidence interval is

where $p$ is the true percentage of all city residents who support the proposal.

Step 3: Compare the interval with the claim

The claim says that more than $(50\%)$ of residents support the proposal.

However, the interval includes $(50\%)$. This means the true support could be exactly half of the population, rather than a majority.

Therefore, the sample provides some evidence of support, but it does not give strong enough evidence to conclude confidently that a majority of all residents support the proposal.

Step 4: State a careful conclusion

A better conclusion is:

In the sample, $(54\%)$ supported the proposal. Because the margin of error is $(4\%)$, the poll does not establish with confidence that more than half of all city residents support it.

The original claim is not fully supported. The random sample and fairly large sample size make the poll useful, but the uncertainty is large enough that the true percentage might be $(50\%)$, not greater than $(50\%)$.