Confidence Interval of the Mean

If the mean is calculated from a sample, one wants to know the probability that the true mean lies within certain limits around the caclulated mean. These limits are established by the confidence interval. The width of the confidence interval  depends on the probability P that the true mean is found within the limits. The probability P is usually specified as = 1-P; is called the level of significance.

The confidence interval of the mean is defined by

,

with t_n-1;1-/2 being the critical value of the t distribution having n-1 degrees of freedom and a level of significance of 1-/2.

Hint 1:

The correct calculation of the confidence interval requires the sample to be normally distributed.

Hint 2:

In many textbooks the confidence interval of the mean is derived by discussing populations and large samples. Consequently the z-transformation is introduced without paying much attention to the t-distribution. In fact, as the t-distribution approaches the normal distribution for large sample sizes, it is much more convenient to use the t-distribution for both small and large samples.