I always tell students that the interpretation of 95% confidence interval, (a, b), should be as follows,

• Base on the same sampling procedure.
• I repeat the procedure 100 times,
• For each sample, a related confidence interval is calculated. (The interval may be different each time.)
• I have confident that approximately 95 out of 100 confidence intervals will cover the true mean.

This is different from saying “There is a .95 probability that the mean is in this interval (a, b).” Understanding how these two interpretations differ is often difficult for students to grasp. Today, an example came to mind and I hope it will be helpful. I’d like to take a note here:

Example

If you see a baby’s picture and you try to guess the gender of the baby. You might say “he is a boy.” What is the chance to get the correct answer? 50%, 95%, or more?

You would not say “this baby is 95% boy (5% girl).” You will say “If one hands me different children’s pictures, I can guess and be right 95 out of  100 times.” It is a similar reasoning for 95% confidence interval. Based on the sample, our statistics can guarantee that 95 out of 100 times the calculated interval will cover the true mean.

P.S. The picture is my younger sister.