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Math Help - Lifetime of Tires

  1. #1
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    Lifetime of Tires

    Here is the question:

    For a tire manufacturing company, it is known that only 7% of their tire
    products last less than 35 months, whereas 11% of them last more tham 63
    months. Assuming the distribution of the lifetime of the manufactured tires
    is normal,

    (a) what is the expected lifetime for a tire manufactured by this company,
    and its standard deviation?
    (b) The manufacturer company gives a warranty that if the tire does not last
    more than 40 months, they will give a replacement for it. The company
    sold 1000 tires during the last year. What is the probability that they
    will have to replace more than 50 tires?
    ---

    So I have
    a) P (X <= 35 months) = 0.07
    P (X >= 63 months) = 0.11


    and for b I've started with...
    b) P ( X <= 40 months) = ...?


    Where do I go from here, for each part?


    Thank you so much in advance : )
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  2. #2
    MHF Contributor matheagle's Avatar
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    Z={X-\mu\over \sigma}, so X=\mu + \sigma Z

    Now match up percentile points.
    When X=35 we have the lower 7 percentile of a Z, known as Z_{.07}\approx -1.47 or -1.48.
    That's via the tables, but you can do better via a link on line.
    And when X=63 we have the upper 11 percentile of a Z, known as Z_{.11}\approx 1.22 or 1.23.

    You now have two equations with two unknowns. Find \mu and \sigma.

    IN part (b) you have a binomial question, where n=1000 and a success is for a tire to fail before 40 months.
    So p=P(X<40)=P\biggl(Z<{40-\mu\over\sigma}\biggr).

    HOWEVER, the question should ask you to APPROXIMATE this via the Central Limit Theorem.
    Otherwise you need to obtain via a summation
    P(BINOMIAL >50)=\sum_{x=51}^{1000}{1000\choose x}p^x(1-p)^{1000-x}
    or you can use the complement.
    BUT I bet they want you to approximate via the normal distribution.

    -----------------------------------------------------------
    I was just surfing and I found...
    http://bayes.bgsu.edu/nsf_web/jscrip...ormal_icdf.htm
    Plug in mean 0, st dev 1, prob .07 and I got....-1.4749
    Plug in mean 0, st dev 1, prob .89 and I got....1.2249
    Last edited by matheagle; March 22nd 2009 at 10:54 PM.
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  3. #3
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    Gotcha, thank you!
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