A man would have to be at least 73.84′′ tall to be in the top 10% of all men.
(ii) Using the calculator, the z -score corresponding to an area of 90% to the left of x is given by
invNorm(0.90) = 1.28. Otherwise, the solution is the same as is given in part (i). See the
following Calculator Tip for a full explanation of the invNorm function.Calculator Tip: invNorm essentially reverses normalcdf . That is, rather than reading from the
margins in, it reads from the table out (as in the example above). invNorm(A) returns the z -score that
corresponds to an area equal to A lying to the left of z . invNorm(A, μ,σ ) returns the value of x that
has area A to the left of x if x has N (μ,σ ).Chebyshev’s Rule (Optional–not part of the AP Curriculum)
The 68-95-99.7 rule works fine as long as the distribution is approximately normal. But what do you do if
the shape of the distribution is unknown or distinctly nonnormal (as, say, skewed strongly to the right)?
Remember that the 68-95-99.7 rule told you that, in a normal distribution, approximately 68% of the data
are within one standard deviation of the mean, approximately 95% are within two standard deviations,
and approximately 99.7% are within three standard deviations. Chebyshev’s rule isn’t as strong as the
empirical rule, but it does provide information about the percent of terms contained in an interval about
the mean for any distribution.
Let k be a number of standard deviations. Then, according to Chebyshev’s rule, for k > 1, at least
of the data lie within k standard deviations of the mean. For example, if k = 2.5, then
Chebyshev’s rule says that at least of the data lie with 2.5 standard deviations of the
mean. If k = 3, note the difference between the 68-95-99.7 rule and Chebyshev’s rule. The 68-95-99.7
rule says that approximately 99.7% of the data are within three standard deviations of . Chebyshev’s
says that at least of the data are within three standard deviations of . This also
illustrates what was said in the previous paragraph about the empirical rule being stronger than
Chebyshev’s. Note that, if at least of the data are within k standard deviations of , it follows
(algebraically) that at most lie more than k standard deviations from .
Knowledge of Chebyshev’s rule is not required in the AP Exam, but its use is certainly okay and is
common enough that it will be recognized by AP readers.
Rapid Review
Describe the shape of the histogram below: