EXAMPLE 5.11

Time between snaps. Snapchat has more than 100 million daily users sending well over 400 million snaps a day.6 Suppose that the time X between snaps received is governed by the exponential distribution with mean μ = 15 minutes and standard deviation σ = 15 minutes. You record the next 50 times between snaps. What is the probability that their average exceeds 13 minutes?

The central limit theorem says that the sample mean time (in minutes) between snaps has approximately the Normal distribution with mean equal to the population mean μ = 15 minutes and standard deviation

The sampling distribution of is, therefore, approximately N(15,2.12). Figure 5.10 shows this Normal curve (solid) and also the actual density curve of (dashed).

The probability we want is . This is the area to the right of 13 under the solid Normal curve in Figure 5.10. A Normal distribution calculation gives

= P (Z > −0.94) = 0.8264

303

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Figure 5.10 The exact distribution (dashed) and the Normal approximation from the central limit theorem (solid) for the average time between snaps received, Example 5.11.

The exactly correct probability is the area under the dashed density curve in the figure. It is 0.8265. The central limit theorem Normal approximation is off by only about 0.0001.