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http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html. Example: IQ. Mean IQ = 100 Standard deviation = 15 What is the probability that a person you randomly bump into on the street has an IQ of 110 or higher?. Step 1: Sketch out question. -3  -2  -1   1  2  3 .

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## http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html

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http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.htmlhttp://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html
Example: IQ
• Mean IQ = 100
• Standard deviation = 15
• What is the probability that a person you randomly bump into on the street has an IQ of 110 or higher?
Step 1: Sketch out question

-3 -2 -1 1 2  3 

Step 1: Sketch out question

110

-3 -2 -1 1 2  3 

Step 2: Calculate Z score

(110 - 100) / 15 = .66

110

-3 -2 -1 1 2  3 

Step 3: Look up Z score in Table

Z = .66; Column C = .2546

110

.2546

-3 -2 -1 1 2  3 

Example: IQ
• You have a .2546 probability (or a 25.56% chance) of randomly bumping into a person with an IQ over 110.
Now. . . .
• What is the probability that the next 5 people you bump into on the street will have a mean IQ score of 110?
• Notice how this is different!
Population
• You are interested in the average self-esteem in a population of 40 people
• Self-esteem test scores range from 1 to 10.
1,1,1,1

2,2,2,2

3,3,3,3

4,4,4,4

5,5,5,5

6,6,6,6

7,7,7,7

8,8,8,8

9,9,9,9

10,10,10,10

Population Scores
What is the average self-esteem score of this population?
• Population mean = 5.5
• What if you wanted to estimate this population mean from a sample?
Group Activity
• Randomly select 5 people and find the average score
Group Activity
• Why isn’t the average score the same as the population score?
• When you use a sample there is always some degree of uncertainty!
• We can measure this uncertainty with a sampling distribution of the mean
Characteristics of a Sampling Distribution of the means
• Every sample is drawn randomly from a population
• The sample size (n) is the same for all samples
• The mean is calculated for each sample
• The sample means are arranged into a frequency distribution (or histogram)
• The number of samples is very large
Sampling Distribution of the Mean
• Notice: The sampling distribution is centered around the population mean!
• Notice: The sampling distribution of the mean looks like a normal curve!
• This is true even though the distribution of scores was NOT a normal distribution
Central Limit Theorem

For any population of scores, regardless of form, the sampling distribution of the means will approach a normal distribution as the number of samples get larger. Furthermore, the sampling distribution of the mean will have a mean equal to  and a standard deviation equal to / N

Mean
• The expected value of the mean for a sampling distribution
• E (X) = 
Standard Error
• The standard error (i.e., standard deviation) of the sampling distribution

x = / N

Standard Error
• The  of an IQ test is 15. If you sampled 10 people and found an X = 105 what is the standard error of that mean?

x = / N

Standard Error
• The  of an IQ test is 15. If you sampled 10 people and found an X = 105 what is the standard error of that mean?

x = 15/ 10

Standard Error
• The  of an IQ test is 15. If you sampled 10 people and found an X = 105 what is the standard error of that mean?

4.74= 15/ 3.16

Standard Error
• The  of an IQ test is 15. If you sampled 10 people and found an X = 105 what is the standard error of that mean? What happens to the standard error if the sample size increased to 50?

4.74= 15/ 3.16

Standard Error
• The  of an IQ test is 15. If you sampled 10 people and found an X = 105 what is the standard error of that mean? What happens to the standard error if the sample size increased to 50?

4.74= 15/ 3.16

2.12 = 15/7.07

Standard Error
• The bigger the sample size the smaller the standard error
• Makes sense!
Question
• For an IQ test
•  = 100
•  = 15
• What is the probability that in a class the average IQ of 54 students will be below 95?
• Note: This is different then the other “z” questions!
Z score for a sample mean

Z = (X - ) / x

Step 2: Calculate the Standard Error

15 / 54 = 2.04

-3 -2 -1 0 1 2 3

Step 3: Calculate the Z score

(95 - 100) / 2.04 = -2.45

-3 -2 -1 0 1 2 3

Step 4: Look up Z score in Table

Z = -2.45; Column C =.0071

.0071

-3 -2 -1 0 1 2 3

Question
• From a sample of 54 students the probability that their average IQ score is 95 or lower is .0071