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Math Help - Normal Distributions

  1. #1
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    Normal Distributions

    This is normal distributions and i dont know anything about how to do this or where to even start so any help would help me Thanks \frac{\mathrm{d} }{\mathrm{d} }

    The time required to finish a test in normally distributed with a mean of 40 minutes and a standard deviation of 8 minutes. What is the probability that a student chosen at random will finish the test in more than 48 minutes?
    A.84%
    B.2%
    C.34%
    D.16%
    2.
    Suppose you have a normally distributed set of data pertaining to a standardized test. The mean score is 1000 and the standard deviation is 200. What is the z-score of 1600 point score?
    A.2.5
    B.3.0
    C.1.5
    D.0.5
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  2. #2
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    Sacramento, CA
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    The z-score is defined as follows:

    z = \frac{x - \mu}{\sigma}

    Where x is the observed mean, \mu is the population mean, and \sigma is the standard deviation. The z number converts the given values into standard units based on the standard deviation. Consider the first question. What is our z-score? The difference of the observed is 8 greater than the mean (= 48 - 40). Thus, the observed mean is one standard deviation from the mean. One thing you should know is that 68% of the population is within one standard deviation of the mean. This means that 100% less 68% (= 32%) of the observations fall outside of one standard deviation from the mean. But we're not concerned with just all observations. We're concerned with those on the positive side: i.e., for those observations \geq 48. Since the normal distribution is symmetric, there is an equal number on both sides of one standard deviation from the mean. Therefore, we only need to consider half. Therefore, 16 (= 32/2) percent of the observations lie positively one standard deviation from the mean. In other words, you have a 16% chance of being observed \geq 48 given this normal distribution.

    Did you understand that? The second question is just a calculation. Interpret it correctly: how many standard deviations is 1600 from the mean of 1000 when \sigma = 200?
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