Illuminating Statistical Analysis Using Scenarios and Simulations

11It Has Been the Normal Distribution All Along

I remarked in an earlier chapter that this “trick” formula

works because the shape of the sampling distribution histograms we have been seeing are expected to become the extra-special bell-shape called the normal distribution when our sample size is large enough.¹

The 1.96 in the formula is there instead of some other number because the distribution is a normal distribution. And with a normal distribution of binomial proportions, we expect 95% of its contents to lie within the interval of times . This is a fact about the normal distribution. It is a fact much like pi () is a fact about the relationship between a circle's diameter and circumference ().

Nature often does not give us nice round numbers. Pi is 3.14159…with seemingly random digits going on and on. For simplicity we often round it and write 3.14 (). The figure 1.96 is rounded ...