Central Limit Theorem Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard
Customers arrive at a store with mean μ=2\mu=2 per minute, σ=1.5\sigma=1.5 per minute (Poisson-like). For 36-minute observation windows, find P(total arrivals>80)P(\text{total arrivals} > 80) using CLT.

Solution

  1. 1
    Total arrivals in 36 min: T=∑i=136XiT = \sum_{i=1}^{36} X_i; mean =36×2=72= 36 \times 2 = 72; SD =1.536=9= 1.5\sqrt{36} = 9
  2. 2
    By CLT: T∼N(72,9)T \sim N(72, 9) approximately
  3. 3
    P(T>80)=P(Z>80−729)=P(Z>0.889)≈1−0.813=0.187P(T > 80) = P\left(Z > \frac{80-72}{9}\right) = P(Z > 0.889) \approx 1 - 0.813 = 0.187
  4. 4
    About 18.7% chance of more than 80 arrivals in a 36-minute window

Answer

P(T>80)≈0.187P(T > 80) \approx 0.187. About 18.7% chance of more than 80 arrivals.
CLT also applies to sums: the sum of n independent identically distributed variables is approximately normal with mean n·μ and SD σ·√n. This enables normal-distribution calculations for Poisson processes, sums of uniform variables, and many other non-normal settings.

About Central Limit Theorem

For sufficiently large sample size (n≥30n \geq 30 as a rule of thumb), the sampling distribution of the sample mean is approximately normal with mean μ\mu and standard deviation σn\frac{\sigma}{\sqrt{n}}, regardless of the shape of the population distribution.

Learn more about Central Limit Theorem →

More Central Limit Theorem Examples

Example 1 medium

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Example 2 hard

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Example 3 easy

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