1. Chapter 7: The Distribution of Sample Means

2. Frequency of Scores 2
Frequency 1 1 2 3 4 5 6 7 8 9
Scores

3. Distribution of Sample Means
The distribution of sample means is the collection of sample means for all the possible random samples of a particular size n that can be obtained from a population.

4. Sampling Distribution
A sampling distribution is a distribution of statistics obtained by selecting all the possible sample of a specific size from a population.

5. Table (samples, score, means)
Scores
Sample Mean
Sample
First
Second X 1 2 4 3 6 8 5 7 9 10 11 12 13 14 15 16

6. Frequency of sample means4 3
Frequency 2 1 1 2 3 4 5 6 7 8 9
Sample Means

7. Central Limit Theorem
For any population with mean µ and standard deviation , the distribution of sample means for sample n will approach a normal distribution with a mean of µ and a standard deviation as n approaches infinity.

8. The distribution of sample means will be almost perfectly normal if either of the following is true:
The population from which the samples are selected is a normal distribution.
The Number of scores (n) in each sample is relatively large, around 30 or more.

9. The mean of the distribution of the sample means will be equal to µ (the population mean) and is called the expected value of x
Mean of all the scores in the population
Mean of all the sample means

10. The standard deviation of the distribution of sample means is called the :
Standard error of X
Standard error =
(standard distance between X and µ)

11. Standard error determined by 2 characteristics:
Variability of the population from which the sample came
The size of the sample

12. Law of Large Numbers
The larger the sample size (n), the more probable it is that the sample mean will be close to the population mean.

13. Probability and Distribution of meansX
500
540 z 1 2

14. Normal curve of means 20
40%
40% X
474.4
500
525.6 z
-1.28
+1.28

15. Graph with grid lines

16. Sample Size and Sampling Error
Assume a population with a mean of 50 and a standard deviation of 15

17. Means presented in a table
Group n
Mean SE
Control 17
32.23
2.31
Experimental 15
45.17
2.78

18. Group A vs. Group B bar graph35 30
X score ( + SE) 25 20 15 10 5
Group A
Group B

19. Group A vs. Group B line graph
X number of mistakes ( + SE) 30 25 20
Group A 15 10
Group B 5 1 2 3 4
Trials

20. Comparison of curves A, B, and C60 64 a 60 64 b 60 64 c
Column B
p =
Column C
p =

21. IQ Scores
 = 15  X
µ = ?

22. Given a population of test scores that is normally distributed with µ = 60 and  = 8
I randomly select a test score. What is the probability that the score will be more than 16 points away from the mean?
(Hint : What proportion of test scores are > 76 or < 44 ?)