Converting to a Standard Normal Distribution

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Converting to a Standard Normal Distribution
Think of me as the measure of the distance from
the mean, measured in standard deviations
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Using Excel to ComputeStandard Normal
Probabilities

• Excel has two functions for computing
  probabilities and z values for a standard normal
  distribution

NORM S DIST
NORM S INV
(The S in the function names reminds us that
they relate to the standard normal probability
distribution.)
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Using Excel to ComputeStandard Normal
Probabilities

• Formula Worksheet

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Using Excel to ComputeStandard Normal
Probabilities

• Value Worksheet

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Using Excel to ComputeStandard Normal
Probabilities

• Formula Worksheet

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Using Excel to ComputeStandard Normal
Probabilities

• Value Worksheet

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Example Pep Zone

• Standard Normal Probability Distribution
• Pep Zone sells auto parts and supplies
• including a popular multi-grade motor
• oil. When the stock of this oil drops to
• 20 gallons, a replenishment order is
• placed.

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Example Pep Zone

• Standard Normal Probability Distribution
• The store manager is concerned that sales are
  being lost due to stockouts while waiting for an
  order. It has been determined that demand during
  replenishment lead time is normally distributed
  with a mean of 15 gallons and a standard
  deviation of 6 gallons.
• The manager would like to know the probability
  of a stockout, P(x 20).

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Solving for Stockout Probability
Step 1 Convert x to the standard normal
distribution
Thus 20 gallons sold during the replenishment
lead time would be .83 standard deviations above
the average of 15.
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Solving for Stockout ProbabilityStep 2
Now we need to find the area under the curve to
the left of z .83. This will give us the
probability that x 20 gallons.
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Example Pep Zone

• Cumulative Probability Table for
• the Standard Normal Distribution

P(z
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Example Pep Zone

• Solving for the Stockout Probability

Step 3 Compute the area under the standard
normal curve to the right of z
.83.
P(z .83) 1 P(z .7967 .2033
Probability of a stockout
P(x 20)
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Example Pep Zone

• Solving for the Stockout Probability

Area 1 - .7967 .2033
Area .7967
z
0
.83
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Example Pep Zone
If the manager of Pep Zone wants the probability
of a stockout to be no more than .05, what should
the reorder point be?
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Example Pep Zone

• Solving for the Reorder Point

Area .9500
Area .0500
z
0
z.05
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Example Pep Zone

• Solving for the Reorder Point

Step 1 Find the z-value that cuts off an area
of .05 in the right tail of the standard
normal distribution.
We look up the complement of the tail area (1 -
.05 .95)
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Solving for the Reorder Point
Step 2 Convert z.05 to the corresponding value
of x
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Solving for the Reorder Point
So if we raising our reorder point from 20 to 25
gallons, we reduce the probability of a stockout
from about .20 to less than .05
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Using Excel to ComputeNormal Probabilities

• Excel has two functions for computing cumulative
  probabilities and x values for any normal
  distribution

NORMDIST is used to compute the
cumulative probability given an x value.
NORMINV is used to compute the x value given a
cumulative probability.
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Using Excel to ComputeNormal Probabilities

• Formula Worksheet

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Using Excel to ComputeNormal Probabilities

• Value Worksheet

Note P(x 20) .2023 here using Excel, while
our previous manual approach using the z table
yielded .2033 due to our rounding of the z value.
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Exercise 18, p. 261

• The average time a subscriber reads the Wall
  Street Journal is 49 minutes. Assume the standard
  deviation is 16 minutes and that reading times
  are normally distributed.
• What is the probability a subscriber will spend
  at least one hour reading the Journal?
• What is the probability a reader will spend no
  more than 30 minutes reading the Journal?
• For the 10 percent who spend the most time
  reading the Journal, how much time do they spend?

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Exercise 18, p. 261

• Convert x to the standard normal
  distributionThus one who read 560 minutes
  would be .69 from the mean. Now find P(z
  .6875). P(z .69) .7549.Thus P(x 60
  minutes) 1 - .7549 .2541.
• Convert x to the standard normal distribution

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P(x 30 minutes)

Red-shaded area is equal to blue shaded area
ThusP(x
z
0
1.19
-1.19
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Exercise 18, p. 261
(c)