# Two Statistics Questions

• Jan 31st 2011, 11:42 AM
maggsmaggs
Two Statistics Questions
1) A tire company has developed a new type of steel-belted radial tire. Extensive testing indicates the population of mileages obtained by all tires of this new type is normally distributed with a mean of 40,000 miles and a standard deviation of 4,000 miles. The company wishes to offer a guarantee providing a discount on a new set of tires if the original tires purchased qualify the guarantee mileage. What should the guaranteed mileage be if the tire company desires that no more than 2 percent of the tires will fail to meet the guaranteed mileage?

-Do you go to the left part of the curve to answer this question, thus a negative z-score? Or is the z-score positive?

-If you do go to the left, what Z-score should you use? If you use the closest to .5-.02, you use -2.05, but that corresponds to .4798, which would be surpassing that 2% limit. Using -2.06 would thus seem to make more sense as a Z-score even though it's further from the .48 than -2.05. That make sense?

• Jan 31st 2011, 12:09 PM
pickslides
Hi there maggsmaggs,

I think you are trying to solve

$\displaystyle \displaystyle P\left( Z<\frac{X-40000}{4000}\right) = 0.02$

$\displaystyle \displaystyle \frac{X-40000}{4000} = -2.0537$

After some further calculations you should get

$\displaystyle \displaystyle X = 31785.2$
• Jan 31st 2011, 12:21 PM
maggsmaggs
So if you just using a simple z-score table with probabilities listed for every hundredth (2.01, 2.02, 2.03, etc.), should you use -2.06 or -2.05?
• Jan 31st 2011, 12:50 PM
pickslides
-2.05, rounded like in regular practice.
• Jan 31st 2011, 01:00 PM
maggsmaggs
But if you use -2.05, then you are inherently making the probability more than 2% ya know? Does that make sense? Like you have two posts above, the actual z-score is between -2.05 and -2.06. Appreciate the help!
• Jan 31st 2011, 01:51 PM
pickslides
Quote:

Originally Posted by maggsmaggs
But if you use -2.05, then you are inherently making the probability more than 2% ya know? Does that make sense? Like you have two posts above, the actual z-score is between -2.05 and -2.06. Appreciate the help!

I know what you are saying but the error is negligible i.e 15 miles.