<\/span><\/h3>\nA confidence interval (CI)<\/strong> is a range of values that is likely to contain the true population parameter.<\/p>\n<\/span>What does a 95% confidence interval mean?<\/span><\/h3>\nA 95% confidence interval means that if you repeated the study 100 times, approximately 95 of those intervals would contain the true value. The width of the interval reflects the precision of your estimate: a narrow interval means a more precise estimate; a wide one reflects more uncertainty, usually due to a small sample size.<\/p>\n
<\/span>How confidence intervals relate to p-values<\/span><\/h3>\nThere is a direct relationship between confidence intervals and p-values. For a two-tailed test at a significance level of \u03b1 = 0.05:<\/p>\n
\n- If a 95% confidence interval does not<\/strong> include zero (for a difference) or one (for a ratio), the result is statistically significant at p<\/em> < 0.05.<\/li>\n
- If the interval does<\/strong> include zero, the result is not statistically significant.<\/li>\n<\/ul>\n
Returning to the clinical trial example: suppose the new drug was found to last an average of 3 hours longer than the standard drug, with a 95% CI of [0.8, 5.2] hours. Because the interval does not include zero, this confirms the statistically significant finding. The interval also tells us something the p-value cannot: the true benefit is plausibly anywhere between just under one hour and over five hours longer.<\/p>\n
<\/span>Why confidence intervals matter<\/span><\/h3>\nA p-value can be very small simply because the sample was very large, even when the true effect is trivial. A confidence interval anchors the result to a real-world scale. Reporting the interval allows readers to judge whether a statistically significant result is also practically<\/em> significant. This is something the p-value alone cannot answer. The American Psychological Association (APA) and most major journals now strongly recommend or require the reporting of confidence intervals alongside p-values.<\/p>\n <\/p>\n
<\/span>Type I and Type II Errors<\/span><\/h2>\nEvery hypothesis test carries the risk of reaching the wrong conclusion. There are two distinct ways this can happen, and understanding them is essential for interpreting p-values correctly.<\/p>\n
<\/span>What is a Type I error?<\/span><\/h3>\nType I error (false positive)<\/strong> occurs when you reject the null hypothesis even though it is actually true. In other words, you conclude that an effect exists when in reality it does not. The probability of making a Type I error is equal to your significance level, \u03b1. If you set \u03b1 = 0.05, there is a 5% chance of a false positive even if the null hypothesis is completely true.<\/p>\n<\/span>What is a Type II error?<\/span><\/h3>\nType II error (false negative)<\/strong> occurs when you fail to reject the null hypothesis even though it is actually false. You conclude there is no effect when in reality one does exist. The probability of a Type II error is denoted \u03b2. It is influenced by sample size, effect size, and the chosen significance level.<\/p>\n