What is the difference between level of significance and confidence interval?

As a domain expert in statistics, I often encounter questions about the nuances between concepts such as the level of significance and confidence intervals. These are fundamental to understanding statistical inference, which is the process of drawing conclusions from data.

Level of Significance refers to the probability of rejecting the null hypothesis when it is true. It is a threshold that researchers set before conducting a statistical test to determine whether the results are statistically significant. The level of significance is denoted by the Greek letter alpha (α) and is often set at 0.05, which means there is a 5% chance of committing a Type I error (false positive). This means that if the p-value (the probability of observing the data given that the null hypothesis is true) is less than the level of significance, the null hypothesis is rejected, and the results are considered statistically significant.

Confidence Interval, on the other hand, provides an estimated range of values that likely contains an unknown population parameter, such as the mean or proportion, with a certain level of confidence. The confidence level is the percentage of possible confidence intervals that contain the true value of the parameter. For instance, a 95% confidence interval means that if we were to take many samples and construct a confidence interval from each, we would expect 95% of those intervals to contain the true population parameter. The confidence interval is constructed using the sample statistic, the standard error, and the critical value from the distribution (often the normal distribution or the t-distribution).

Now, let's address the statement provided: "So, if your significance level is 0.05, the corresponding confidence level is 95%." This is a common misconception. While it is true that a significance level of 0.05 corresponds to a confidence level of 95% in the context of a two-tailed test, this does not mean that the concepts are equivalent. The significance level is about the probability of making a Type I error, while the confidence level is about the proportion of confidence intervals that include the true parameter value.

Another point to clarify is: "If the P value is less than your significance (alpha) level, the hypothesis test is statistically significant." This is correct. A p-value that is less than the significance level indicates that the observed results are unlikely to have occurred by chance if the null hypothesis were true, leading to its rejection.

Lastly, "If the confidence interval does not contain the null hypothesis value, the results are statistically significant." This statement is also correct. If the confidence interval for a parameter does not include the value specified by the null hypothesis (often zero or some other baseline value), it suggests that the observed effect is unlikely to be zero or that value, and thus the results are statistically significant.

In summary, while the level of significance and confidence intervals are related in that they both deal with the probability of a certain event occurring, they serve different purposes and are not interchangeable. The level of significance is used to make a decision about the null hypothesis, whereas the confidence interval provides an estimate of the parameter's range.

So, if your significance level is 0.05, the corresponding confidence level is 95%. If the P value is less than your significance (alpha) level, the hypothesis test is statistically significant. If the confidence interval does not contain the null hypothesis value, the results are statistically significant.Apr 2, 2015