4. 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?

5. Step 1: Sketch out question
-3 -2 -1  1 2  3 

6. Step 1: Sketch out question
110
-3 -2 -1  1 2  3 

7. Step 2: Calculate Z score
( ) / 15 = .66
110
-3 -2 -1  1 2  3 

8. Step 3: Look up Z score in Table
Z = .66; Column C = .2546
110
.2546
-3 -2 -1  1 2  3 

9. Example: IQ
You have a probability (or a 25.56% chance) of randomly bumping into a person with an IQ over 110.

10. 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!

12. Population
You are interested in the average self-esteem in a population of 40 people
Self-esteem test scores range from 1 to 10.

13. 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

14. Histogram

15. 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?

16. Group Activity
Randomly select 5 people and find the average score

17. 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

18. EXCEL

19. 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

20. INTERNET EXAMPLE

21. 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

22. 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

23. Mean The expected value of the mean for a sampling distribution

24. Standard Error
The standard error (i.e., standard deviation) of the sampling distribution
x = / N

25. 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

26. 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

27. 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

28. 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

29. 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

30. Standard Error
The bigger the sample size the smaller the standard error
Makes sense!

31. 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!

32. Z score for a sample mean
Z = (X - ) / x

33. Step 1: Sketch out question

34. Step 2: Calculate the Standard Error
15 / 54 = 2.04

35. Step 3: Calculate the Z score
( ) / 2.04 = -2.45

36. Step 4: Look up Z score in Table
Z = -2.45; Column C =.0071
.0071

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