Transcript Normal Distributions and Z

Normal Distributions
and Z-Scores
Accelerated Math 3
Normal Distributions
Last year we learned about normal distributions and the Empirical
Rule.
It would still be a good idea to have these
probabilities memorized!
Remember 68% – 95% – 99.7%
Normal Distributions
• The Empirical Rule tells us the probability that a certain range
of data falls within one, two, or three standard deviations of
the mean.
•
is the mean and is the standard deviation of the data set.
x

( xi  x )

n
2
Ex. 1: Word Problem
• The data for the SAT is normally distributed with a mean of 1000
and standard deviation 180. What is the probability that a
randomly selected student scored between a 640 and 1180?
Normal Distributions and Z-Scores
• What happens when a value doesn’t quite fit in exactly one,
two or three standard deviations?
• We can use z-scores and z-tables!
• Z-scores tell us exactly how many standard deviations away a
value is from the mean and the z-table gives us the
probability a value is below that amount.
Z-Scores
• We can find Z-Scores by using the formula:
z
xx

Example 2:
• Looking back at our SAT example, lets find the probability that a
student scores under 1200.
Example 3:
• What if I wanted to know the percent of test takers who score
above a 1200?
Example 4:
• What if I wanted to know the z-score for a score between 900 and
1300?
Ex 5: Working Backwards
• You run a Tootsie Pop factory and want to make a guarantee that
your Tootsie Pops will last a certain number of licks. If your mean is
964 licks with a standard deviation of 51 licks, how many licks
should you guarantee your Tootsie Pops will last so that 90% last
that long?