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Thread: probability and random samples

  1. #1
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    probability and random samples

    Hi I have some uni problems in my exam revision i can't sort out.
    m=215mg
    sd=15mg
    normal distribution
    215mg of nicotine in a standard cigarette

    a) find the probability a single cigarette has a nicotine content more than 220mg

    b) if a random sample of 25 cigarettes is selected, what is the probability that the mean is more than 220mg

    I tried using Z=y-u/sd but the z score from the table was 0.629 which can be correct right?
    PLEASE help i dont know what model i should be using!!

    Also, i have no idea about part b! HELP!
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  2. #2
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    Quote Originally Posted by kate204 View Post
    Hi I have some uni problems in my exam revision i can't sort out.
    m=215mg
    sd=15mg
    normal distribution
    215mg of nicotine in a standard cigarette

    a) find the probability a single cigarette has a nicotine content more than 220mg

    b) if a random sample of 25 cigarettes is selected, what is the probability that the mean is more than 220mg

    I tried using Z=y-u/sd but the z score from the table was 0.629 which can be correct right?

    Mr F says: Obviously NOT since $\displaystyle {\color{red}220 > \mu = 215}$.

    PLEASE help i dont know what model i should be using!!

    Also, i have no idea about part b! HELP!
    The model you're using is plainly the following:

    X ~ Normal($\displaystyle \mu = 215, ~ \sigma = 15$) where X is the random variable nicotine content of a single cigarette.

    (a) $\displaystyle \Pr(X > 220) = \Pr \left( Z > \frac{220 - 215}{15} \right)$.

    (b) You need to know the distribution of the sample mean:

    $\displaystyle \overline{X}$ ~ Normal$\displaystyle \left( \mu = 215, ~ \sigma = \frac{15}{\sqrt{25}}\right)$

    assuming an 'infinite' population of cigarettes. You must have a theorem in your textbook or class notes that shows where this has come from.

    Now find $\displaystyle \Pr (\overline{X} > 220) = \Pr(Z > .......)$.
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  3. #3
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    Quote Originally Posted by kate204 View Post
    Hi I have some uni problems in my exam revision i can't sort out.
    m=215mg
    sd=15mg
    normal distribution
    215mg of nicotine in a standard cigarette

    a) find the probability a single cigarette has a nicotine content more than 220mg

    b) if a random sample of 25 cigarettes is selected, what is the probability that the mean is more than 220mg

    I tried using Z=y-u/sd but the z score from the table was 0.629 which can be correct right?
    PLEASE help i dont know what model i should be using!!

    Also, i have no idea about part b! HELP!

    (a) $\displaystyle z=\frac{220-215}{15}=0.3333..$

    Now look this up to find $\displaystyle P(Z<0.3333)\approx 0.631$, so to within the limits of accuracy of your table your answer looks OK.

    (b) for the second part you need to use the result that the SD of a sample mean is the SD of an individual divided by the square root of the sample size.

    RonL
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  4. #4
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    Quote Originally Posted by CaptainBlack View Post
    (a) $\displaystyle z=\frac{220-215}{15}=0.3333..$

    Now look this up to find $\displaystyle P(Z<0.3333)\approx 0.631$, so to within the limits of accuracy of your table your answer looks OK.
    [snip]
    Sorry to be disagreeable but the answer is not OK. Not as a final answer anyway, which is how it was presented ....

    The question asks "find the probability a single cigarette has a nicotine content MORE than 220mg" so the answer is 1 - 0.631 ......
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  5. #5
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    sorry im still a little confused,
    why is the final answer 1-0.631 instead of 0.631?
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  6. #6
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    Quote Originally Posted by kate204 View Post
    sorry im still a little confused,
    why is the final answer 1-0.631 instead of 0.631?
    $\displaystyle \Pr (Z < 0.3333) \approx 0.631$. But the question is not asking this.

    It should be clear (and my posts make this abundantly clear) that the question wants $\displaystyle \Pr(Z > 0.3333)$ .....

    As I said at the start, 220 > 215 (the mean) ...... it must be obvious then that Pr(X > 220) cannot possibly be larger than 0.5 .....
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