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IBMathsDPAnalysis & Approaches (AA)SLFlashcards4. Statistics & ProbabilityNormal Distribution

Normal Distribution (DP IB Analysis & Approaches (AA))

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  • What is a continuous random variable?

    A continuous random variable is a random variable that can take any value within a range of values.

  • True or False?

    The probability of a continuous random variable being exactly equal to a specific value is always zero.

    True.

    The probability of a continuous random variable being exactly equal to a specific value is always zero.

  • What does the notation X tilde straight N open parentheses mu comma space sigma squared close parentheses mean?

    X tilde straight N open parentheses mu comma space sigma squared close parentheses means that the random variable X follows a normal distribution with mean mu and variance sigma squared.

  • True or False?

    The random variable X tilde straight N open parentheses 330 comma space 16 close parentheses has a standard deviation of 16.

    False.

    The random variable X tilde straight N open parentheses 330 comma space 16 close parentheses has a standard deviation of 4.

    The 16 is the variance of the distribution, sigma squared. To find the standard deviation, sigma, you have to take the square root: square root of 16 equals 4.

  • What is the shape of the graph for a normal distribution?

    The graph for a normal distribution is bell-shaped.

    A typical bell-shaped graph for a normal distribution.

  • True or False?

    Increasing the variance of a normal distribution makes its graph taller and narrower.

    False.

    Increasing the variance of a normal distribution makes its graph wider and shorter.

  • What percentage of data lies within one standard deviation of the mean in a normal distribution?

    Approximately 68% of the data lies within one standard deviation of the mean in a normal distribution.

  • What interval contains approximately 95% of the data in a normal distribution?

    The interval mu plus-or-minus 2 sigma contains approximately 95% of the data in a normal distribution.

    I.e., approximately 95% of the data is within two standard deviations of the mean.

  • True or False?

    Nearly all the data lies within three standard deviations of the mean in a normal distribution.

    True.

    Nearly all of the data (99.7%) lies within three standard deviations of the mean in a normal distribution.

  • True or False?

    The normal distribution is asymmetrical.

    False.

    The normal distribution is symmetrical about its mean.

  • What is the relationship between mean, median, and mode in a normal distribution?

    In a normal distribution, the mean, median, and mode are all equal.

  • True or False?

    straight P open parentheses a less than X less than b close parentheses is equal to straight P open parentheses a less or equal than X less or equal than b close parentheses in a normal distribution.

    True.

    straight P open parentheses a less than X less than b close parentheses is equal to straight P open parentheses a less or equal than X less or equal than b close parentheses in a normal distribution.

    For continuous probability distributions, strict inequalities (less than comma space greater than) and the equivalent weak inequalities (less or equal than comma space greater or equal than) are interchangeable in probabilities.

  • If X is a normal variable, what is straight P open parentheses X equals 3 close parentheses equal to?

    If X is a normal variable, straight P open parentheses X equals 3 close parentheses equals 0.

    The probability for any single value in a normal distribution is always zero.

  • If X is a normal variable, what four pieces of information would you need to enter into your calculator to find a probability of the form straight P open parentheses a less or equal than X less or equal than b close parentheses?

    To find a probability of the form straight P open parentheses a less or equal than X less or equal than b close parentheses in your calculator, you would need to enter:

    • the value of mu (the mean for the distribution)

    • the value of sigma (the standard deviation for the distribution)

    • the value of a

    • the value of b

    Remember to use the standard deviation (sigma) and not the variance (sigma squared)!

  • If X is a normal random variable, what upper bound should you use to calculate straight P open parentheses X greater than a close parentheses using a calculator?

    If X is a normal random variable, then to calculate straight P open parentheses X greater than a close parentheses using a calculator you should select a very large number as the upper bound.

    E.g. 9999999999 or 1 cross times 10 to the power of 99.

    Make sure that the upper bound you use is at least 4 standard deviations above the mean.

  • If X is a normal random variable, what lower bound should you use to calculate straight P open parentheses X less than a close parentheses using a calculator?

    If X is a normal random variable, then to calculate straight P open parentheses X less than a close parentheses using a calculator you should select a very large negative number as the lower bound.

    E.g. negative 9999999999 or negative 1 cross times 10 to the power of 99.

    Make sure that the lower bound you use is at least 4 standard deviations below the mean.

  • True or False?

    straight P open parentheses X less than mu close parentheses is always equal to 0.25 in a normal distribution.

    False.

    straight P open parentheses X less than mu close parentheses is always equal to 0.5 in a normal distribution.

    For a normal distribution, straight P open parentheses X less than mu close parentheses equals straight P open parentheses X less or equal than mu close parentheses equals straight P open parentheses X greater than mu close parentheses equals straight P open parentheses X greater or equal than mu close parentheses equals 0.5

  • For a normal distribution, state the equation for straight P open parentheses X greater than a close parentheses in terms of straight P open parentheses X less than a close parentheses.

    For a normal distribution straight P open parentheses X greater than a close parentheses equals 1 minus straight P open parentheses X less than a close parentheses

  • What is the inverse normal distribution function on a GDC used for?

    The inverse normal distribution function on a GDC is used to find the bound that gives a particular probability.

    E.g. the value of a that gives straight P open parentheses X less than a close parentheses equals 0.65 or straight P open parentheses X greater than a close parentheses equals 0.1.

  • What is a quick way to check if your answer makes sense when using the inverse normal distribution function on a GDC?

    To check if your answer makes sense when using the inverse normal distribution function on a GDC, verify that

    • if straight P open parentheses X less than a close parentheses is less than 0.5, then a is smaller than the mean

    • if straight P open parentheses X less than a close parentheses is more than 0.5, then a is larger than the mean

    • if straight P open parentheses X greater than a close parentheses is less than 0.5, then a is larger than the mean

    • if straight P open parentheses X greater than a close parentheses is more than 0.5, then a is smaller than the mean

  • What is the standard normal distribution?

    The standard normal distribution is a normal distribution where the mean is 0 and the standard deviation is 1.

  • What is Z tilde straight N open parentheses 0 comma space 1 squared close parentheses the notation for?

    Z tilde straight N open parentheses 0 comma space 1 squared close parentheses is the notation for the standard normal distribution.

  • True or False?

    Any normal distribution curve can be transformed to the standard normal distribution curve.

    True.

    Any normal distribution curve can be transformed to the standard normal distribution curve.

  • What is a z-value?

    A z-value is a standardised value that tells you how many standard deviations a value is away from the mean in a normal distribution.

  • True or False?

    A negative z-value indicates that the x-value is greater than the mean.

    False.

    A negative z-value indicates that the x-value is less than the mean.

  • State the formula for calculating a z-value from an x-value, in terms of the mean, mu and standard deviation, sigma.

    If X tilde straight N open parentheses mu comma space sigma squared close parentheses and Z tilde straight N open parentheses 0 comma space 1 squared close parentheses, then the formula for calculating a z-value from an x-value is z equals fraction numerator x minus mu over denominator sigma end fraction.

    Where:

    • muis the mean of the distribution of the variable X

    • straight sigma is the standard deviation of the distribution of the variable X

    This formula is in the exam formula booklet.

  • Why are z-values useful?

    z-values are useful because they allow us to compare values from different normal distributions.

  • What is a good first step in finding an unknown parameter (mean, mu, or standard deviation, sigma) in a normal distribution?

    A good first step in finding an unknown parameter (mean, mu, or standard deviation, sigma) in a normal distribution is to sketch the normal curve.

  • True or False?

    The inverse normal distribution function can be used to find the z-value for a given probability.

    True.

    The inverse normal distribution function can be used to find the z-value for a given probability.

  • How many probabilities are needed to find the mean, bold italic mu and standard deviation, bold italic sigma when both parameters are unknown?

    Two probabilities are needed to find the mean, bold italic mu and standard deviation, bold italic sigma when both parameters are unknown.

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