Sampling Distributions & Standard Error

1. Sampling Distributions & Standard Error Lesson 7

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

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

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

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

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

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

8. Parameters: Distribution of X • Results in narrower distribution • Has m and s • Find exact values • take all possible samples • or apply Central Limit Theorem ~

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

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

11. Distributions: Xi vs X m = 100 s= 15 n = 9 f 5 70 85 100 115 130 105 110 95 90 IQ Score mean IQ Score

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

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

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

15. Calculating z scores

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

17. .34 .02 .02 .14 -2 -1 0 1 2 Distribution of Sample Means is Normal f .34 .14 standard error of mean

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

19. -2 -1 0 1 2 Distribution of Sample Means is Normal Boundaries for middle 95% (or .95) of sample means? for middle 99% (or .95) of sample means? f -1.96 +1.96 -2.58 +2.58 z scores

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