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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
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
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
Step 1: Sketch out question

-3 -2 -1 1 2  3 

step 1 sketch out question6
Step 1: Sketch out question

110

-3 -2 -1 1 2  3 

step 2 calculate z score
Step 2: Calculate Z score

(110 - 100) / 15 = .66

110

-3 -2 -1 1 2  3 

step 3 look up z score in table
Step 3: Look up Z score in Table

Z = .66; Column C = .2546

110

.2546

-3 -2 -1 1 2  3 

example iq9
Example: IQ
  • You have a .2546 probability (or a 25.56% chance) of randomly bumping into a person with an IQ over 110.
slide10
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
Population
  • You are interested in the average self-esteem in a population of 40 people
  • Self-esteem test scores range from 1 to 10.
population scores
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
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
Group Activity
  • Randomly select 5 people and find the average score
group activity17
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
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
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
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

slide23
Mean
  • The expected value of the mean for a sampling distribution
  • E (X) = 
standard error
Standard Error
  • The standard error (i.e., standard deviation) of the sampling distribution

x = / N

standard error25
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 error26
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 error27
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 error28
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 error29
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 error30
Standard Error
  • The bigger the sample size the smaller the standard error
  • Makes sense!
question
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 score for a sample mean

Z = (X - ) / x

step 2 calculate the standard error
Step 2: Calculate the Standard Error

15 / 54 = 2.04

-3 -2 -1 0 1 2 3

step 3 calculate the z score
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
Step 4: Look up Z score in Table

Z = -2.45; Column C =.0071

.0071

-3 -2 -1 0 1 2 3

question37
Question
  • From a sample of 54 students the probability that their average IQ score is 95 or lower is .0071