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Justifying a Claim Based on a Confidence Interval for a Population Proportion
When making claims about an unknown population proportion, a confidence interval provides a plausible range of values for that proportion. Instead of simply stating a point estimate, a confidence interval incorporates the uncertainty inherent in Sampling Distributions for Sample Proportions. This note page focuses on how to use a Constructing a Confidence Interval for a Population proportion to support or refute a specific claim.
Understanding the Confidence Interval
A confidence interval for a population proportion $ p $ is typically expressed as:
$$ \hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}
$$
Where:
- $ \hat{p} $ is the sample proportion.
- $ z^* $ is the critical value from the standard normal distribution corresponding to the desired confidence level.
- $ n $ is the sample size.
This interval gives a range of values within which we are, for example, 95% confident the true population proportion $ p $ lies.
The Role of the Claimed Value
To justify a claim using a confidence interval, we compare the claimed value of the population proportion (often denoted as $ p_0 $ ) with the calculated interval. The claim could be one of two types:
- A specific value claim: The true proportion is equal to some value, e.g., $ p = 0.5 $ .
- A directional claim: The true proportion is greater than, less than, or not equal to some value, e.g., $ p > 0.5 $ , $ p < 0.5 $ , or $ p \neq 0.5 $ .
Justification Rules
The decision rule for justifying a claim based on a confidence interval is straightforward:
| Claim Type | Confidence Interval Result | Conclusion |
|---|---|---|
| $ p = p_0 $ (specific value) | $ p_0 $ is within the interval | We do not have convincing evidence to reject the claim that $ p = p_0 $ . $ p_0 $ is a plausible value. |
| $ p_0 $ is outside the interval | We have convincing evidence to reject the claim that $ p = p_0 $ . $ p_0 $ is not a plausible value. | |
| $ p > p_0 $ (directional) | The entire interval is above $ p_0 $ | We have convincing evidence to support the claim that $ p > p_0 $ . |
| The interval overlaps or is entirely below $ p_0 $ | We do not have convincing evidence to support the claim that $ p > p_0 $ . | |
| $ p < p_0 $ (directional) | The entire interval is below $ p_0 $ | We have convincing evidence to support the claim that $ p < p_0 $ . |
| The interval overlaps or is entirely above $ p_0 $ | We do not have convincing evidence to support the claim that $ p < p_0 $ . | |
| $ p \neq p_0 $ (directional) | $ p_0 $ is outside the interval | We have convincing evidence to support the claim that $ p \neq p_0 $ . |
| $ p_0 $ is within the interval | We do not have convincing evidence to support the claim that $ p \neq p_0 $ . $ p_0 $ is a plausible value. |
Interpreting the Conclusion
When concluding, it’s crucial to phrase your justification carefully:
- Avoid definitive statements: Never say “we have proven” or “the true proportion is.” Instead, use phrases like “we have convincing evidence,” “it is plausible,” or “we do not have convincing evidence to reject/support.”
- Contextualize: Always state your conclusion in the context of the problem.
- Relationship to Significance Tests: A 95% confidence interval is equivalent to a two-sided Setting Up a Test for a Population Proportion at the $ \alpha = 0.05 $ significance level. If the null hypothesized value ( $ p_0 $ ) falls outside the 95% confidence interval, then a two-sided test would reject the null hypothesis at $ \alpha = 0.05 $ . This connection is part of Concluding a Test for a Population Proportion.
Example Scenario
A company claims that 80% of its customers are satisfied. A random sample of 200 customers reveals 150 are satisfied. A 95% confidence interval for the true proportion of satisfied customers is calculated as $ (0.697, 0.803) $ .
Claim: The true proportion of satisfied customers is $ p = 0.80 $ .
Justification: The claimed value, $ p_0 = 0.80 $ , falls within the 95% confidence interval $ (0.697, 0.803) $ . Therefore, we do not have convincing evidence to reject the company’s claim that 80% of its customers are satisfied. The value 0.80 is a plausible proportion for satisfied customers based on this sample.
Claim: The true proportion of satisfied customers is $ p < 0.85 $ .
Justification: The entire 95% confidence interval $ (0.697, 0.803) $ is below 0.85. Therefore, we have convincing evidence to support the claim that the true proportion of satisfied customers is less than 0.85.