1. Sampling Distributions & Standard Error Lesson 7

2. Populations & Samples n Research goals l Learn about population l Characteristics that widely apply l Impossible/impractical to directly study n Research methods l Study representative sample l Introduce sampling error l ~

3. Sampling Error n l Difference between sample statistic and population parameter l result of choosing random sample n Many potential samples l With different ~

4. Sampling Distributions n Samples from a single population l Repeatedly draw random samples l Every possible combination n Calculate a test statistic (e.g., t test) l One-sample: l or l Independent samples: n Results  sampling distribution  and  ~

5. The Distribution of Sample Means n Distribution of means for many samples from a single population l Repeatedly draw random samples l Calculate n Sampling variation (or sampling error) l will differ from population l different shape l similar mean larger sample  closer to  ~

6. Samples: n=10 #1 #2 #3 #4

7. Law of Large Numbers n Large sample size (n) l give better estimates of parameters l i.e., better fit l:l: l:l: l:l: l:l:

8. Parameters: Distribution of X n Results in narrower distribution Has  and  n Find exact values l take all possible samples l or apply Central Limit Theorem ~

9. Central Limit Theorem n 1. n 2. l APA style: SE l Also SEM ~ or

10. Central Limit Theorem n 3. As sample size (n) increases l the sampling distribution of means approaches a normal distribution l even if parent population not normal distribution of variable (or X) n Very Important! In n ≥ 6, then… l probabilities from standard normal distribution useful l Because we study samples ~

11. Distributions: X i vs X 1301008570 f IQ Score  = 100  = 15 n = 9 5 115 95 105 mean IQ Score 90 110

12. Standard Error of the Mean: Magnitude n Small standard error  better fit l sample means close m l More representative sample Depends on n and  large sample size & small  little control  l can increase sample size increase value of denominator ~

13. Using the distribution of X n Use samples to describe populations l is it representative of population? l how close is ? n Sample means normally distributed n Use z table l find area under curve l only slight difference in z formula ~

14. Conducting an experiment n Same as randomly selecting... l n For a sample size n with mean =  l & standard error

15. Calculating z scores

16. How close is X to  ? n means are normally distributed n Use area under curve l between mean and 1 standard error above the mean l 34% n Same rules as any normal distribution l compute z score ~

17. Distribution of Sample Means is Normal 120-2.34.14 f standard error of mean.02.34.14.02

18. z scores & Distribution of X n What are z scores that define boundaries of middle 95% of ? l p in left & right tails =.025 +.025 l Look up z scores l Left tail = - 1.96; right tail = + 1.96 n Boundaries for middle 99% of ? ~

19. Distribution of Sample Means is Normal 120-2 f z scores Boundaries for middle 95% (or.95) of sample means? -1.96+1.96 for middle 99% (or.95) of sample means? -2.58+2.58

20. Using z scores Sample Mean z score area under curve or proportion Or probability or percentage Table: large/smaller portion column Table: z column