THE DAY BEFORE THANKSGIVING

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THE DAY BEFORE THANKSGIVING

• WE WILL HAVE CLASS AS USUAL
• WE WILL HAVE A QUIZ
• WE WILL LEARN NEW MATERIAL
• WE WILL HAVE A QUIZ ON THE NEW MATERIAL AFTER
  THANKSGIVING

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STATISTICS IS A MEAN SUBJECT
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4.1 The Central Limit Theorem for
We use a large sample from a population whose
histogram is unknown. But suppose we know the
mean µ and standard deviation ? of the
population, and the sample size n, which is at
least 30. Then (a) The sample mean is a
random variable with mean
and

, the standard error of the mean.
standard deviation

• The sample mean is a nearly normal random
  variable.
• (c) The standard form of the sample mean
  is the random variable

which is standard normal.
M155 L26 The Central Limit Theorem -- Slide 1
4
M155 L26 The Central Limit Theorem -- Slide 2
Figure 2
_
P(a ? X ? b)
_
X
?
a
b
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EXAMPLE G A large shipment of bolts has mean
weight of .15 ounces and a (weight) standard
deviation of .018 ounces. Suppose a random
sample of 36 bolts is to be taken, and consider
the random variable , the mean weight of the
sample. What is the probability of the event
? .155 ? SOLUTION G We note that the shipment
(population) is relatively "large" (hopefully at
least 20x36720). Knowing the mean µ and
standard deviation ? of the population, we
compute the mean and standard deviation of

M155 L26 The Central Limit Theorem -- Slide 3
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Figure 3
M155 L26 The Central Limit Theorem -- Slide 4
.0475
.4525
_
X
.155
.15
.005/.0030
Z
1.67
0
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McClave Section 4.9, page 217 Exercises 97,
98a
POPULATION
SAMPLE
RANDOM VARIABLE
large
M155 L26 The Central Limit Theorem -- Slide 6
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M155 L26 The Central Limit Theorem -- Slide 7
.0228
.4772
_
X
20
16
- 4 / 2
Z
0
-2.00
9
4.1 The Central Limit Theorem for
We use a large sample from a population whose
histogram is unknown. But suppose we know the
mean µ and standard deviation ? of the
population, and the sample size n, which is at
least 30. Then (a) The sample mean is a
random variable with mean
and

, the standard error of the mean.
standard deviation

• The sample mean is a nearly normal random
  variable.
• (c) The standard form of the sample mean
  is the random variable

which is standard normal.
M155 L26 The Central Limit Theorem -- Slide 1
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CENTRAL LIMIT THEOREM
All large-sample methods (using Z) are based on
the Central Limit Theorem, which says the
larger the sample, the more nearly the sample
mean random variable is normally distributed.
They assume accurate knowledge of sigma, such
as s from a large sample.