Answer:
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
<u>Empirical rule solution</u>
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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<u>Calculator solution</u>
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%.
