1. Probability and the Sampling Distribution Quantitative Methods in HPELS 440:210

2. Agenda Introduction Distribution of Sample Means Probability and the Distribution of Sample Means Inferential Statistics

3. Introduction Recall:  Any raw score can be converted to a Z-score Provides location relative to µ and   Assuming NORMAL distribution: Proportion relative to Z-score can be determined Z-score relative to proportion can be determined  Previous examples have looked at single data points  Reality  most research collects SAMPLES of multiple data points Next step  convert sample mean into a Z- score  Why? Answer probability questions

4. Introduction Two potential problems with samples: 1. Sampling error  Difference between sample and parameter 2. Variation between samples  Difference between samples from same taken from same population  How do these two problems relate?

5. Agenda Introduction Distribution of Sample Means Probability and the Distribution of Sample Means Inferential Statistics

6. Distribution of Sample Means Distribution of sample means = sampling distribution is the distribution that would occur if:  Infinite samples were taken from same population  The µ of each sample were plotted on a FDG Properties:  Normally distributed  µ M = the “mean of the means”   M = the “SD of the means” Figure 7.1, p 202

8. Distribution of Sample Means Sampling error and Variation of Samples Assume you took an infinite number of samples from a population  What would you expect to happen?  Example 7.1, p 203

9. Assume a population consists of 4 scores (2, 4, 6, 8) Collect an infinite number of samples (n=2)

11. Total possible outcomes: 16 p(2) = 1/16 = 6.25%p(3) = 2/16 = 12.5% p(4) = 3/16 = 18.75%p(5) = 4/16 = 25% p(6) = 3/16 = 18.75%p(7) = 2/16 = 12.5% p(8) = 1/16 = 6.25%

12. Central Limit Theorem For any population with µ and , the sampling distribution for any sample size (n) will have a mean of µ M and a standard deviation of  M, and will approach a normal distribution as the sample size (n) approaches infinity If it is NORMAL, it is PREDICTABLE!

13. Central Limit Theorem The CLT describes ANY sampling distribution in regards to: 1. Shape 2. Central Tendency 3. Variability

14. Central Limit Theorem: Shape All sampling distributions tend to be normal Sampling distributions are normal when:  The population is normal or,  Sample size (n) is large (>30)

15. Central Limit Theorem: Central Tendency The average value of all possible sample means is EXACTLY EQUAL to the true population mean  µ M = µ If all possible samples cannot be collected?  µ M approaches µ as the number of samples approaches infinity

16. µ = 2+4+6+8 / 4 µ = 5 µ M = 2+3+3+4+4+4+5+5+5+5+6+6+6+7+7+8 / 16 µ M = 80 / 16 = 5

17. Central Limit Theorem: Variability The standard deviation of all sample means is denoted as  M  M =  /√n Also known as the STANDARD ERROR of the MEAN (SEM)

18. SEM  Measures how well statistic estimates the parameter  The amount of sampling error between M and µ that is reasonable to expect by chance Central Limit Theorem: Variability

19. SEM decreases when:  Population  decreases  Sample size increases Other properties:  When n=1,  M =  (Table 7.2, p 209)  As SEM decreases the sampling distribution “tightens” (Figure 7.7, p 215)  M =  /√n

21. Agenda Introduction Distribution of Sample Means Probability and the Distribution of Sample Means Inferential Statistics

22. Probability  Sampling Distribution Recall:  A sampling distribution is NORMAL and represents ALL POSSIBLE sampling outcomes  Therefore PROBABILITY QUESTIONS can be answered about the sample relative to the population

23. Probability  Sampling Distribution Example 7.2, p 209 Assume the following about SAT scores:  µ = 500   = 100  n = 25  Population  normal What is the probability that the sample mean will be greater than 540? Process: 1. Draw a sketch 2. Calculate SEM 3. Calculate Z-score 4. Locate probability in normal table

24. Step 1: Draw a sketch Step 2: Calculate SEM SEM =  M =  /√n SEM = 100/√25 SEM = 20 Step 3: Calculate Z-score Z = 540 – 500 / 20 Z = 40 / 20 Z = 2.0 Step 4: Probability Column C p(Z = 2.0) = 0.0228

25. Agenda Introduction Distribution of Sample Means Probability and the Distribution of Sample Means Inferential Statistics

26. Looking Ahead to Inferential Statistics Review:  Single raw score  Z-score  probability Body or tail  Sample mean  Z-score  probability Body or tail What’s next?  Comparison of means  experimental method

28. Textbook Assignment Problems: 13, 17, 25 In your words, explain the concept of a sampling distribution In your words, explain the concept of the Central Limit Theorum