Download 5.3 The Central Limit Theorem

Statistical Reasoning
for everyday life
Intro to Probability and
Statistics
Mr. Spering – Room 117
5.3 The Central Limit Theorem
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Is it Normal?
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Times in the 100-meter dash…
YES--NORMAL
The weights of new Casino Dice…
NO--UNIFORM
Number of candy bars in a Twix package…
NO--UNIFORM
Heights of Sequoia trees…
YES--NORMAL
Weights of 5000 randomly selected Boston Terriers…
YES--NORMAL
5.3 The Central Limit Theorem
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THE CENTRAL LIMIT THEOREM
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When we take many random samples of size n for a
variable with any distribution and record the distribution
of the means of each sample. Then,
1.
The distribution of means will be approximately a
normal distribution for large sample sizes.
The mean of the distribution of means approaches
the population mean, μ, for large sample sizes.
The standard deviation of the distribution of
means approaches σ/√n for large sample sizes,
where σ is the standard deviation of the
population.
2.
3.
practical purposes--the distribution will need more than 30 valesccccccp… Ooops, I
think I feel asleep…> 30 values. Recall, If normal then the 68-95-99.7 rule holds true.
5.3 The Central Limit Theorem
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Summary:
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Within the central limit theorem, we always start with a
particular variable, such as the outcomes of a die roll or
weights of people, that varies randomly over a
population. The variable has a certain mean, μ, and
standard deviation, σ, which we may or may not know.
This variable can have any sort of distribution. When we
take many samples of that variable, with n items on each
sample, and make a histogram of the means from the
many samples, we will see a distribution that is close to
“normal”. The larger the sample size, n, the more closely
the distribution of means approximates a normal
distribution.
Check out this website on rolling dice & the C.L.T.
--- http://www.stat.sc.edu/~west/javahtml/CLT.html
5.3 The Central Limit Theorem
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VISUAL EXAMPLE:
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Pictures of a distribution being "smoothed out" by summation (showing original density of
distribution and three subsequent summations, obtained by convolution “averaging” of
density functions)
Works Cited:
http://en.wikipedia.org/wiki/Ce
ntral_limit_theorem
More visuals: Page 218
TheCentralLimitTheorem.nbp
5.3 The Central Limit Theorem
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EXAMPLE:
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Suppose you are a principal of a middle school and your 100
eighth-graders are about to take a national standardized test.
(sound familiar?) The test is designed so that the mean score
is μ = 400 with a standard of deviation σ = 70. Assume the
scores are normally distributed.
A) What is the likelihood that one of your eighth-graders,
selected at random, will score below 380 on the exam?
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How do we start? Percentage?
Z-scores correspond to a percentage from the table 5.1 on page
232. Therefore, the standard score of 380 is x  x 380  400
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 0.29
-0.29 is approximately 38th percentile, or
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70
38% of all students (about 4 out of 10).
5.3 The Central Limit Theorem
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EXAMPLE:
B) Your performance as a principal depends on how well your entire
group of eighth-graders scores on the exam. What is the
likelihood that your group of 100 eighth-graders will have a mean
score below 380?
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How do we start? Central Limit Theorem?
According to the C.L.T. if we take random groups of say 100 students
and study their means, then the means distribution will approach
normal. Hence, the μ = 400 and its standard of deviation is according
to the C.L.T. 
70

7
n
100
x  x 380  400
 2.9
Therefore, the z-score for a mean of 380 is  
7
The percentile for the z-score on page 232 is 0.19th percentile or
0.0019, which is very small, about 1 in 500.
EMPHASIS POINT: THERE IS MORE VARIATION IN THE SCORES OF
INDIVIDUALS THAN IN THE MEANS OF GROUPS OF INDIVIDUALS
5.3 The Central Limit Theorem
When listening to corn pop, you are hearing
normal distribution and the C.L.T.
-- William A. Massey {Princeton}
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HOMEWORK:
Pg 220 # 1-20
all
-
Z-Score
Percentile
≈ -1.2 and -1.3
12th
≈ -0.7 and -0.65
25th
≈ -0.35 and -0.30
37th
≈ 0.0 and 0.0
50th
≈ 0.30 and 0.35
62nd
≈ 0.65 and 0.70
75th
≈ 1.1 and 1.2
87th
≈ 3.5 and above
99.98th
The larger the sample size,
the  of the means approaches:

n