Question 1159022: Suppose that the lifetimes of tires of a certain brand are normally distributed with a mean of 75,000 miles and a standard deviation of ~ miles. These tires come with a 55,000 -mile warranty. The manufacturer of the tires can adjust ~ during the production process, but the adjustment of ~ is quite costly. The manufacturer wants to set once and for all so that only 1% of the tires will fail before warranty expires. Find the standard deviation to be set. Carry your intermediate computations to at least four decimal places. Round your answer to at least one decimal place.
Answer by ikleyn(53996)   (Show Source):
You can put this solution on YOUR website! .
Suppose that the lifetimes of tires of a certain brand are normally distributed with a mean of 75,000 miles
and a standard deviation of ~ miles. These tires come with a 55,000 -mile warranty.
The manufacturer of the tires can adjust ~ during the production process, but the adjustment of ~ is quite costly.
The manufacturer wants to set once and for all so that only 1% of the tires will fail before warranty expires.
Find the standard deviation to be set. Carry your intermediate computations to at least four decimal places.
Round your answer to at least one decimal place.
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Let' solve it step by step.
Step 1. For the standard normal curve with the mean value 0 and the standard deviation 1 find
the z-mark such that the area under this normal curve on the right of this z-mark is 0.01.
It means that the area under the standard curve on the left of this z-mark is 0.99.
You can use the inverse normal function for it
mean SD area on the left <<<---=== formatting pattern
z = InvNorm( 0, 1, 0.99).
To get the value, you may use either the standard table, or function InvNorm in the calculator
like TI-84, or some of numerous online calculators, for example, the most popular from website
https://onlinestatbook.com/2/calculators/inverse_normal_dist.html/
In any way, you will get the value z = 2.3263 correct with 4 decimals.
Step 2. So now you write this equation for the unknown standard deviation, which the company wants to establish
 = 2.3263.
Solve this equation to find y
75000 - y = 2.3263y,
75000 = 2.3263y + y,
75000 = 3.3263y,
y =  = 22547.57538.
Since they want the answer with one decimal, round this value accordingly.
Notice that the correct rounding in this problem is not to the greater value of SD.
The correct rounding is to the lesser number.
Thus we get after rounding the value for the desired standard deviation of 22547.5 miles (with one decimal).
ANSWER. The desired standard deviation to establish is 22547.5 miles.
Solved.
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