The useful life of a radial tire is normally distributed with a mean of 30,000 miles and a standard deviation of 5000 miles. The company makes 10,000 tires a month. What is the probability that if a radial tire is purchased at random, it will last

between 20,000 and 35,000 miles?

empirical rule: 81.5%

table or calculator: 81.9%

Step-by-step explanation:

The lower limit of the life range of interest has a z-score of ...

z = (x - μ) / σ = (20,000 - 30,000) / 5,000 = - 2

The upper limit has a z-score of ...

z = (35,000 - 30,000) / 5,000 = 1

Empirical rule solution

The empirical rule tells you that 95% of the distribution lies within 2 standard deviations of the mean, so (100% - 95%) / 2 = 2.5% lie below z = - 2. It also tells you 68% lie within 1 standard deviation of the mean, so (100% - 68%) / 2 = 16% lie above z = 1.

The fraction that lies within - 2 to 1 standard deviations of the means is thus ...

(100% - 2.5% - 16%) = 81.5%

The probability the tire has a life in the desired range is about 81.5%.

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Calculator solution

A probability calculator for the Normal distribution tells you that ...

P (-2 < z < 1) ≈ 0.8185946

The probability the tire has a life in the desired range is about 81.9%.