Download Example for Binomial Distribution (Problem 3.7, page 140)

Review of Binomial Concepts
A single experiment, with
probability of success p, is
repeated n independent times
X = number of successes
the discrete random variable
1
Probability
distribution
for X when
n=10 and
p=0.5
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
P(X = k)
0
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
1
Cumulative
distribution
for X when
n=10 and
p=0.5
P(X  k)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Excel Commands for Binomial Probabilities
X is a binomial random variable
with binomial parameters n and p
P(X = k): BINOMDIST(k, n, p, 0)
P(X  k): BINOMDIST(k, n, p, 1)
Important Formulas for
Binomial Random Variable
E[X] = n * p
X = sqrt [ n * p * (1 – p) ]
Example for Binomial Distribution
A study by one automobile manufacturer indicated
that one out of every four new cars required repairs
under the company’s new-car warranty, with an
average cost of $50 per repair.
a) For 100 new cars, what is the expected cost of
repairs? What is the standard deviation?
b) Provide a reasonable estimate of the most it might
cost the manufacturer for a specified group of 100
cars. (Hint: Ensure a 95% probability of not
exceeding amount.)
Example for Binomial Distribution
Suppose you are in charge of hiring new undergraduate
accounting majors for Coopers and Young (C&Y), an accounting
firm headquartered in Chicago. This year your goal is to hire 20
graduates. On the average, about 40% of the people you make
offers to will accept. Unfortunately, offers have to go out
simultaneously this year, so you plan to make more than 20
offers.
a) How many offers should you make so that your expected
number of hires is 20?
b) How many offers should you make if you want to have an
80% chance of hiring at least 20 people?
c) Find a 95% probability interval if you make 50 offers.
Normal Example
Otis Elevator in Bloomington, Indiana, reported that the
number of hours lost per week last year due to employees’
illnesses was approximately normally distributed, with a
mean of 60 hours and a standard deviation of 15 hours.
Determine, for a given week, the following probabilities:
a) The number of hours lost will exceed 85 hours.
b) The number of hours lost will be between 45 and
55 hours.
Normal Example
A mail-order company has estimated one-year orders for
a popular item to be normally distributed with a mean of
180,000 units and a standard deviation of 15,000 units.
a) What is the probability of selling all of the stock on
hand if inventory equals 200,000 units?
b) What inventory should the company have on hand if
they want the probability of running out of stock to
be 5%?
The Normal Distribution (preview)
• The normal distribution is related to a particular type of
continuous random variable (as opposed to “discrete
random variable”)
• It is the “bell-shaped” curve that you may have heard about
• It is used widely in statistics and appears in lots of practical
problems
• It has an expected value (or “mean”) and a standard
deviation just like other random variables
• We’ll discuss it in detail shortly, but a little background
first….
Continuous Random Variables
Recall
a random variable takes a probability event
and assigns a number, or value, to it
Recall
a discrete random variable deals with a
finite number of events and numbers
A continuous random variable can take on any
value within a specified interval of numbers
There are an infinite number of events
and an infinite number of values
Examples of Continuous
Random Variables
X = the precise time that a train arrives, when it is
scheduled to arrive at 8:00 PM
X = the residual chemical levels in the blood 24 hours
after taking a specific medication
X = the precise amount of soda placed in a bottle by a
soda filling machine (or should it be “pop”?)
X = the amount of energy used in a home during one hour
Continuous or Discrete? Or Both?
Sometimes a discrete random variable has so many
possible outcomes (still finite, however) that we
consider it to be continuous
Sometimes a continuous random variable is actually
discrete when we measure it
X = salary of an office manager
X = price of a therm of natural gas in January
X = grades of a student on the SAT
X = length of a manufactured part (limits of our
measurement techniques)
How to Talk about
Continuous Random Variables
Since a continuous random variable has so many possible
outcomes, we do not consider things like:
P(X = 1.0)
P(X = 0.3349)
P(X = -0.5)
P(X  0.3349)
P(-0.5  X  0.3349)
Instead, we consider:
P(X  1.0)
equalities (no), intervals (yes)
( In fact, P(X = k) = 0 for a continuous random variable.
Again, this is “equal to.” )
Example: A Delivery Truck
Suppose the warehouse says that the delivery truck will
arrive sometime “between 8 AM and noon.”
What is the probability that the truck will arrive between
9 AM and 10 AM?
0.50
0.25
8
9
10
11
12
The total area of the
red block equals 1.0,
the total probability.
So P(9  X  10) =
0.25
Probability density function (distribution);
similar to probability mass function in discrete random variables
The Normal Distribution
• Bell-shaped
• Horizontal axis
represents values of
the random variable
• Expected value, or
mean, is 
• Symmetrical about 
• Total blue area is 1.0
• Vertical axis is not
very important
The Normal Distribution (cont’d)
 is the standard
deviation of the
normal random
variable
P(  -   X   +  ) = 0.6826
“Probability that X is within 1 std dev of its mean is 68.26%”
The Normal Distribution (cont’d)
P(  - 2  X   + 2 ) = 0.9544
“Probability that X is within 2 std dev of its mean is 95.44%”
The Normal Distribution (cont’d)
P(  - 3  X   + 3 ) = 0.9973
“Probability that X is within 3 std dev of its mean is 99.73%”
The Normal Distribution (summary)
A continuous random variable X is “normally distributed
with expectation/mean  and standard deviation ” if its
probability distribution is
1. Bell-shaped
2. Symmetrical about the value 
3. The following are true:
i. P(  -   X   +  ) = 0.6826
ii. P(  - 2  X   + 2 ) = 0.9544
iii. P(  - 3  X   + 3 ) = 0.9973
We say X is N(  , 2 )
Excel’s Normal Functions
If X is N(  , 2 ):
P(X  k) = NORMDIST(k, , , 1)
NORMINV( prob, ,  ) gives the
value of k so that P(X  k) = prob
“normal inverse”
The Standardized Normal Distribution
X is N(  , 2 )
Z is N( 0, 12 )
The standardized normal
1. For comparison of several different
normal distributions
2. For calculations without a computer
Formula relating standard Z and any given X:
Z=(X-)/
“Given the value for X, the corresponding
value for Z is given by the above formula.”
Compare your
performance in all
courses…
Class
Your
Score
Mean
Std Dev
1
92
76.3
10.2
2
75
81.1
5.1
3
166
152.8
21.4
4
158
134.5
16.7
The standardized normal is also useful for hand
calculations of normal probabilities…
Suppose X is normally distributed with mean 20 and
standard deviation 4.3.
What is P(X  23)? What is P(X  16.5)?
Z=(X-)/
P(Z  0.698) = ?
P(Z  -0.814) = ?
(23 – 20) / 4.3 = 0.698
(16.5 – 20) / 4.3 = -0.814
(See Excel)
… = 0.7580
… = 0.2090
The Normal Approximation
to the Binomial
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
9
10
Both distributions have the same shape…
Suppose X is a binomial variable with n = 900 and p = 0.35
E[X] =  = np = 315
 = sqrt( np(1-p) ) = 14.31
Let Y be the normal variable with  = 315 and  = 14.31
P(X  300) = BINOMDIST(300, 900, 0.35, 1) = 0.155
P(X  300)  P(Y  300) = NORMDIST(300, 315, 14.31, 1) = 0.147
P(X = 300) = BINOMDIST(300, 900, 0.35, 0) = 0.0162
P(X = 300)  P(299.5  Y  300.5) =
NORMDIST(300.5, 315, 14.31, 1) – NORMDIST(300.5, 315, 14.31, 1) = 0.0161
The second is called the “continuity correction”