4) The “Cool Breeze” company sells an air conditioner model with a mean life of

Question

4) The “Cool Breeze” company sells an air conditioner model with a mean life of 9.25 years and a standard deviation of 42 months. The company will replace any air conditioner that malfunctions during the guarantee period.

If the guarantee period is 6 years, what fraction of the air conditioner units it produces will it have to replace? (to 3 decimal places)

A Business Analyst has determines that no more than 15% of the units for this model should be replaced if its profit target is to be met. How long should the guarantee run (at the most) to meet this goal?

Explanation / Answer

Given, mean life = 9.25 years SD=42 months =3.5 years and theguarantee period is 6 years.

The fraction of the air conditioners to be replaced = P(x<6 years).

By converting the distribution to Standard Normal Distribution, we get

P(x<6 years) = P[ (x-9.25)/3.5 < (6-9.25)/3.5]

= P(z< -0.92857) = 0.176556 [ According to biometrika table]

The required fraction is 0.177

For the number of replacements to be at the most 15%, take P(z < New guarantee period in years) = 0.15

According to the biometrika table for p = 0.15 the value of z is -1.036433 that is 1.036433 times the SD below the mean.

Therefore, New guarantee period = 9.25 - 1.036433 * 3.5 = 9.25 -3.6275155

= 5.6224845 years = 5.6224845 * 12 months

= 67.469814 months

Therefore New guarantee period must not be more than 5 years and 7 months.

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