1 and 2 are equations and 3 is a solution
Answer:
The minimum guaranteed mileage is ![x = 71489.55](https://tex.z-dn.net/?f=x%20%3D%2071489.55)
Step-by-step explanation:
From the question we are told that
The mean is ![\mu = 67900](https://tex.z-dn.net/?f=%5Cmu%20%20%3D%20%2067900)
The standard deviation is ![\sigma = 2050](https://tex.z-dn.net/?f=%5Csigma%20%3D%20%202050)
Generally for the probability to be 4% the minimum guaranteed mileage is evaluated as
![P( X \ge x ) =1- P( \frac{X - \mu }{\sigma } < \frac{ x - 67900 }{ 2050} ) = 0.04](https://tex.z-dn.net/?f=P%28%20X%20%20%5Cge%20x%20%29%20%3D1-%20%20P%28%20%5Cfrac%7BX%20-%20%5Cmu%20%7D%7B%5Csigma%20%7D%20%3C%20%5Cfrac%7B%20x%20-%20%2067900%20%7D%7B%202050%7D%20%20%29%20%3D%200.04)
![\frac{X -\mu}{\sigma } = Z (The \ standardized \ value\ of \ X )](https://tex.z-dn.net/?f=%5Cfrac%7BX%20-%5Cmu%7D%7B%5Csigma%20%7D%20%20%3D%20%20Z%20%28The%20%20%5C%20standardized%20%5C%20%20value%5C%20%20of%20%20%5C%20X%20%29)
![P( X \ge x ) =1- P( Z](https://tex.z-dn.net/?f=P%28%20X%20%20%5Cge%20x%20%29%20%3D1-%20%20P%28%20Z%3Cz%29%20%3D%200.04)
=> ![P( X \ge x ) = P( Z](https://tex.z-dn.net/?f=P%28%20X%20%20%5Cge%20x%20%29%20%3D%20P%28%20Z%3Cz%29%20%3D%200.96)
=> ![z = \frac{ x - 67900 }{ 2050}](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7B%20x%20-%20%2067900%20%7D%7B%202050%7D)
From the z table, the critical value corresponding to 0.96 to the left of the curve is
![z = 1.751](https://tex.z-dn.net/?f=z%20%3D%201.751)
So
![1.751 = \frac{ x - 67900 }{ 2050}](https://tex.z-dn.net/?f=1.751%20%3D%20%5Cfrac%7B%20x%20-%20%2067900%20%7D%7B%202050%7D)
=> ![x = 71489.55](https://tex.z-dn.net/?f=x%20%3D%2071489.55)
The square root of 2.25 is 1.5