Title: Chapter 6 ~ Normal Probability Distributions
1Chapter 6 Normal Probability Distributions
2Chapter Goals
 Learn about the normal, bellshaped, or Gaussian
distribution
 How probabilities are found
 How probabilities are represented
 How normal distributions are used in the real
world
36.1 Normal Probability Distributions
 The normal probability distribution is the most
important distribution in all of statistics
 Many continuous random variables have normal or
approximately normal distributions
 Need to learn how to describe a normal
probability distribution
4Normal Probability Distribution
 1. A continuous random variable
2. Description involves two functions a. A
function to determine the ordinates of the graph
picturing the distribution b. A function to
determine probabilities
4. The probability that x lies in some interval
is the area under the curve
5The Normal Probability Distribution
6Probabilities for a Normal Distribution
7Notes
 The definite integral is a calculus topic
 We will use the TI83/84 to find probabilities for
normal distributions
 We will learn how to compute probabilities for
one special normal distribution the standard
normal distribution
 We will learn to transform all other normal
probability questions to this special distribution
 Recall the empirical rule the percentages that
lie within certain intervals about the mean come
from the normal probability distribution
 We need to refine the empirical rule to be able
to find the percentage that lies between any two
numbers
8Percentage, Proportion Probability
 Basically the same concepts
 Percentage (30) is usually used when talking
about a proportion (3/10) of a population
 Probability is usually used when talking about
the chance that the next individual item will
possess a certain property
 Area is the graphic representation of all three
when we draw a picture to illustrate the situation
96.2 The Standard Normal Distribution
 There are infinitely many normal probability
distributions
 They are all related to the standard normal
distribution
 The standard normal distribution is thenormal
distribution of the standard variable z(the
zscore)
10Standard Normal Distribution
 Properties
 The total area under the normal curve is equal to
1  The distribution is mounded and symmetric it
extends indefinitely in both directions,
approaching but never touching the horizontal
axis  The distribution has a mean of 0 and a standard
deviation of 1  The mean divides the area in half, 0.50 on each
side  Nearly all the area is between z 3.00 and z
3.00
 Notes
 Table 3, Appendix B lists the probabilities
associated with the intervals from the mean (0)
to a specific value of z  Probabilities of other intervals are found using
the table entries, addition, subtraction, and
the properties above
11Table 3, Appendix B Entries
 The table contains the area under the standard
normal curve between 0 and a specific value of z
12Example
 Example Find the area under the standard normal
curve between z 0 and z 1.45
13Using the TI 83/84
 To find the area between 0 and 1.45, do the
following  2nd DISTR 2 which is normalcdf(
 Enter the lower bound of 0
 Enter a comma
 Then enter 1.45
 Close the parentheses if you like or hit Enter
 The value of .426 is shown as the answer!
 Interpretation of the result The probability
that Z lies between 0 and 1.45 is 0.426
14Example
 Example Find the area under the normal curve to
the right of z 1.45 P(z gt 1.45)
15Using the TI 83/84
 To find the area between 1.45 and 8, do the
following  2nd DISTR 2 which is normalcdf(
 Enter the lower bound of 1.45
 Enter a comma
 Then enter 1 2nd EE 99
 Close the parentheses if you like or hit Enter
 The value of .074 is shown as the answer!
 Interpretation of result The probability that Z
is greater than 1.45 is 0.074
16Example
 Example Find the area to the left of z 1.45
P(z lt 1.45)
17Using The TI 83/84
 To find the area between  8 and 1.45, do the
following  2nd DISTR 2 which is normalcdf(
 Enter the lower bound of 1 2nd EE 99
 Enter a comma
 Then enter 1.45
 Close the parentheses if you like or hit Enter
 The value of 0.926 is shown as the answer!
 Interpretation of result The probability that Z
is less than 1.45 is 0.926
18Notes
 The addition and subtraction used in the previous
examples are correct because the areas
represent mutually exclusive events
 The symmetry of the normal distribution is a key
factor in determining probabilities associated
with values below (to the left of) the mean. For
example the area between the mean and z 1.37
is exactly the same as the area between the mean
and z 1.37.
 When finding normal distribution probabilities, a
sketch is always helpful
19Example
 Example Find the area between the mean (z 0)
and z 1.26
20Using the TI 83/84
 Find the area to the left of z 0.98
 Use 1E99 for  8 and enter 2nd DISTR
 Normalcdf (1e99, 0.98) which gives .164
21Example
 Example Find the area between z 2.30 and z
1.80
22Using the TI 83/84
 Find the area between z 2.30 and z 1.80
 Enter 2nd DISTR, normalcdf (2.3, 1.80) and press
enter  .953 is given as the answer.
 Remember, the function normalcdf is of the form
 Normalcdf(lower limit, upper limit, mean,
standard deviation) and if youre working with
distributions other than the standard normal
(recall mean 0, stddev 1), you must enter the
values for mean and standard deviation
23Normal Distribution Note
 The normal distribution table may also be used to
determine a zscore if we are given the area
(working backwards)
 Example What is the zscore associated with the
85th percentile?
24Using the TI 83/84
 There is another function in the DISTR list that
is used to find the value of z (or x) when the
probability is given. For the previous problem,
we are actually asking what is the value of z
such that 85 of the distribution lies below it.
25Using the TI 83/84
 Use 2nd DISTR invNorm( to calculate this value
 2nd DISTR invNorm(.85) ENTER gives us a value
of 1.036 which is shown
26Example
 Example What zscores bound the middle 90 of a
standard normal distribution?
27Using the TI 83/84
 The TI 83/84 calculates areas from 8 to the
value of z we are interested in. Therefore, we
must get a little creative to solve some
problems.  Using the idea that the total area equals one
comes in very handy here!  For the example given, where we are interested in
the value of z that bounds the middle 90, the
tails therefore represent a total of 10. Divide
this in two since it is symmetric and this gives
5 in each tail.
28Using the TI 83/84
 Now use the 2nd DISTR invNorm with .05 in the
argument like this  Which gives an answer of 1.645
 Since the distribution is symmetric, the upper
limit is 1.645, so 90 of the distribution lies
between  (1.645, 1.645)
29Using the TI 83/84
 Now lets work the problems on page 279
306.3 Applications of Normal Distributions
 Apply the techniques learned for the z
distribution to all normal distributions
 Start with a probability question in terms
ofxvalues
 Convert, or transform, the question into an
equivalent probability statement
involvingzvalues
31Standardization
 Suppose x is a normal random variable with mean m
and standard deviation s
32Example
 Example A bottling machine is adjusted to fill
bottles with a mean of 32.0 oz of soda and
standard deviation of 0.02. Assume the amount
of fill is normally distributed and a bottle is
selected at random
1) Find the probability the bottle contains
between 32.00 oz and 32.025 oz 2) Find the
probability the bottle contains more than 31.97 oz
33Solution Continued
34Example, Part 2
2)
35Notes
 The normal table may be used to answer many kinds
of questions involving a normal distribution
 Often we need to find a cutoff point a value of
x such that there is a certain probability in a
specified interval defined by x
 Example The waiting time x at a certain bank is
approximately normally distributed with a mean
of 3.7 minutes and a standard deviation of 1.4
minutes. The bank would like to claim that 95
of all customers are waited on by a teller
within c minutes. Find the value of c that
makes this statement true.
36Solution
37Example
 Example A radar unit is used to measure the
speed of automobiles on an expressway during
rushhour traffic. The speeds of individual
automobiles are normally distributed with a mean
of 62 mph. Find the standard deviation of all
speeds if 3 of the automobiles travel faster
than 72 mph.
38Solution

m
x
z
s
.
1
88
10
s
39Notation
 If x is a normal random variable with mean m and
standard deviation s, this is often denoted x
N(m, s)
 Example Suppose x is a normal random variable
with m 35 and s 6. A convenient notation to
identify this random variable is x N(35, 6).
406.4 Notation
 zscore used throughout statistics in a variety
of ways
 Need convenient notation to indicate the area
under the standard normal distribution
 z(a) is the algebraic name, for the zscore
(point on the z axis) such that there is a of the
area (probability) to the right of z(a)
41Illustrations
42Example
 Example Find the numerical value of z(0.10)
z(0.10) 1.28
43Example
 Example Find the numerical value of z(0.80)
 Use Table 3 look for an area as close as
possible to 0.3000  z(0.80) 0.84
44Notes
 The values of z that will be used regularly come
from one of the following situations
1. The zscore such that there is a specified
area in one tail of the normal distribution
2. The zscores that bound a specified middle
proportion of the normal distribution
45Example
 Example Find the numerical value of z(0.99)
 Because of the symmetrical nature of the normal
distribution, z(0.99) z(0.01)
46Example
 Example Find the zscores that bound the middle
0.99 of the normal distribution
476.5 Normal Approximation of the Binomial
 Recall the binomial distribution is a
probability distribution of the discrete random
variable x, the number of successes observed in n
repeated independent trials
 Binomial probabilities can be reasonably
estimated by using the normal probability
distribution
48Background Histogram
 Background Consider the distribution of the
binomial variable x when n 20 and p 0.5
The histogram may be approximated by a normal
curve
49Notes
 The normal curve has mean and standard deviation
from the binomial distribution
 Can approximate the area of the rectangles with
the area under the normal curve
 The approximation becomes more accurate as n
becomes larger
50Two Problems
 1. As p moves away from 0.5, the binomial
distribution is less symmetric, less
normallooking
Solution The normal distribution provides a
reasonable approximation to a binomial
probability distribution whenever the values of
np and n(1  p) both equal or exceed 5
2. The binomial distribution is discrete, and the
normal distribution is continuous
Solution Use the continuity correction factor.
Add or subtract 0.5 to account for the width of
each rectangle.
51Example
 Example Research indicates 40 of all students
entering a certain university withdraw from a
course during their first year. What is the
probability that fewer than 650 of this years
entering class of 1800 will withdraw from a
class?
52Solution
 Use the normal approximation method
53Random Number Generation
 With each rand execution, the TI84 Plus
generates the same randomnumber sequence  for a given seed value. The TI84 Plus
factoryset seed value for rand is 0. To generate
a  different randomnumber sequence, store any
nonzero seed value to rand. To restore  the factoryset seed value, store 0 to rand or
reset the defaults (Chapter 18).  Note The seed value also affects randInt(,
randNorm(, and randBin( instructions.