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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

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ch 6 the 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.
slide2
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

X

a

b

(Note that the probability of any individual value is zero)

the normal distribution
The Normal Distribution
  • Bell Shaped
  • Perfectly Symmetrical
  • Mean, Median and Mode

are Equal

  • 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  

f(X)

σ

+

μ

 

Mean

= Median

= Mode

slide4
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
  • 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 Normal Distribution Density Function:
  • The formula for the normal probability density function is: (the curve is generated by):

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).

slide6
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.

Changingμ shifts the distribution left or right.

Changing σ increases or decreases the spread.

standard normal distribution
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 X-values below the mean of will have negative Z-values
  • The transformation equation is:
slide8
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 = $83is: Z= (83-65)/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.

X

(μ = 65, σ =9)

47

56

65

74

83

Z

(μ = 0, σ =1)

-2

-1

0

1

2

slide9
Applications: Use of Standard Normal Distribution Table, Pages 737-738
  • What is the probability that a randomly selected bill is between $56 and $74?
assessing normality
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 bell-shaped?
  • 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