Sampling Distributions Standard Error Lesson 7 Populations Samples

Sampling Distributions & Standard Error Lesson 7

Populations & Samples 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 ~ n

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

Sampling Distributions 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 l m and s ~ n

The Distribution of Sample Means 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 l larger sample closer to m ~ n

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

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

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

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

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 ~

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

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

Using the distribution of X 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 ~ n

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

Calculating z scores

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

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

z scores & Distribution of X 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 ? ~ n

Distribution of Sample Means is Normal Boundaries for middle 95% (or. 95) of sample means? for middle 99% (or. 95) of sample means? f -2 -2. 58 -1. 96 -1 0 z scores 1 2 +1. 96 +2. 58

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