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Justifying a Claim About the Difference of Two Means Based on a Confidence Interval
When comparing two population means, $ \mu_1 $ and $ \mu_2 $ , a confidence interval for their difference, $ \mu_1 - \mu_2 $ , is a powerful tool for making a justified claim. This method allows us to infer whether there’s a statistically significant difference between the two means, or if one mean is larger or smaller than the other, based on sample data. This builds upon the concepts introduced in Confidence Intervals for the Difference of Two Means.
The Purpose of Justification
The primary goal is to use the constructed confidence interval (e.g., a 95% confidence interval) to support or refute a claim about the true difference between the two population means. Common claims often revolve around whether the means are equal, or if one is greater than the other.
Relating Confidence Intervals to Hypotheses
Though we’re not performing a formal Setting Up a Test for the Difference of Two Population Means, the interpretation of the confidence interval is closely related to potential null hypotheses. The most common null hypothesis is that there is no difference between the two population means, i.e., $ H_0: \mu_1 - \mu_2 = 0 $ .
A $ 100(1-\alpha)% $ confidence interval for $ \mu_1 - \mu_2 $ provides a range of plausible values for the true difference.
Decision Rule Based on the Confidence Interval
The key to justifying a claim lies in observing whether a specific value (most commonly 0, representing no difference) is included within the interval, or where the entire interval lies relative to that value.
Let $ (L, U) $ be the calculated confidence interval for $ \mu_1 - \mu_2 $ , where $ L $ is the lower bound and $ U $ is the upper bound.
| Condition | Interpretation | Conclusion (Claim) |
|---|---|---|
| **0 is contained in $ (L, U) $ ** | The confidence interval includes 0. This means that a difference of zero between the two population means is a plausible value. | We do not have convincing evidence to claim a difference between $ \mu_1 $ and $ \mu_2 $ . It is plausible that $ \mu_1 = \mu_2 $ . |
| ** $ L > 0 $ ** | The entire confidence interval is above 0. This means that all plausible values for $ \mu_1 - \mu_2 $ are positive. | We have convincing evidence to claim that $ \mu_1 - \mu_2 > 0 $ , or equivalently, $ \mu_1 > \mu_2 $ . The first mean is significantly larger than the second. |
| ** $ U < 0 $ ** | The entire confidence interval is below 0. This means that all plausible values for $ \mu_1 - \mu_2 $ are negative. | We have convincing evidence to claim that $ \mu_1 - \mu_2 < 0 $ , or equivalently, $ \mu_1 < \mu_2 $ . The first mean is significantly smaller than the second. |
Example Justification
Suppose a 95% confidence interval for the difference in mean test scores ( $ \mu_{\text{Method A}} - \mu_{\text{Method B}} $ ) is $ (2.5, 7.8) $ .
Justification: “We are 95% confident that the true difference in mean test scores between Method A and Method B (Method A - Method B) lies between 2.5 and 7.8 points. Since the entire interval is above 0, we have convincing evidence to claim that the mean test score for students using Method A is significantly higher than for students using Method B.”
If the interval was $ (-3.1, 1.2) $ :
Justification: “We are 95% confident that the true difference in mean test scores between Method A and Method B (Method A - Method B) lies between -3.1 and 1.2 points. Since the interval includes 0, we do not have convincing evidence to claim a significant difference in mean test scores between the two methods. It is plausible that the mean scores are equal.”
Importance of Context
Always state your conclusion in the context of the problem. This means referring to the specific populations, variables, and units involved.
Relationship to Hypothesis Testing
Justifying a claim using a confidence interval is directly related to a two-sided Carrying Out a Test for the Difference of Two Population Means at the corresponding significance level $ \alpha $ . If a $ 100(1-\alpha)% $ confidence interval for $ \mu_1 - \mu_2 $ does not contain 0, then a hypothesis test for $ H_0: \mu_1 - \mu_2 = 0 $ versus $ H_A: \mu_1 - \mu_2 \neq 0 $ would result in rejecting $ H_0 $ at the $ \alpha $ significance level. Conversely, if the confidence interval does contain 0, then we would fail to reject $ H_0 $ .
For one-sided claims, while confidence intervals can provide strong evidence, a formal one-sided hypothesis test is generally more appropriate for specific claims like $ \mu_1 > \mu_2 $ . However, if the entire interval is above (or below) 0, it certainly provides evidence for a one-sided claim.
This method of justification is a critical skill for Selecting, Implementing, and Communicating Inference Procedures.