Justifying a Claim Based on a Confidence Interval for a Population Proportion
Introduction
After learning how to construct confidence intervals, the next step is interpreting them correctly and using them to evaluate claims about a population proportion. Confidence intervals provide a range of plausible values for an unknown parameter and serve as statistical evidence when assessing whether a claim is reasonable. Understanding how to justify conclusions is a core skill in statistical inference.
What a Confidence Interval Tells Us
A confidence interval either contains the true population parameter or it does not. The parameter is fixed but unknown.
Correct interpretation: "We are C% confident that the true population proportion is between the lower and upper bounds."
Main idea: The confidence level refers to the reliability of the method. If the same sampling process were repeated many times, about C% of the intervals produced would capture the true parameter.
Interpreting Confidence Intervals Properly
A valid interpretation must include:
- The confidence level
- The population parameter (proportion)
- The population in context
- Both interval bounds
Avoid these common mistakes:
- Treating the confidence level as a probability about the parameter (e.g., "There is a 95% chance...")
- Saying the sample proportion (p-hat) lies in the interval (it is always at the center of the interval!)
- Claiming a certain percentage of data falls within the interval.
- Using absolute language such as "proves" or "definitely."
Confidence intervals provide evidence, not certainty.
Using Confidence Intervals to Justify Claims
Confidence intervals define a range of plausible parameter values.
The Decision Rule:
- Claimed value inside the interval: The claim is plausible (consistent with the data).
- Claimed value outside the interval: There is convincing evidence against the claim.
When writing conclusions, use statistical language such as:
- "Provides convincing evidence..."
- "Is consistent with..."
- "Is not supported by..."
Confidence Level and Interval Width
For a fixed sample size:
- Increasing the confidence level increases the critical value ().
- A larger critical value increases the margin of error.
- A larger margin of error produces a wider interval.
The Trade-off: Higher confidence = wider interval = less precision. Lower confidence = narrower interval = greater precision.
Sample Size and Interval Width
Sample size (n) directly affects the standard error. As sample size increases:
- Standard error decreases.
- Margin of error decreases.
- Interval width decreases.
Important Relationship: The width is approximately proportional to . To cut the width in half, you must multiply the sample size by 4. Larger samples improve precision but require substantially more data.
Variability and Interval Width
Interval width is also influenced by variability in the sample proportion (p-hat).
- Proportions near 0.5 create the greatest variability (wider intervals).
- Proportions near 0 or 1 create less variability (narrower intervals).