Statistics Chapter 7

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STANDARD NORMAL DISTRIBUTION
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STANDARD NORMAL DISTRIBUTIONRELATIONSHIP BETWEEN
x AND z

• If a random variable X is normally distributed
  with mean ? and standard deviation ?, then

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STANDARD NORMAL DISTRIBUTION TABLE, z-VALUES,
AREA AND PROBABILITY
Example 1.1 Table 3, Appendix B, p. 837 shows
the area under the curve from z0 to some
positive z value. For example, the area from z0
to z1.3.041.34 is 0.4099.
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STANDARD NORMAL DISTRIBUTION TABLE, z-VALUES,
AREA AND PROBABILITY
Example 1.2 The area shown on the table can be
used to get many other areas. For example, using
the fact that the area on each of the left and
right side of the mean is 0.5, the area from
z1.34 to z? is 0.5-0.40990.0901.
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STANDARD NORMAL DISTRIBUTION TABLE, z-VALUES,
AREA AND PROBABILITY
Example 1.3 Using symmetry, we can get the area
under the curve from z0 to some negative z
value. For example, the area from z-1.34 to z0
is 0.4099.
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STANDARD NORMAL DISTRIBUTION TABLE, z-VALUES,
AREA AND PROBABILITY
Example 1.4 Using symmetry, we can also get the
area under the curve from z-? to some negative z
value. For example, the area from z-? to z-1.34
is 0.0901.
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STANDARD NORMAL DISTRIBUTION TABLE, z-VALUES,
AREA AND PROBABILITY
Example 1.5 Using the fact that the area on each
of the left and right side of the mean is 0.5,
the area from z-? to z1.34 is 0.50.40990.9099.
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STANDARD NORMAL DISTRIBUTION TABLE, z-VALUES,
AREA AND PROBABILITY
Example 1.6 Using symmetry, the area from
z-1.34 to z? is 0.9099.
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STANDARD NORMAL DISTRIBUTION GIVEN z, FIND
PROBABILITY
Example 2 Find the following
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STANDARD NORMAL DISTRIBUTION GIVEN z, FIND
PROBABILITY EXCEL
Excel function NORMSDIST(z) provides the area
under the standard normal distribution curve on
the left side of z. For example, NORMSDIST(1.34)
0.9099
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STANDARD NORMAL DISTRIBUTION GIVEN PROBABILITY,
FIND z NOTATION zA
represents that value of z for which area
on the right is A. So,
For example, the area on the right side of
z1.96 is 0.025. Hence, z.0251.96.
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STANDARD NORMAL DISTRIBUTION GIVEN PROBABILITY,
FIND z

• Example 3 Determine zA and locate its value on
  the graph of the standard normal distribution
  each of the following values of A
• 1. 0.003
• 2. 0.05

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STANDARD NORMAL DISTRIBUTION GIVEN PROBABILITY,
FIND z EXCEL
Excel function NORMSINV(1-A) provides the value
of zA. For example, NORMSINV(1-0.025)1.96z.025
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NORMAL DISTRIBUTION GIVEN x, FIND PROBABILITY

• Example 4.1 A retailer has observed that the
  monthly demand of an item is normally distributed
  with a mean of 650 and standard deviation of 50
  units. What is the probability that the demand of
  the item in the next month will not exceed 700
  units?
• 1. Compute z
• 2. Find area from the Table
• 3. Find probability

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NORMAL DISTRIBUTION GIVEN x, FIND PROBABILITY
Example 4.2 A retailer has observed that the
monthly demand of an item is normally distributed
with a mean of 650 and standard deviation of 50
units. What is the probability that the demand of
the item in the next month will exceed 600
units? 1. Compute z 2. Find area from the
Table 3. Find probability
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NORMAL DISTRIBUTION GIVEN x, FIND PROBABILITY

• Example 4.3 A retailer has observed that the
  monthly demand of an item is normally distributed
  with a mean of 650 and standard deviation of 50
  units. What is the probability that the demand of
  the item in the next month will be between 600
  and 700 units?
• 1. Compute z1 and z1
• 2. Find areas from the Table
• 3. Find probability

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NORMAL DISTRIBUTION GIVEN x, FIND PROBABILITY
EXCEL
Excel function NORMDIST(x,?,?,TRUE) provides the
area under the standard normal distribution curve
on the left side of x. For example,
NORMDIST(700,650,50,TRUE) 0.8413
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NORMAL DISTRIBUTION GIVEN PROBABILITY, FIND x
Example 5 A retailer has observed that the
monthly demand of an item is normally distributed
with a mean of 650 and standard deviation of 50
units. If the retailer wants to limit the
probability of stock out to 0.10, how many units
should be ordered for the next month? Assume that
there is no units in the inventory. 1. Find zA
from the table 2. Find x
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NORMAL DISTRIBUTION GIVEN PROBABILITY, FIND x
EXCEL
Excel function NORMINV(p,?,?) provides that value
of x for which the area under the standard normal
distribution curve on the left side of x is p.
For example, NORMINV(0.90,650,50) 714.
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NORMAL APPROXIMATION TO BINOMIAL CONTINUITY
CORRECTION

• The normal distribution can be used to
  approximate other distributions including
  binomial distribution.
• The approximation is useful for large number of
  trials, n for which use of formula is not
  convenient and table is not available.
• From our previous discussion, if X is a binomial
  random variable, the mean, ? and the standard
  deviation, ? of X are
• n is the number of trials
• p is the probability of success on each trial
• q is the probability of failure on each trial
  1-p

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NORMAL APPROXIMATION TO BINOMIAL CONTINUITY
CORRECTION

• Continuity correction
• Binomial probability P(Xa) is approximated by
  normal probability P(a-0.5 ? X ? a0.5)
• This simple rule gives rise to many adjustments
• Binomial Normal
• P(Xa) P(a-0.5 ? X ? a0.5)
• P(a ? X ? b) P(a-0.5 ? X ? b0.5)
• P(X ? b) P(X ? b0.5)
• P(X ? a) P(X ? a-0.5)

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NORMAL APPROXIMATION TO BINOMIAL CONTINUITY
CORRECTION
Example 6.1 If you toss a fair coin 36 times,
what is the probability of getting 18 heads?
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NORMAL APPROXIMATION TO BINOMIAL CONTINUITY
CORRECTION
Example 6.2 If you toss a fair coin 36 times,
what is the probability of getting 12 to 24 heads?
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NORMAL APPROXIMATION TO BINOMIAL CONTINUITY
CORRECTION
Example 6.3 If you toss a fair coin 36 times,
what is the probability of getting at most 24
heads?
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NORMAL APPROXIMATION TO BINOMIAL CONTINUITY
CORRECTION
Example 6.4 If you toss a fair coin 36 times,
what is the probability of getting at least 12
heads?
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READING AND EXERCISES

• Reading pp. 262-274
• Exercises 7.6, 7.8, 7.12, 7.22