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Section 8.6: Normal Random
Variables
Understanding this section will be very
important in several ways in this course!
Standardization
• If the distribution of X has mean m and
standard deviation s, then letting
Z=(X-m)/s is called standardization.
• After standardization, the distribution of Z
has mean 0 and standard deviation 1.
• The standardized version of X is sometimes
called a z-score.
Why Standardize?
• If X~N(m,s), then Z~N(0,1).
• Everything we need to know about the
N(0,1) distribution is contained in Table A.1
on pages 538-539!
Using Table A.1 to find
proportions
• Example: If the heights of adult females are
normally distributed with mean 65.5 inches
and standard deviation 2.5 inches,
– What proportion of women are more than 70
inches tall?
– What proportion are between 66 and 68 inches
tall?
Using Table A.1 to find
percentiles
• Example: If the heights of adult females are
normally distributed with mean 65.5 inches
and standard deviation 2.5 inches,
– What is the 75th percentile of women’s heights?
– What is the 5th percentile?
Using Table A.1 to find a
confidence interval multiplier
• For a normal distribution, 95% of the
distribution is found within ______ standard
deviations on either side of the mean.
• 99% of the distribution is found within
_____ standard deviations on either side of
the mean.
• 90% is found within _____ standard
deviations.