However, when predicting the height of individuals, the length of the interval does not decrease with larger n . Assume that the 55 men were a random sample from the population to which we wish to generalize. Now for an example with discrete data. Suppose I interview 20 students and learn that 13 of them prefer cola to orange pop. A different sample would probably give some other estimate of the proportion of students who prefer cola.
Statisticians construct confidence intervals all the time.
Then about 150 of them will have provided accurate information and the others would have been totally misleading. The justification can be found in any discussion of standard normal tables, where you will learn that the number two we have been using is actually a nice approximation to 1.96. The values 2, 1.65 and 2.58 work well in all kinds of situations where you might want to estimate an average. A statistician handles the problem as follows. Now, here is the crunch.