Statistics 221

1
Statistics 221

• Chapter 6 Part B
• Continuous Probability Distributions

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Solving for x when p is known

• Using the normal distribution / z tables, we can
  find the answer to another kind of question.
• Suppose for a given frequency distribution, we
  want to know what values would be within a
  certain percentage (p) of the mean.
• The next two examples demonstrate how we can use
  the normal distribution /z tables to answer this
  kind of question.

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Solving for x Example 1

• Lets assume that Grear Tire wants to provide a
  guarantee that the tires will last some
  pre-specified minimum number of miles. Their
  guarantee states that they will replace any tire
  that wears out before x number of miles.
• But they want to set x low enough that they wont
  have to replace more than 10 of the tires.
• What should they set the guarantee at so that no
  more than 10 of the tires will have to be
  replaced?

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Solve for x (where p .10)

• If we have probability distribution for all the
  possible tire lives (in number of miles) whose
  mean is 36,500, what tire-lives are in the bottom
  10 of that distribution?

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Start by drawing a picture to visualize the
problem. Place p in the area
P .90
10
Area 90
Area (p) .90
Miles driven until tire wears out
36,500
? 36,500
36500
x ?
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2. Use p to (reverse) lookup z.
When p .10, z -1.282
.1000
-1.2
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3. Transform z to an x
10
Area 90
Miles driven until tire wears out
36500

• To find x from z
• If we know that z (x ?) / ?
• Then if we solve for x x ? (z ?)
• Therefore x 36500 (-1.282)(5000) 30,092

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4. Refer to the picture and fill in a value for X
10
Area 90
Miles driven until tire wears out
36500
X 30,092
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5. Make a conclusion statement

• If they set the guaranteed minimum mileage at
  30,092, then they wont have to replace more than
  10 of the tires because they wore out too soon.

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Open the file DataSetsForCh6 and click on the
worksheet tab Grear Tire II
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1. Fill in the values for p, ?, and ? C3
.10 C4 36500 C5 5000
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2. Use Excels normsinv( ) formula to calculate z
from p C6 normsinv(C3)
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3. Multiply z by ? and add the product to the
mean to calculate what x-value segregates the
bottom 10 of the area under the curve C7 C4
(C6 C5)
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4. Write a conclusion statement.
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Solving for x Example 2

• When designing the placement of a CD player in a
  new model car, engineers must consider the
  forward grip reach of the driver. If a driver
  needs to lean forward too far to put a CD in the
  CD player, they might have an accident.
• Design engineers decide that the CD player should
  be within comfortable reach of 95 of women.
• The forward reach of a sample of women was
  obtained and it was found to have a normal
  distribution with a mean of 27 inches and a
  standard deviation of 1.3 inches.
• Find the forward grip reach that separates the
  TOP 95 from the bottom 5.

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1. Start by drawing a picture to visualize the
problem. Place p in the area
Area (p) .95
x ?
? 27.0
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2. Use p to (reverse) lookup z.
When p .05, z -1.645
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3. Transform z to an x

• To find x from z
• If we know that z (x ?) / ?
• Then solving for x x ? (z ?)
• Therefore x 27.0 (-1,645)(1.3) 24.86

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4. Refer to the picture and fill in a value for X
X 24.86
? 27.0
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5. Make a conclusion statement

• If a CD player is 24.86 inches away from a seated
  driver, 95 of women can reach the CD player
  without having to lean forward.
• But 5 of the women would have to lean forward to
  reach it (if it was 24.86 inches away).
• The designer might want to place the CD player on
  the dashboard a little closer to the driver.

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Open the file DataSetsForCh6 and click on the
worksheet tab CD reach
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1. Fill in the values for p, ?, and ? C3
.05 C4 27 C5 1.3
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2. Use Excels normsinv( ) formula to calculate z
from p C6 normsinv(C3)
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3. Multiply z by ? and add the product to the
mean to calculate what x-value segregates the
bottom 5 of the area under the curve C7 C4
(C6 C5)
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4. Write a conclusion statement.
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Normal approximation of binomial probabilities

• The methodology of calculating-area-to-find-
  expected-probability can be applied to discrete
  or continuous random variables.
• This methodology is universally applied when
  calculating the probability of continuous
  variables.
• But its not usually applied when calculating the
  probability of discrete variables because
  more-precise formula-based methods (e. g.,
  Binomial formula) are available.
• But now we will see that the areaprobability
  methodology can be applied to approximating the
  probability of discrete variables.

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When would we use the areaprobability strategy
for discrete variables?

• For a binomial experiment, when the number of
  trials (n) is large ( 20), the calculation of
  probabilities using the binomial formula can be
  complex because the values can be very large.
• But the areaprobability methodology (hereafter
  referred to as the normal probability method)
  can provide an easy-to-use approximation of
  binomial probabilities if these conditions are
  true (np 5) and (nq 5).

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The conditions (np 5) and (nq 5)

• If n is small (success (p) is close to 1.0 or close to 0, then
  (np) or (nq) may be less than 5 and the
  probability distribution of that binomial
  variable will not closely resemble a normal
  (symmetric) distribution.
• But if n is large (20), even if the probability
  of one success (p) is close to 1.0 or close to 0,
  (np) and (nq) will still be greater than 5 and
  the probability distribution of a binomial
  variable will closely resembles a normal
  (symmetric) distribution.
• The probability distribution must be (close to)
  normal or the z-values will not provide good
  approximations of p-values (probabilities).

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Examples of binomial distributions that are
normal and not normal (n 14 not large)
normal
Not normal
In this example, p .5 and q.5
In this example, p .9 and q.1
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Applying the normal probability method to
approximate the probabilities of discrete
variables

• A sample of 100 invoices has been taken and we
  want to compute the probability that exactly 12
  of those invoices contain errors.
• The probability that any one invoice has an error
  is 10.
• So p .10 and q .90 and n 100

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Using the normal method to calculate the
probability of a binomial variable

• For a binomial variable
• ? np (100) (.1) 10
• ? vnpq v (100) (.1) (.9) 3
• But in a normal distribution, the probability of
  x being exactly 12 is 0. So we must find the
  probability that
• P(11.5

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What is P(11.5 What is this area?
11.5
12.5
10
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Open the file DataSetsForCh6 and click on the
worksheet tab Invoices
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Enter the appropriate values into the cells to
compute the P(11.5 conclusion statement.
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The Exponential Distribution

• The exponential probability distribution is
  useful in describing the time it takes to
  complete a task.
• Some examples of random variables that typically
  have exponential probability distributions
• The time between vehicle arrivals at a toll
  booth.
• The time required to complete a questionnaire
• Distance between major defects on a highway.
• The time it requires a driver to pick up a
  package or load a suitcase on to an airplane.

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An exponential probability distribution looks
like this
The x-axis need not start at 0.
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Example the wait time between cars arriving at a
tollbooth
Most of the time there will be 0 seconds (or
could be minutes) wait between car arrivals
longer wait times are increasingly more unlikely.
Rarely would there be say, 5 or more seconds (or
minutes) between car arrivals.
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Calculating probabilities for events that have an
exponential probability distributions

• As with any continuous probability distribution,
  an interval area under the curve corresponds to
  the probability that the random variable assumes
  a value in that interval.
• To find an area / calculate the probability that
  a random variable x is less than or equal to some
  value, the exponential cumulative probability
  formula can be used

Where e 2.71828

• Or we can use Excels expondist( ) formula

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Example using the Exponential Probability
Distribution

• The time between arrivals of cars at Als
  full-service gas pump follows an exponential
  probability distribution with a mean time between
  arrivals of 3 minutes. Al would like to know the
  probability that the time between two successive
  arrivals will be 2 minutes or less.

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Exponential Probability Distribution
f(x)
P(x

3

x 2

x
1 2 3 4 5 6 7 8 9 10
Time Between Successive Arrivals (mins.)
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Exponential Probability Distribution

• A property of the exponential distribution is
  that the mean, m, and standard deviation, s, are
  equal.
• The standard deviation, s, and variance, s2 for
  the time between arrivals at Als full-service
  pump are

s m 3 minutes
s 2 (3)2 9
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Example

• The time it takes for an employee at the Schips
  loading dock to load up a truck has an
  exponential distribution.
• In an average eight-hour shift, 32 trucks will be
  loaded, or one every 15 minutes.
• What is the probability that a truck will take
  6-12 minutes to load?

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1. Draw a picture to visualize the area you are
trying to find
Truck loading time
What is this area as a percentage of the total
area under the curve?
0 1 2 3 4 5 6 7 8 9 10 11 12
13 14 15 16 17 18 19
minutes
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2. Use the exponential cumulative formula to
calculate the areas

• P(x
• 1 - .6703 .3297
• P(x
• 1 - .4493 .5506
• P(6

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3. Write conclusion statement

• 22 of the time, it will take between 6 and 12
  minutes to load a truck.

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Open the file DataSetsForCh6 and click on the
worksheet tab Truck load
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1. Fill in the values for x and ? C3 6 C4 15
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2. Use Excels expondist( ) formula to find the
area to the left of the line C5 expondist(C3,
1/C4, true)
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3. Fill in the values for x and ? D3 12 D4 15
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4. Use Excels expondist( ) formula to find the
area to the left of the line C5 expondist(C3,
1/C4, true)
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5. Subtract the areas to find the area in the
interval C6 D5-C5
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6. Write a conclusion statement
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Homework 11

• 18 (c only) on page 241
• normal distribution
• 20 (c only) on page 242
• normal distribution
• 24 (d only) on page 242
• normal distribution
• 26 on page 245
• Using normal to approximate binomial
• 30 on page 245
• Using normal to approximate binomial
• 34 on page 249
• Using the exponential distribution