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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.

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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.

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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.

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.

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

.

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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

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.