EXAMPLE 5.9

The central limit theorem in action. Figure 5.8 shows the central limit theorem in action for another very non-Normal population. Figure 5.8(a) displays the density curve of a single observation from the population. The distribution is strongly right-skewed, and the most probable outcomes are near 0. The mean μ of this distribution is 1, and its standard deviation σ is also 1. This particular continuous distribution is called an exponential distributionexponential distribution. Exponential distributions are used as models for how long an iPhone will function properly and for the time between snaps you receive on Snapchat.

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Figure 5.8 The central limit theorem in action: the sampling distribution of sample means from a strongly non-Normal population becomes more Normal as the sample size increases, Example 5.9. (a) The distribution of 1 observation. (b) The distribution of for 2 observations. (c) The distribution of for 10 observations. (d) The distribution of for 25 observations.

Figure 5.8(b), (c), and (d) are the density curves of the sample means of 2, 10, and 25 observations from this population. As n increases, the shape becomes more Normal. The mean remains at μ = 1, but the standard deviation decreases, taking the value . The density curve for 10 observations is still somewhat skewed to the right but already resembles a Normal curve having μ = 1 and . The density curve for n = 25 is yet more Normal. The contrast between the shape of the population distribution and of the distribution of the mean of 10 or 25 observations is striking.