X = life of motor
x = guarantee period
We need to find the value (in years) that will give us the bottom 3% of the distribution. These are the motors that we are willing to replace under the guarantee.
`P(X < x) = 0.03`
The area that we can find from the z-table is
`0.5 - 0.03 = 0.47`
The corresponding z-score is `z = -1.88`.
Since `Z=(x-mu)/sigma`, we can write:
Solving this gives `x = 6.24.`
So the guarantee period should be `6.24` years.
Here's a graph of our situation. Our normal curve has μ = 10, σ = 2.
The yellow portion represents the 47% of all motors that we found in the z-table (that is, between 0 and −1.88 standard deviations).
The light green portion on the far left is the 3% of motors that we expect to fail within the first 6.24 years.
The left-most portion represents the 3% of motors that we are willing to replace.
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