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5/7/14
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Sta*s*cs with Probability Lesson #18
Slides available as supplemental material for the Pearson text
Elementary Sta*s*cs: Picturing the World, 5th edi*on, by Ron
Larsen and Betsy Farber.
The Central Limit Theorem 1. If samples of size n 30 are drawn
from any popula*on
with mean = and standard devia*on = ,
x
xx xx
xxxx x
xxx x
then the sampling distribution of sample means approximates a
normal distribution. The greater the sample size, the better the
approximation.
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The Central Limit Theorem 2. If the popula*on itself is
normally
distributed,
then the sampling distribution of sample means is normally
distribution for any sample size n.
x
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x
xx xx
xxxx x
xxx
The Central Limit Theorem In either case, the sampling
distribu*on of sample means has a mean equal to the popula*on
mean.
The sampling distribu*on of sample means has a variance equal to
1/n *mes the variance of the popula*on and a standard devia*on
equal to the popula*on standard devia*on divided by the square root
of n.
Variance
Standard deviation (standard error of the mean)
x =
x n =
22x n
=
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Mean
The Central Limit Theorem 1. Any Popula*on Distribu*on 2. Normal
Popula*on Distribu*on
Distribution of Sample Means, n 30
Distribu*on of Sample Means, (any n)
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Example: Interpre*ng the Central Limit Theorem
Cellular phone bills for residents of a city have a mean of $63
and a standard devia*on of $11. Random samples of 100 cellular
phone bills are drawn from this popula*on and the mean of each
sample is determined. Find the mean and standard error of the mean
of the sampling distribu*on. Then sketch a graph of the sampling
distribu*on of sample means.
2012 Pearson Education, Inc. All rights reserved. 6 of 105
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5/7/14
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Solu*on: Interpre*ng the Central Limit Theorem
The mean of the sampling distribu*on is equal to the popula*on
mean
The standard error of the mean is equal to the popula*on
standard devia*on divided by the square root of n.
63x = =
11 1.1100x n
= = =
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Solu*on: Interpre*ng the Central Limit Theorem
Since the sample size is greater than 30, the sampling
distribu*on can be approximated by a normal distribu*on with
$63x = $1.10x =
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Example: Interpre*ng the Central Limit Theorem Suppose the
training heart rates of all 20-year-old
athletes are normally distributed, with a mean of 135 beats per
minute and standard devia*on of 18 beats per minute. Random samples
of size 4 are drawn from this popula*on, and the mean of each
sample is determined. Find the mean and standard error of the mean
of the sampling distribu*on. Then sketch a graph of the sampling
distribu*on of sample means.
2012 Pearson Education, Inc. All rights reserved. 9 of 105
Solu*on: Interpre*ng the Central Limit Theorem
The mean of the sampling distribu*on is equal to the popula*on
mean
The standard error of the mean is equal to the popula*on
standard devia*on divided by the square root of n.
135x = =
18 94x n
= = =
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Solu*on: Interpre*ng the Central Limit Theorem
Since the popula*on is normally distributed, the sampling
distribu*on of the sample means is also normally distributed.
135x = 9x =
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Probability and the Central Limit Theorem
To transform x to a z-score Value MeanStandard error
x
x
x xzn
= = =
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Lecture #18 - Learning about the world through surveys
Some important definitions:
1. Population - A group of objects or people we wish to
study.
2. Parameter - A numerical value that characterizes some aspect
of this population.
3. Census - A survey in which EVERY member of the population is
measured.
4. Sample - A collection of people or objects taken from the
population of interest.
5. Statistic - A numerical characteristic of a sample data.
Statistics are used to estimateparameters. Statistics are sometimes
called estimators and the numbers that result arecalled
estimates.
6. Bias is measured using the center of the sampling
distribution: It is the distancebetween the center and the
population parameter value.
7. Precision is measured using the standard deviation of the
sampling distribution, whichis called the standard error. When the
standard error is small, we say the estimatoris precise.
8. Sampling Distribution - the special name for the probability
distribution of a statis-tic. Used to make inferences about a
population.
Facts:
1. No matter how many different samples we take, the value of
(the population mean)is always the same, but the value of x changes
from sample to sample.
2. The precision of an estimator does NOT depend on the size of
a population; it dependsonly on the sample size.
3. Surveys based on larger sample sizes have smaller standard
error and therefore betterprecision. Increasing sample size
improves precision.
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Keeping track of parameters and statistics:
Parameters (typically unknown)
1. - population mean
2. - population standard deviation
3. 2 - population variance
4. p - population proportion
Statistics (based on data)
5. x - sample mean
6. s - sample standard deviation
7. s2 - sample variance
8. p - sample proportion
THE CENTRAL LIMIT THEOREM - Three ways
1. The Central Limit Theorem for a Sample PROPORTION tells us
that if we take arandom sample from a population, and if the sample
size n is large and the populationsize is much larger than the
sample size, then the sampling distribution of the sampleproportion
p is approximately normal with mean p and standard deviation
p(1 p)n
(If you dont know the value of p, then you can substitute the
value of p to calculatethe estimated standard error.)
2. The Central Limit Theorem for Sample SUM tells us that if we
take a random sampleX1, X2, . . . , Xn from a population, and if
the sample size n is large and the populationsize is much larger
than the sample size, then the sampling distribution of the sumX1 +
X2 + + Xn is approximately normal with mean n and standard
deviationn.
3. The Central Limit Theorem for Sample MEAN tells us that if we
take a randomsample from a population, and if the sample size n is
large and the population sizeis much larger than the sample size,
then the sampling distribution of the mean X is
approximately normal with mean and standard deviationn
.
Page 2
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In Class Activity #3 Does Reaction Distance Depend on Gender?
(from Gould, Robert and Colleen Ryan. 2013. Essential Statistics:
Exploring the World Through Data. Pearson.) Work in groups of two
or three. One person holds the meter stick vertically, with one
hand near the top of the stick, so that the 0-centimeter mark is at
the bottom. The other person then positions his or her thumb and
index finger about 5 cm apart (2 inches apart) on opposite sides of
the meter stick at the bottom. Now the first person drops the meter
stick without warning, and the other person catches it. Record the
location of the middle of the thumb of the catcher. This is the
distance the stick traveled and is called the reaction distance,
which is related to reaction time. A student who records a small
distance has a fast reaction time, and a student with a larger
distance has a slower reaction time. Now switch tasks. Each person
should try catching the meter stick twice, and the better (shorter)
distance should be reported for each person. Then record the gender
of each catcher. Your instructor will collect your data and combine
the class results. Before the Activity
1. Imagine that your class has collected data and you have 25
men and 25 women. Sketch the shape of the distribution you expect
to see for the men and the distribution you expect to see for the
women. Explain why you chose the shape you did.
2. What do you think would be a reasonable value for the typical
reaction distance for the women? Do you think it will be different
from the typical reaction distance for the men?
After the Activity 1. Now that you have actual data, how do the
shapes of the distributions for men and women compare to the
sketches you made before you collected data? 2. What measures of
center and spread are appropriate for comparing men and women's
reaction distances?
Why? 3. How do the actual typical reaction distances compare to
the values you predicted? 4. Using the data collected from the
class, write a short paragraph (a couple of sentences) comparing
the
reaction distances of men and women. You should also talk about
what group you could extend your findings to, and why. For example,
do your findings apply to all men and women? Or do they apply only
to college students?
Results'of'Activity'#3'1'Does'Reaction'Distance'Depend'on'Gender?Reaction)distance)measured)in)inchesMen
bins Women bins
9 4 9 69.5 5 6 78 6 9 85 7 11 95 8 11.5 10
6.5 9 6.5 116 10 10 127 97 1177 Mean'=' 9.222222
6.5 Median'= 96.5 Modes'=' 910 S.d.'=' 1.93828495
Mean'=' 7.125Median'= 7Modes'=' 7S.d.'=' 1.586401
4 5 6 7 8 9 10 Frequency 0 3 1 7 1 2 2 0 2 4 6 8
Freq
uency
bins
Reaction Distance for Men
0 0.5 1 1.5 2 2.5 3 3.5
6 7 8 9 10 11 12
Freq
uency
bins
Reaction distance for women
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Works Cited, References and Links: Larsen, Ron, and Betsy
Farber. 2012. Powerpoint Lecture Slides for Elementary Statistics:
Picturing the World, 5th edition. Pearson.ISBN-10: 0321693728. Can
be downloaded at:
http://www.pearsonhighered.com/educator/product/Elementary-Statistics-Picturing-the-World-5E/9780321693624.page#dw_resources
Gould, Robert and Colleen Ryan. 2013. Essential Statistics:
Exploring the World Through Data. Pearson. ISBN-10: 0321322150.
Emerson, Tisha L. N., and Beck A. Taylor. 2004. Comparing Student
Achievement Across Experimental and Lecture-Oriented Sections of a
Principles of Microeconomics Course. Southern Economic Journal 70:
672-93. Knight, Jennifer K., and William B. Wood. 2005. Teaching
More by Lecturing Less. Cell Biology Education 4: 298-310. Prince,
Michael. 2004. Does Active Learning Work? A Review of the Research.
Journal of Engineering Education 93: 223-31. Robinson, Carole F.,
and Peter J. Kakela. 2006. Creating a Space to Learn: A Classroom
of Fun, Interaction and Trust. College Teaching 54: 202-06.