Transcript Chapter 6, Normal Distribution

Ch. 6 The Normal Distribution
• A continuous random variable is a variable that
can assume any value on a continuum (can assume
an uncountable number of values)
–
–
–
–
thickness of an item
time required to complete a task
temperature of a solution
height, in inches
• These can potentially take on any value,
depending only on the ability to measure
accurately.
• The polygon of the distribution become a
smooth curve called Probability Density
Function (this is equivalent to Probability
Distribution for discrete random variable)
f(X)
P (a ≤ X ≤ b )
=P ( a < X < b )
a
b
X
(Note that the probability of any individual value is zero)
The Normal Distribution
•
Bell Shaped
•
Perfectly Symmetrical
•
Mean, Median and Mode
are Equal
f(X)
• Location on the value axis
is determined by the mean, μ
Spread is determined by the
standard deviation, σ
• The random variable has an
infinite theoretical range:
+  to  
σ

μ
Mean
= Median
= Mode
+
What can we say about the distribution of values around the mean? There are
some general rules:
• Probability is measured by the area under the curve
• The total area under the curve is 1.0, and the curve is symmetric, so half is
above the mean, half is below
P(    X  μ)  0.5
P(μ  X  )  0.5
• The height of the curve at a certain value below the mean is equal to the
height of the curve at the same value above the mean
• The area between:
μ ± 1σ encloses about 68% of X’s
μ ± 2σ covers about 95% of X’s
μ ± 3σ covers about 99.7% of X’s
The Normal Distribution Density Function:
• The formula for the normal probability density function is:
(the curve is generated by):
1
f(X) 
e
2π
1  (X μ) 
 

2  
2
Where e = the mathematical constant approximated by 2.71828
π = the mathematical constant approximated by 3.14159
μ = the population mean
σ = the population standard deviation
X = any value of the continuous variable
Note: f (X) is the same concept as P(X).
• By varying the parameters μ and σ, we obtain different
normal distributions. That is to say that any particular
combination of μ and σ, will produce a different normal
distribution. This means that we have to deal with
numerous distribution tables.
Changing μ shifts the distribution left or right.
Changing σ increases or decreases the spread.
Standard Normal Distribution
• Any normal distribution (with any mean and standard deviation
combination) can be transformed into the standardized normal
distribution (Z).
• To do so, we need to transform all X units (values) into Z units
(values)
• Then the resulting standard normal distribution has a mean=zero and a
standard deviation =1
• X-Values above the mean of X will have positive Z-value and Xvalues below the mean of will have negative Z-values
• The transformation equation is:
Xμ
Z 
σ
• EXAMPLE:
•
A sample of 19 apartment’s electricity bill in December is distributed normally
with mean of $65 and std. dev. Of $9. Z- values for X = $83 is: Z= (8365)/9=18/9=2. That is to say that X=83 is 2 standard deviation above the
mean or $65. Note that the distributions are the same, only the scale has
changed.
47
-2
56
-1
65
0
74
1
83
2
X
Z
(μ = 65, σ =9)
(μ = 0, σ =1)
• Applications: Use of Standard Normal
Distribution Table, Pages 737-738
• What is the probability that a randomly selected bill is
between $56 and $74?
• What is the value of the lower 10% of the
bills? (finding X value for a known probability)
• What is the value of the highest 10% of the
bills? (finding X value for a known probability)
• What is the range of the values that contain
the middle 95% of the bills?
Assessing Normality
• Not all continuous random variables are normally distributed
• It is important to evaluate how well the data set is approximated by a
normal distribution
• Construct charts or graphs
– For small- or moderate-sized data sets, do stem-and-leaf display
and box-and-whisker plot look symmetric?
– For large data sets, does the histogram or polygon appear bellshaped?
• Compute descriptive summary measures
– Do the mean, median and mode have similar values?
– Is the interquartile range approximately 1.33 σ?
– Is the range approximately 6 σ?
• Observe the distribution of the data set within certain std. Dev.
• Evaluate normal probability plot
– Is the normal probability plot approximately linear with positive
slope? Page 209