Transcript Normal Distributions

Normal Distributions
MM2D1d Compare the means and standard deviations
of random samples with the corresponding population
parameters, including those population parameters for
normal distributions. Observe that the different
sample means vary from one sample to the next.
Observe that the distribution of the sample means
has less variability than the population distribution.
Normal Distributions
Normal Distribution – modeled by a bellshaped curve called a normal curve.
The total area under the curve is 1.
Normal Distributions
Another breakdown
Normal Distributions
Empirical Rule-only applies for normal
distributions and applies to the normal bell
curve.
68% of the data will be located within one
____
standard deviation symmetric to the mean.
95%
____
of the data will be located within two
standard deviations ….
99.7%
____of the data will be located within three
standard deviations….
Normal Distributions
Find a normal probability
For these samples REMEMBER
1 standard deviation is 68% which means
34% for -1 and 34% for +1.
2 standard deviations is 95% which means
47.5% for -2 and 47.5% for +2.
3 standard deviations is 99.7% which means
49.85% for -3 and 49.85% for +3.
Normal Distributions
Construct a bell curve
1. Find the mean of the data.
2. Find the standard deviation of the
data.
3. The mean goes in the middle of the
curve.
4. The standard deviation is added or
subtracted for each interval.
Normal Distribution
Bell curves
Test scores
86, 88, 90, 99, 70
Mean 86.6
Standard deviation 9.41
The bottom of the curve will be labeled
58.37 67.78 77.19 86.6 96.01 105.42 114.83
x-3σ
x-2σ x-1σ
x
x+1σ x+2σ x+3σ
Normal Distributions
Normal Probability
Mean +2 standard deviations mean how
much percent?
47.5%
What is that in decimal form?
.475
Normal Distributions
Normal Probability
Try one:
84%
Normal Distributions
Interpret Normally Distributed Data
The heights (in feet) of fully grown white oak
trees are normally distributed with a mean of
90 ft and a standard deviation of 3.5 feet.
About what percent of white oak trees have
heights less than 97 feet?
97.5%
How about between 83 feet and 90 feet?
47.5%
Normal Distributions
Using z-scores and the standard normal table
Standard normal distribution-mean 0 and standard
deviation 1.
Formula that can turn x values from
normal distributions into z values
Z
X 

Z values for a particular x value is called the zscore (the number of standard deviations the xvalue lies above or below the mean.)
Normal Distributions
Using z-scores and the standard normal table
Find the probability that a randomly
selected white oak tree has a height of
at most 94 feet. (remember 90 is the
mean from a previous problem.
Find the z-score corresponding to the xvalue of 94 feet. (Page 265)
Use the formula
X 
Z
94-90/3.5=1.1

Normal Distributions
Using z-scores and the standard normal table
Try one:
Find the probability that a randomly
selected white oak tree has a height
of at most 85 feet. Use the table on
page 296 in your book.
About 0.0808
Normal Distributions
Page 267 (1-20)
Normal Distributions
Exploration
Materials:
Laptops
Instructions:
Go to at least three of the websites listed. Take notes on the
presentations. Make sure your notes include at least three
examples of problems worked out.
Write a review of the websites by ranking them from 1 to 3
with 3 being the most helpful. Tell why you ranked them as
you did.
Find two more web based resources on the lesson and include
them on your sheet to turn in.
Normal Distribution
Web search
• http://mathforum.org/library/drmath/view/5760
8.html
• http://www.gifted.uconn.edu/siegle/research/Nor
mal/Normal%20DistributionShow.pps
• http://cda.morris.umn.edu/~benw/ppt/1.3.normal.
ppt
• http://gunston.gmu.edu/healthscience/597/Norm
alDistribution4.PPT
• http://www.d.umn.edu/~rregal1/documents/stat2
411_sp07/ch9.ppt