Download 24 C Standard Normal Distributions

Name: _______________________
Class: __________
Date:_____________
Math SL: 24C Z-Distribution
Review:
1. Given that sin B =
2
π
and
 B  , find cos B and cos 2B
3
2
2. From January to September, the mean number of car accidents per month was 630. From
October to December, the mean was 810 accidents per month.
What was the mean number of car accidents per month for the whole year?
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24 C Standard Normal Distributions
Today’s Objectives:
(1) to understand how the standard normal distribution can be found by
transforming any normal distribution
(2) to find probabilities for a standard normal distribution
Standard Normal Distribution
A standard normal distribution (Z-distribution) has m = 0 and s = 1. Thus we can
write
. The random variable Z is equivalent to the number of standard
deviations from the mean.
Finding Probabilities using a Graphics Calculator
To find P(Z £ a) or P(Z < a) use normalcdf(-E99,upper bound)
To find P(Z ³ a) or P(Z > a) use normalcdf(lower bound,E99)
To find P(a £ Z £ b) or P(a < Z < b) use normalcdf(lower bound,upper
bound)
Example 1: Given that
find
a) P(-2 < Z < 1)
b) P(Z < 1)
c)
P(Z > -1.5)
d) P(Z < 0)
Standardizing Any Normal Distribution
Every normal X-distribution can be transformed into the standard normal
distribution or Z-distribution using the transformation:
z=
x-m
s
If x is an observation from a normal distribution with mean m and standard
deviation s , the Z-score of x is the number of standard deviations x is from the
mean.
For example:
 If z1 = 1.34 then z1 is 1.34 standard deviations to the right of the mean
 If z2 = -1.92 then z2 is 1.92 standard deviations to the left of the mean
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A z-score can also be used to compare x-scores from two different normal
distributions.
Steps for converting an x value into a z-score:
(1) Convert x-values to z-values using z =
x-m
s
(2) Sketch the standard normal curve and shade the required region.
(3) Use your calculator to find the probability.
Example 2: A random variable X is normally distributed with mean 70 and standard
deviation 4. By converting to the standard variable Z and then using your calculator,
find P( X £ 68) .
Example 3: The length L of a nail is normally distributed with mean 50.2 mm and
standard deviation 0.93 mm. Find Z to determine P(L ³ 48).
**Note** Your curriculum no longer requires the tables that are found in your text
book. All normal distribution work from todays lesson must be done on calculator or
the calculations must result in integer multiples of the standard deviation. i.e. 68%,
95%, 99.7%.
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Hmwk#56:
24C The standard normal distribution
pg. 657 # 1 – 3, 5, 6
pg. 658 # 1 – 3(a,c)
USE GRAPHING CALCULATOR USE GRAPHING CALCULATOR and always draw a picture
where possible.
By now, most of you should have an idea of what you want
to explore. Take this opportunity to look for relevant
math… You do NOT want to start your exploration from
scratch next year during college research, college
applications, IB papers, SATs, IGCSEs, extracurricular
activities, regular SAS school demands, etc…
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