Question

Michelin Tire Company wishes to set a minimum mileage guarantee on its new MX100 tire. Tests reveal the mean mileage is 67,900 with a standard deviation of 2,050 miles and that the distribution of miles follows the normal probability distribution. Michelin wants to set the minimum guaranteed mileage so that no more than 4 percent of the tires will have to be replaced. What minimum guaranteed mileage should Michelin announce

Answer 1

The minimum guaranteed mileage is
`x = 71489.55`

Explanation:

From the question we are told that

The mean is
`\mu = 67900`

The standard deviation is
`\sigma = 2050`

Generally for the probability to be 4% the minimum guaranteed mileage is evaluated as

`P( X \ge x ) =1- P( (X - \mu )/(\sigma ) < ( x - 67900 )/( 2050) ) = 0.04`

`(X -\mu)/(\sigma )  =  Z (The  \ standardized \  value\  of  \ X )`

`P( X \ge x ) =1- P( Z<z) = 0.04`

=>
`P( X \ge x ) = P( Z<z) = 0.96`

=>
`z = ( x - 67900 )/( 2050)`

From the z table, the critical value corresponding to 0.96 to the left of the curve is

`z = 1.751`

So

`1.751 = ( x - 67900 )/( 2050)`

=>
`x = 71489.55`