Download 11.3 Notes - Normal Probability Distributions

Name: __________________________________________ Per _____ Normal Probability Distribution
Assignment: Section 11.3 #1-5, 7, 8, 10
Many continuous variables follow a distribution that is symmetric and mounded in a “bell” shape.
Graphs of distributions for large populations, such as heights, clothing sizes, and test scores often have
this shape. The bell-shaped distribution is so common that it is called the NORMAL DISTRIBUTION and
its graph is called the NORMAL CURVE.
RECALL:
The formula you’ve learned for the SAMPLE MEAN and SAMPLE STANDARD DEVIATION are both
statistics because they are numbers that describe a sample. We use these two statistics as estimates
for the values in the population. Write the formula for both of these statistics if the sample size is n=3.
Sample Mean
𝑥̅ =____________________________________
Sample Standard Deviation
s = ____________________________________
If you find the mean and standard deviation for an entire population, they are called population
parameters. We use Greek Letters to distinguish between population parameter and sample statistics.
Population Mean is pronounced “mew” and written with the Greek Letter, 
Population Standard deviation is pronounced “sigma” and is written with the Greek letter, .
How do to graph a Normal Probability Distribution?
You will explore how changing either the mean or the standard deviation (or both) affects the shape of
the “bell-curve”.
The parent graph of the Normal Probability Distribution is called the STANDARD NORMAL Distribution
and has a mean equal to =0 and standard deviation equal to =1.
1
y
 2
Mean = 
 e
 x 


  
2
Standard deviation = 
The Standard Normal has   0 and   1
 x2
1
normPdf ( x, 0,1)
y
e
2
 
Two other Normal Distributions:
y
1
0.5 2
 e
 
 x2


 .5 


 x
normPdf ( x, 0, 0.5)
2
 
1
normPdf ( x, 0, 2)
y
e 2
2 2
Which standard deviation creates a graph that is MORE spread out? __________
Which standard deviation creates a graph that is LESS spread out?
__________
If you want to graph a Normal Distribution with a different mean value (keeping =1) you shift the
graph right or left by the mean value.
1
y
 2
Mean = 
 e
 x 


  
2
Standard deviation = 
The Standard Normal has   0 and   1
 x2
1
normPdf ( x, 0,1)
y
e
2
 
Two other Normal Distributions:
 ( x  2)2
1
normPdf ( x, 2,1)
y
e
2
 ( x  2) 2
1
normPdf ( x, 2,1)
y
e
2
When the standard normal graph’s mean was changed to +2, the graph shifted ______________.
 
 
When the standard normal graph’s mean was changed to -2, the graph shifted ______________.
1
y
 2
y
y
 e
1
0.5 2
1
0.5 2
1
y
3 2
 x 


  
Mean = 
2
normPdf ( x,  ,  )
2
 e
 x 3 


 0.5 
 e
 x4 


 0.5 
 e
 x 1 


 3 
Standard
Deviation= 
normPdf ( x,3, 0.5)
2
normPdf ( x, 4, 0.5)
2
normPdf ( x,1,3)
Compare the graphs. You should notice that all
Normal Distributions are “Bell-Shaped”.
Changing the mean only changes the LOCATION of the
graph (shifted right or shifted left).
Changing the standard deviation changes the SHAPE of
the “bell”
(taller and skinner if the standard deviation is smaller
and shorter and fatter if the standard deviation is
larger).
Write down what you have noticed about the graphs of Normal Distributions.
Matches
(a) How does changing the mean affect the graph?
(b) How does changing the standard deviation affect the graph?
Calculating Probabilities using the Normal Probability Distribution (aka, The Bell-Curve)
In Sections 11-2 we were able to calculate probabilities using the area under the probability
distribution curves (see examples 2 & 3 from yesterday’s notes). Recall that we used formulas from
geometry to find the areas we needed. Since there are no formulas from geometry that we can use to
calculate the area under the curve for the Bell-Curve, we use the 68-95-99.7% Rule
68% of the data will fall within ______ standard
deviation away from the mean.
Demonstrate on the calculator how to find these and
other area under the curve.
95% of the data will fall within ______ standard
deviations away from the mean.
99.7% will fall within ______ standard deviations
away from the mean.
Example #1: From the equation of the Normal Distribution, estimate the mean and standard deviation.
(a) 𝑦 =
1
3√2𝜋
−(
(√𝑒)
(𝑥−10) 2
)
3
(b) 𝑦 =
0.4
15
+(
(0.60653)
(𝑥−100) 2
)
15
Example #2: Use the graph to estimate the mean and
standard deviation.
Example #3: Sketch these on the back of this paper since there is no room to show work here. Assume
the mean height of an adult male gorilla is 6.7 feet with a standard deviation of 0.6 feet.
(a) Sketch the graph of the normal distribution of gorilla heights.
(b) Give the range of heights for 65% of the gorillas
(c) Give the range of heights for 95% of the gorillas.
(d) Give the range of heights for 99.7% of the gorillas.