Finding Probabilities from Normal Distributions (College Board AP® Statistics): Study Guide

Finding probabilities from normal distributions

What is the standard normal table?

• The proportion of a distribution of a variable $X$ within a particular interval can be determined using the standard normal table given to you in the exam

• The standard normal table shows the proportion of the distribution (or percentage of values) that lie below a particular $z$-score on a standard normal distribution

  • A $z$-score is the number of standard deviations that a value is from the mean

• A $z$-score on the table is made up from

  • the row that corresponds to the correct units and tenths of the $z$-score

  • and the column that corresponds to the correct hundredths of the $z$-score

    • e.g. a $z$-score of -1.32 would use the row -1.3 and the column 0.02

• You can also calculate proportions using your calculator

How do I find a probability that a normal variable is below a given value using a standard normal table?

• The proportion of a distribution of a variable $X$ that lies below a given value is the area under the curve to the left of the given value

  • $\mathrm{P}\left(X<a\right)=p$

• To find this proportion using the standard normal table:

  • find the $z$-score for the value $a$

  • use this $z$-score to locate the appropriate row and column in the standard normal table

  • identify the cell that corresponds to this row and column

    • the value in this cell is $\mathrm{P}\left(Z<z\right)$, the proportion of the distribution that lies below the $z$-score

    • this is equal to $\mathrm{P}\left(X<a\right)$, the proportion of the distribution that lies below $a$

How do I find a probability that a normal variable is below a given value using a calculator?

• You need a calculator that can calculate cumulative normal probabilities

• You want to use the Normal Cumulative Distribution function

  • This is sometimes shortened to NCD, Normal CD or Normal Cdf

• When using a calculator to find a proportion you can use the given values

  • You do not need to enter the associated $z$-scores first

    • However you should write down the $z$-score in the exam to support your work

• To find $\mathrm{P}\left(X<a\right)=p$, you will need to enter:

  • The lower bound

    • this needs to be a value that is sufficiently smaller than the mean

    • an easy option is to input lots of 9's for the lower bound with a negative sign (-99999999... or -10⁹⁹)

  • The upper bound

    • this will be the value $a$

  • The mean, $\mu$

  • The standard deviation, $\sigma$

How do I find a probability that a normal variable is above a given value using a standard normal table?

• The proportion of a distribution of a variable $X$ that lies above a given value is the area under the curve to the right of the given value

  • $\mathrm{P}\left(X>b\right)=p$

• The total area under the curve is 1

  • So $\mathrm{P}\left(X>b\right)=1-\mathrm{P}\left(X<b\right)$

• To find this proportion using the standard normal table:

  • find the $z$-score for the value $b$

  • use this $z$-score to locate the appropriate row and column in the standard normal table

  • identify the cell that corresponds to this row and column

  • subtract the value in this cell from 1

    • the result is $\mathrm{P}\left(Z>z\right)$, the proportion of the distribution that lies above the $z$-score

    • this is equal to $\mathrm{P}\left(X>b\right)$, the proportion of the distribution that lies above $b$

How do I find a probability that a normal variable is above a given value using a calculator?

• You will still use the Normal Cumulative Distribution function

• To find $\mathrm{P}\left(X>b\right)=p$, you will need to enter:

  • The lower bound

    • this will be the value $b$

  • The upper bound

    • this needs to be a value that is sufficiently bigger than the mean

    • an easy option is to input lots of 9's for the upper bound with a positive sign (99999999... or 10⁹⁹)

  • The mean, $\mu$

  • The standard deviation, $\sigma$

How do I find a probability that a normal variable is within an interval using a standard normal table?

• The proportion of a distribution that lies in a given interval is the area under the curve between the lower bound value, $a$, and the upper bound value, $b$

  • $\mathrm{P}\left(a<X<b\right)=p$

• To find this proportion using the standard normal table:

  • find the $z$-score for both the value $a$ and the value $b$

  • use these $z$-scores in the standard normal table to find the proportion of the distribution that lies below each value

  • subtract the proportions of the two $z$-scores

    • $\mathrm{P}\left(Z<z_b\right)-\mathrm{P}\left(Z<z_a\right)$

    • This is equal to $\mathrm{P}\left(X<b\right)-\mathrm{P}\left(X<a\right)$,the proportion of the distribution that lies between $a$ and $b$

How do I find a probability that a normal variable is within an interval using a calculator?

• You will still use the Normal Cumulative Distribution function

• To find $\mathrm{P}\left(a<X<b\right)=p$, you will need to enter:

  • The lower bound

    • this will be the value $a$

  • The upper bound

    • this will be the value $b$

  • The mean, $\mu$

  • The standard deviation, $\sigma$