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| Math Central | Quandaries & Queries |
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Question from Lorie, a student: 3. The amount of annual snowfall in a certain mountain range is normally distributed with a mean of 109 inches, and a standard deviation of 10 inches. a. What is the probability that the mean annual snowfall during 40 randomly picked years will exceed 111.8 inches? |
Hi Lorie,
Let X be the amount of annual snowfall in inches then X is distributed as the normal random variable with μ = 109 and σ = 10. Let X-bar be the mean annual snowfall of n = 40 years then X-bar is distributed as the normal random variable with mean μX-bar = μ = 109 and variance σ2X-bar = σ2/n = 100/40. You want to find Pr(X-bar > 111.8).
You first need to convert X-bar to the standard normal distribution by Z = (X-bar - μ)/(σ/√n) and then evaluate Pr[(X-bar - μ)/(σ/√n) > (118 - μ)/(σ/√n)] = Pr(Z > (111.8 - 109)/(10/√40)]. Finally evaluate (111.8 - 109)/(10/√40) and use the standard normal table to evaluate the probability.
Harley
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