Can confidence intervals overlap and still be significant?

As a statistical expert with a deep understanding of confidence intervals and their role in hypothesis testing, I can provide a comprehensive answer to your question. Confidence intervals are a critical tool in statistical analysis, offering a range that is likely to contain the true population parameter. When we talk about the overlap of confidence intervals, we're essentially discussing the potential for two intervals to share common values, which can have implications for the significance of the difference between the means of two populations.

### Understanding Confidence Intervals

Confidence intervals are constructed around sample statistics using a formula that includes the sample mean, the standard error, and a critical value from a distribution (often the normal or t-distribution) that corresponds to the desired level of confidence. A 95% confidence interval, for instance, suggests that if we were to take many samples and construct such intervals, 95% of these would contain the true population mean.

### Overlapping Intervals and Statistical Significance

The overlap of confidence intervals does not inherently indicate whether the difference between two means is statistically significant. The significance is determined by conducting a hypothesis test, such as a t-test or an ANOVA, which compares the observed difference between the means to the variability within and between the groups. The p-value resulting from this test indicates the likelihood that the observed difference could have occurred by chance if there were no true difference.

When two 95% confidence intervals for the means of two independent populations do not overlap, it suggests that the sample means are sufficiently far apart that the true means may not be equal. However, this does not automatically mean that the difference is statistically significant. The lack of overlap raises the likelihood of a significant difference, but it is the p-value from the hypothesis test that provides the definitive answer.

### The Role of Sample Size and Variability

The sample size and the variability within the samples play crucial roles in determining the width of the confidence intervals. Larger samples with less variability will produce narrower confidence intervals, making it more likely that intervals from two groups will not overlap if there is a true difference in the population means.

### The Importance of Hypothesis Testing

Even if confidence intervals do not overlap, it is essential to perform a hypothesis test to confirm statistical significance. The test accounts for the possibility of a Type I error (incorrectly rejecting a true null hypothesis) or a Type II error (failing to reject a false null hypothesis). The significance level (α), typically set at 0.05, is the threshold for deciding whether the results are statistically significant.

### Conclusion

In conclusion, while overlapping confidence intervals suggest that the means of two populations may not be significantly different, it is not a definitive indicator. The overlap can be a preliminary observation that prompts further investigation through hypothesis testing. It is the p-value from the test that ultimately determines the statistical significance of the difference between the means.

Now, let's proceed with the translation into Chinese.

When 95% confidence intervals for the means of two independent populations don't overlap, there will indeed be a statistically significant difference between the means (at the 0.05 level of significance).May 23, 2014