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

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okpalawalter8 okpalawalter8 · Answer 1 · 2020-12-06T03:41:30+01:00

Answer:

The minimum guaranteed mileage is [tex]x = 71489.55[/tex]

Step-by-step explanation:

From the question we are told that

The mean is [tex]\mu = 67900[/tex]

The standard deviation is [tex]\sigma = 2050[/tex]

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

[tex]P( X \ge x ) =1- P( \frac{X - \mu }{\sigma } < \frac{ x - 67900 }{ 2050} ) = 0.04[/tex]

[tex]\frac{X -\mu}{\sigma } = Z (The \ standardized \ value\ of \ X )[/tex]

[tex]P( X \ge x ) =1- P( Z<z) = 0.04[/tex]

=> [tex]P( X \ge x ) = P( Z<z) = 0.96[/tex]

=> [tex]z = \frac{ x - 67900 }{ 2050}[/tex]

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

[tex]z = 1.751[/tex]

So

[tex]1.751 = \frac{ x - 67900 }{ 2050}[/tex]

=> [tex]x = 71489.55[/tex]