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Math Help - Need help with probability

  1. #1
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    Need help with probability

    I guess I was hoping if anybody could check my work? This is a 4 part question, I hope I am not breaking rules by posting so much?? I took most of them as far as I could, a couple I am completely clueless on....!

    Suppose that for a school, distances between students' homes and school observe a normal distribution, with mean 4.76 miles and standard deviation being 1.76 miles.

    a) What percentage of students live farther than 6.78 miles from the school?

    P(X>6.78 miles)
    Z=X-mean/std --> 6.78-4.76/1.74= 1.16
    P(0<z<1.16)= 0.3770
    0.5- 0.3770= 0.123
    Thus, 12.3% of students live farther than 6.78 miles from the school.
    (yes? no? i think i may have got the right answer for that one)

    b) A survey shows that 8% of students who live closest to the school chose to walk to the school. What is the maximum walking distance of these 8% of students? In other words, what is the distance below which these 8% of students walk to school?

    (I really don't know how to start on this one, could anybody set me in the right direction?)

    c) Suppose there is a new policy that allows students living beyond 4.50 miles to take a bus to school. There are 3,567 students enrolled. How many students are not eligible to take school buses?

    Z=4.5-4.76/ 1.74= -0.15
    P(-0.15<z<0)= 0.0596
    0.50 + 0.0596 = 0.5596 or 55.96%
    1-0.5596= 0.4404
    0.4404 * 3567= 1570.91 = 1571
    Thus, approximately 1,571 students are not eligible to take school buses.

    d) Suppose all samples of size 12 are taken. What percentage of sample means has a value larger than 6.78 miles?

    *Since this one is talking about sample means, to find the z-score is:

    Z=Mean-Mean of sample means/ std of sample means, so

    Z=6.78-4.76/ (1.74/[sqrt]12)= 4.02

    I get lost here, because how am I supposed to find a z-score of 4.02 in my table? *bangs head* Anybody?
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  2. #2
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    Hello !

    Quote Originally Posted by peanutbutter View Post
    I guess I was hoping if anybody could check my work? This is a 4 part question, I hope I am not breaking rules by posting so much?? I took most of them as far as I could, a couple I am completely clueless on....!
    Breaking the rules ? You're kidding eh ?
    Well, I personnally don't see why it would break the rules

    Suppose that for a school, distances between students' homes and school observe a normal distribution, with mean 4.76 miles and standard deviation being 1.76 miles.

    a) What percentage of students live farther than 6.78 miles from the school?

    P(X>6.78 miles)
    Z=X-mean/std --> 6.78-4.76/1.74= 1.16
    P(0<z<1.16)= 0.3770
    0.5- 0.3770= 0.123
    Thus, 12.3% of students live farther than 6.78 miles from the school.
    (yes? no? i think i may have got the right answer for that one)
    Good
    But please use parenthesis, because it's (X-mean)/std

    b) A survey shows that 8% of students who live closest to the school chose to walk to the school. What is the maximum walking distance of these 8% of students? In other words, what is the distance below which these 8% of students walk to school?

    (I really don't know how to start on this one, could anybody set me in the right direction?)
    That is to say P(0<x<d_{max})=0.08
    I guess you can take it from here, can't you ?

    c) Suppose there is a new policy that allows students living beyond 4.50 miles to take a bus to school. There are 3,567 students enrolled. How many students are not eligible to take school buses?

    Z=4.5-4.76/ 1.74= -0.15
    P(-0.15<z<0)= 0.0596
    0.50 + 0.0596 = 0.5596 or 55.96%
    1-0.5596= 0.4404
    0.4404 * 3567= 1570.91 = 1571
    Thus, approximately 1,571 students are not eligible to take school buses.
    Good !

    d) Suppose all samples of size 12 are taken. What percentage of sample means has a value larger than 6.78 miles?

    *Since this one is talking about sample means, to find the z-score is:

    Z=Mean-Mean of sample means/ std of sample means, so

    Z=6.78-4.76/ (1.74/[sqrt]12)= 4.02

    I get lost here, because how am I supposed to find a z-score of 4.02 in my table? *bangs head* Anybody?
    I'm sorry for this one, I can't help because I have never studied it...

    If you are sure you're correct, then the z-score of 4.02 is something very very near of 0.5, considered as 0.5.
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  3. #3
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    so..

    P(0<x<Xmax)=0.08
    Z-score corresponding to 0.08 is 0.2
    X=mean + z*std

    X=4.76 + 0.2*1.74=5.108

    yes? no?
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  4. #4
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    ok, since i have z score for 0.08 you gave me -2.74, then is the Zmax that you are telling me I need to find, the Zscore for 0.42? since 0.5-0.08=0.42?
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  5. #5
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    I'm sorry, I don't know what happened to my answer

    P(0<x<Xmax)=0.08
    Z-score corresponding to 0.08 is 0.2
    X=mean + z*std

    X=4.76 + 0.2*1.74=5.108

    yes? no?
    This is actually correct. I just hope I'm not mistaking once again... It seems that I'm tired >.<
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  6. #6
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    Quote Originally Posted by peanutbutter View Post
    I guess I was hoping if anybody could check my work? This is a 4 part question, I hope I am not breaking rules by posting so much?? I took most of them as far as I could, a couple I am completely clueless on....!

    Suppose that for a school, distances between students' homes and school observe a normal distribution, with mean 4.76 miles and standard deviation being 1.76 miles.

    [snip]

    d) Suppose all samples of size 12 are taken. What percentage of sample means has a value larger than 6.78 miles?

    *Since this one is talking about sample means, to find the z-score is:

    Z=Mean-Mean of sample means/ std of sample means, so

    Z=6.78-4.76/ (1.74/[sqrt]12)= 4.02

    I get lost here, because how am I supposed to find a z-score of 4.02 in my table? *bangs head* Anybody?
    See here: http://www.mathhelpforum.com/math-he...need-help.html
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  7. #7
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    Quote Originally Posted by peanutbutter View Post
    I guess I was hoping if anybody could check my work? This is a 4 part question, I hope I am not breaking rules by posting so much?? I took most of them as far as I could, a couple I am completely clueless on....!

    Suppose that for a school, distances between students' homes and school observe a normal distribution, with mean 4.76 miles and standard deviation being 1.76 miles.

    [snip]

    b) A survey shows that 8% of students who live closest to the school chose to walk to the school. What is the maximum walking distance of these 8% of students? In other words, what is the distance below which these 8% of students walk to school?

    (I really don't know how to start on this one, could anybody set me in the right direction?)

    [snip]
    Quote Originally Posted by peanutbutter View Post
    P(0<x<Xmax)=0.08
    Z-score corresponding to 0.08 is 0.2
    X=mean + z*std

    X=4.76 + 0.2*1.74=5.108

    yes? no?
    No. Pr(0 < X < 5.108) = 0.575, not 0.08.

    I think you're probably expected to consider Pr(X < Xmax) = 0.08 rather than Pr(0 < X < Xmax) = 0.08. If you draw a simple picture of the distribution for X and put Xmax in a spot where Pr(X < Xmax) is 0.08, it's very clear that Xmax lies well left of the mean. So the corresponding z-value has to be less than 0.

    You should get the z-value corresponding to Pr(Z < zmax) = 0.08 to be zmax = -1.41 and so Xmax = 2.29.
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  8. #8
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    Quote Originally Posted by mr fantastic View Post
    No. Pr(0 < X < 5.108) = 0.575, not 0.08.

    I think you're probably expected to consider Pr(X < Xmax) = 0.08 rather than Pr(0 < X < Xmax) = 0.08. If you draw a simple picture of the distribution for X and put Xmax in a spot where Pr(X < Xmax) is 0.08, it's very clear that Xmax lies well left of the mean. So the corresponding z-value has to be less than 0.

    You should get the z-value corresponding to Pr(Z < zmax) = 0.08 to be zmax = -1.41 and so Xmax = 2.29.
    Ok, I have a question for this...

    x can't be < 0. It's a nonsense.

    So isn't P(0<x<X_{max})=P(x<X_{max}) ?

    Because I once helped someone for that, and we had to consider this, id est not including the extreme values of definition...
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  9. #9
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    Quote Originally Posted by Moo View Post
    Ok, I have a question for this...

    x can't be < 0. It's a nonsense.

    So isn't P(0<x<X_{max})=P(x<X_{max}) ?

    Because I once helped someone for that, and we had to consider this, id est not including the extreme values of definition...
    I agree. But:

    "distances between students' homes and school observe a normal distribution, with mean 4.76 miles and standard deviation being 1.76 miles"

    so the 'nonsense' is due to a defect in the given model.

    I think the problem is more difficult than intended if Xmax such that Pr(0 < X < Xmax) = 0.08 is attempted.

    In light of previous questions, I think Pr(X < Xmax) = 0.08 is the level of difficulty the question intends.

    Peanutbutter should seek clarification on this matter from her professor.
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  10. #10
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    Quote Originally Posted by mr fantastic View Post
    I agree. But:

    "distances between students' homes and school observe a normal distribution, with mean 4.76 miles and standard deviation being 1.76 miles"

    so the 'nonsense' is due to a defect in the given model.

    I think the problem is more difficult than intended if Xmax such that Pr(0 < X < Xmax) = 0.08 is attempted.

    In light of previous questions, I think Pr(X < Xmax) = 0.08 is the level of difficulty the question intends.

    Peanutbutter should seek clarification on this matter from her professor.
    Ok, so answer should be :

    P(x<X_{max})=P(z<Z_{max})=0.08=1-0.92 \implies Z_{max}={\color{red}-1.41}, etc... ?


    This looks weird... and it's confusing :'(
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  11. #11
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    Quote Originally Posted by Moo View Post
    Ok, so answer should be :

    P(x<X_{max})=P(z<Z_{max})=0.08=1-0.92 \implies Z_{max}={\color{red}-1.41}, etc... ?
    [snip]

    Quote Originally Posted by mr fantastic in post #7 View Post
    [snip]
    You should get the z-value corresponding to Pr(Z < zmax) = 0.08 to be zmax = -1.41 and so Xmax = 2.29.
    To expand:

    Z_{\text{max}} = \frac{X_{\text{max}} - 4.76}{1.76} \Rightarrow -1.41 = \frac{X_{\text{max}} - 4.76}{1.76} \Rightarrow X_{\text{max}} = 2.28.

    Note that Pr(X < 2.28) = 0.08 so all is well.

    Quote Originally Posted by Moo View Post
    [snip]
    This looks weird... and it's confusing :'(
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  12. #12
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    Quote Originally Posted by Moo View Post
    Ok, I have a question for this...

    x can't be < 0. It's a nonsense.

    So isn't P(0<x<X_{max})=P(x<X_{max}) ?

    Because I once helped someone for that, and we had to consider this, id est not including the extreme values of definition...
    Just to cover all bases:

    \Pr(0 < X < X_{\text{max}}) = 0.08 \Rightarrow \Pr(X < X_{\text{max}}) - \Pr(X < 0) = 0.08

    \Rightarrow \Pr(X < X_{\text{max}}) - 0.003 = 0.08 \Rightarrow \Pr(X < X_{\text{max}}) = 0.083.

    So the error in making the approximation is small. (For the record it turns out that Xmax = 2.32).

    I'd say the difference is probably comparable to the error in the tables ....
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