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Math Help - How do I calculate the probability of Z-score 4?

  1. #1
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    Unhappy How do I calculate the probability of Z-score 4?

    Ok I know that the probability of a Z-score of 4 is .49997. I simply want to find out HOW this number was calculated from a Z-score chart that only goes up to like 3.49.

    Z-score charts typically only go up to 3.4 and 3.49, so how do I find the probability of a Z-score equaling 4?

    *NOTE* this is finite math. please be descriptive as possible and refrain from statistics and calculus solutions.

    much thanks!
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    Re: How do I calculate the probability of Z-score 4?

    Quote Originally Posted by sharkfood4 View Post
    Ok I know that the probability of a Z-score of 4 is .49997

    Are you telling us that \displaystyle P(Z<4) = 0.49997 ?
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    Re: How do I calculate the probability of Z-score 4?

    IDK. The respective area for Z=4 is .49997
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    Re: How do I calculate the probability of Z-score 4?

    The area to the left or right of Z=4? Saying the area for Z=4 does not make a lot of sense given the normal distribution is continuous.
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    Re: How do I calculate the probability of Z-score 4?

    Its to the right of the mean, which has a raw score of 16oz. The Z-score of 4 coresponds to the raw score of 16.5oz to the right.

    sample size 64, standard error of the mean= .125
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    Re: How do I calculate the probability of Z-score 4?

    Ok, now you have provided more information, this is helpful.

    The value 16.5 is to the right of 16, but to answer the entire question we need to know even more. Lets say its looking at the probability that the mean is less than 16.5 then

    P(\bar{X} <16.5) = P\left( Z< \frac{16.5-16}{0.125}\right) = P(Z<4) = 0.\dot{9}

    In the case that

    P(\bar{X} >16.5) = P\left( Z> \frac{16.5-16}{0.125}\right) = P(Z>4) =1-P(Z<4) = 1-0.\dot{9} \approx 0

    How did I get this value for P(Z<4) = 0.\dot{9} ? Well consider the 68-95-99.7 percent rule for the normal distribution. It states that P(-3<Z<3)= 0.997 you should now be able to convince yourself that P(Z<4) = 0.\dot{9}
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    Re: How do I calculate the probability of Z-score 4?

    Thank You.

    How about finding the probability that the sample mean is between 15.7 and 16.5.

    random sample of 64
    15.7 has Z=-2.4
    16.5 has Z=4
    Mean=16
    Standard error of the mean=.125
    Sample variance=.015625
    population variance=1
    population SD=1

    Looking at the Z-chart, I find the area between the raw scores of 15.7 and 16 to be .4918
    Now I must find the area between 16 and 16.5 so I can add that value to .4918 and get the sum of the two areas.

    How should this area be found if the raw score of 16.5 gives a z-score of 4. Am I to use the 68-95-99.7 percent rule?
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    Re: How do I calculate the probability of Z-score 4?

    Another thought,

    given \displaystyle n = 64 and  \displaystyle 0.125 = \frac{\sigma}{\sqrt{64}} \implies \sigma = 0.125\times 8 = 1

    then,

    P(X<16.5) = P\left( Z< \frac{16.5-16}{1}\right) = P(Z<0.5)

    I have a feeling this is what is required.
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    Re: How do I calculate the probability of Z-score 4?

    Given the above questions have a 'raw score' in them my information in post #8 applies when you have a raw score.


    Quote Originally Posted by sharkfood4 View Post

    How about finding the probability that the sample mean is between 15.7 and 16.5.

    random sample of 64
    15.7 has Z=-2.4
    16.5 has Z=4

    Correct, as you have stated, this is for the 'sample mean', then P(-2.4<Z<4) = P(Z<4) - P(Z<-2.5)
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    Re: How do I calculate the probability of Z-score 4?

    your probably right. Im just new to it all.

    I found the the area between -2.4 and 0 to be .4918. This was done first by converting the raw score of 15.7 to a z-score of -2.4 by using the formula Z= X-M/standard error of the mean

    Now, to be consistant, I used the same formula to convert our other raw score of 16.5 to a z-score of 4.

    Ok now finding the area between -2.4 and 0 was simply a matter of looking at the z-score chart, finging the 2.4 and observing the coresponding area of .4918

    Now, Its understood that we need to find the area between 0 and 4. The problem was that the z-chart ONLY goes up to Z-score 3.49, not 4.

    The area between 0 and 4 is .49997. I only know this much because the answer in the back of the book is .99177, which means that
    .4918 + .49997 = .99177

    Its not a matter of me finding the answer though. I need to find out a more direct way of looking at the z-chart and finding the area when the z-score is 4. From the computation .4918 + .49997 =.99177 , its already clear that my math is correct since .99177 is the answer listed in the back of the book. And the fact the the z-score of -2.4 yeilds .4918, as clearly seen on the z chart.

    The mystery to me wrapped around computing the area between 0 and 4, since the z-chart ONLY goes up to 3.49
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    Re: How do I calculate the probability of Z-score 4?

    It only goes to 3.49 b/c the area between 0 & 3.49 is approx 0.5 which will be the same for all values >3.49.
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    Re: How do I calculate the probability of Z-score 4?

    Z=Score of 4 gives us .49997. BUT, only armed with a sub-par understanding of finite math and a z-chart going up to 3.49, how would I go about finding that 4 = ('.49997')? thats my hang up.
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    Re: How do I calculate the probability of Z-score 4?

    I can see this is frustrating, just don't read too much into it.

    The normal distribution is a total contridiction. Consdering You have a contiuous function all > 1 on (-\infty, \infty) with a 'finite' area of 1 underneath, you can never understand it fully.

    So my advice is to be at peace with it. There are bigger fish to fry in the big world that is probability
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    Re: How do I calculate the probability of Z-score 4?

    1.3.6.7.1. Cumulative Distribution Function of the Standard Normal Distribution

    The link above is what I was looking for; The area between the mean and a Z-score of 4 is listed as .49997 thank you for your patience though. I am still pretty new at this ...obviously.
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    Re: How do I calculate the probability of Z-score 4?

    I think you understand it better than most newbies.
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