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5/7/14 The Central Limit Theorem 1.  If samples of size n ≥ 30 are drawn from any popula*on with mean = µ and standard devia*on = σ, Sta*s*cs with Probability Lesson #18 x
µ
then the sampling distribution of sample means
approximates a normal distribution. The greater the
sample size, the better the approximation.
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. xx
x x
x x x
x x x x x
© 2012 Pearson Education, Inc. All rights reserved.
The Central Limit Theorem µx = µ
x
σx =
x
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© 2012 Pearson Education, Inc. All rights reserved.
The Central Limit Theorem Any Popula*on Distribu*on Distribution of Sample Means,
n ≥ 30
© 2012 Pearson Education, Inc. All rights reserved.
2. 
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. 2
σ
σ x2 =
Variance
n
σ
then the sampling distribution of sample means is
normally distribution for any sample size n.
1. 
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•  In either case, the sampling distribu*on of sample means has a mean equal to the popula*on mean. µ
µ
x
The Central Limit Theorem 2.  If the popula*on itself is normally distributed, xx
x x
x x x
x x x x x
µ
Normal Popula*on Distribu*on Distribu*on of Sample Means, (any n) 5 of 105
n
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Standard deviation (standard
error of the mean)
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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.
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1 5/7/14 Solu*on: Interpre*ng the Central Limit Theorem Solu*on: Interpre*ng the Central Limit Theorem •  The mean of the sampling distribu*on is equal to the popula*on mean •  Since the sample size is greater than 30, the sampling distribu*on can be approximated by a normal distribu*on with µ x = µ = 63
n
100
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© 2012 Pearson Education, Inc. All rights reserved.
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. 9 of 105
© 2012 Pearson Education, Inc. All rights reserved.
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. µ x = 135
© 2012 Pearson Education, Inc. All rights reserved.
σ x = $1.10
µx = $63
•  The standard error of the mean is equal to the popula*on standard devia*on divided by the square root of n. σ x = σ = 11 = 1.1
σx = 9
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© 2012 Pearson Education, Inc. All rights reserved.
Solu*on: Interpre*ng the Central Limit Theorem •  The mean of the sampling distribu*on is equal to the popula*on mean µ x = µ = 135
•  The standard error of the mean is equal to the popula*on standard devia*on divided by the square root of n. σ x = σ = 18 = 9
n
4
© 2012 Pearson Education, Inc. All rights reserved.
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Probability and the Central Limit Theorem •  To transform x to a z-­‐score z=
Value − Mean x − µ x x − µ
=
=
σ
Standard error
σx
n
© 2012 Pearson Education, Inc. All rights reserved.
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2 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 estimate
parameters. Statistics are sometimes called estimators and the numbers that result are
called estimates.
6. Bias is measured using the center of the sampling distribution: It is the distance
between the center and the population parameter value.
7. Precision is measured using the standard deviation of the sampling distribution, which
is called the standard error. When the standard error is small, we say the estimator
is precise.
8. Sampling Distribution - the special name for the probability distribution of a statistic. Used to make inferences about a population.
Facts:
1. No matter how many different samples we take, the value of µ (the population mean)
is always the same, but the value of x̄ changes from sample to sample.
2. The precision of an estimator does NOT depend on the size of a population; it depends
only on the sample size.
3. Surveys based on larger sample sizes have smaller standard error and therefore better
precision. Increasing sample size improves precision.
Keeping track of parameters and statistics:
Parameters (typically unknown)
Statistics (based on data)
1. µ - population mean
5. x̄ - sample mean
2. σ - population standard deviation
6. s - sample standard deviation
3. σ 2 - population variance
7. s2 - sample variance
4. p - population proportion
8. p̂ - sample proportion
THE CENTRAL LIMIT THEOREM - Three ways
1. The Central Limit Theorem for a Sample PROPORTION tells us that if we take a
random sample from a population, and if the sample size n is large and the population
size is much larger than the sample size, then the sampling distribution of the sample
proportion p̂ is approximately normal with mean p and standard deviation
r
p(1 − p)
n
(If you don’t know the value of p, then you can substitute the value of p̂ to calculate
the estimated standard error.)
2. The Central Limit Theorem for Sample SUM tells us that if we take a random sample
X1 , X2 , . . . , Xn from a population, and if the sample size n is large and the population
size is much larger than the sample size, then the sampling distribution of the sum
X√
1 + X2 + · · · + Xn is approximately normal with mean nµ and standard deviation
σ n.
3. The Central Limit Theorem for Sample MEAN tells us that if we take a random
sample from a population, and if the sample size n is large and the population size
is much larger than the sample size, then the sampling distribution of the mean X̄ is
σ
approximately normal with mean µ and standard deviation √ .
n
Page 2
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?
Reaction Distance for Men Frequency 8 6 4 2 0 Frequency 4 5 6 7 8 9 10 0 3 1 7 1 2 2 bins Reaction distance for women 3.5 3 Frequency Results'of'Activity'#3'1'Does'Reaction'Distance'Depend'on'Gender?
Reaction)distance)measured)in)inches
Men
bins
Women bins
9
4
9
6
9.5
5
6
7
8
6
9
8
5
7
11
9
5
8
11.5
10
6.5
9
6.5
11
6
10
10
12
7
9
7
11
7
7
Mean'=' 9.222222
6.5
Median'=
9
6.5
Modes'='
9
10
S.d.'='
1.938284
9
5
Mean'='
7.125
Median'=
7
Modes'='
7
S.d.'='
1.586401
2.5 2 1.5 1 0.5 0 6 7 8 9 bins 10 11 12 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-World5E/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.