Samplig Distributios, Z-Tests, Power We draw ifereces about populatio parameters from sample statistics Sample proportio approximates populatio proportio Sample mea approximates populatio mea Sample variace (usig -1) approximates populatio variace Etc. Statistics vary from oe sample to the ext If a statistic is ubiased, the o average over may samples, it will equal the populatio parameter There will be variability aroud that average The distributio of a statistic is the samplig distributio We will cosider samplig distributios for Sample meas Sample proportios assumig Beroulli processes I.e., possible outcomes, ad the biomial distributio applies Note that for a Beroulli process, if we use the radom variable (0, 1) to deote a failure ad a success, respectively, the the sample proportio is the mea of Thus, the sample proportio is also a sample mea The shape of the samplig distributio depeds o the populatio distributio ad o the sample size 1
For the samplig distributio of the mea Samplig distributio mea populatio mea E ( ) µ µ Samplig distributio variace (populatio variace)/sample size Stadard deviatio of the samplig distributio is called the stadard error of the mea The Cetral Limit Theorem For samples of size take from a populatio with mea µ ad stadard deviatio E µ ( ) µ As icreases, the samplig distributio approaches the ormal distributio Holds for all populatio distributios The fact that decreases with is very importat ad useful
Apply this reasoig to the biomial distributio Recall µ E () s p ad p( 1 p) Istead of umber of successes, let s work with the proportio of successes Divide s by to get Divide expected mea by ad divide variace by to get p( 1 p) p( 1 p) s E E ( ) p The Cetral Limit Theorem applies to the samplig distributio for the proportio of successes i Beroulli trials s The biomial distributio applied to sample proportios for samples of size s E E ( ) p ( 1 p) As icreases, the distributio of sample proportios approaches the ormal p ( 1 p) Kowig or assumig the populatio proportio, we ca use the table of the stadard ormal distributio to determie the probability of obtaiig sample proportios withi ay iterval or beyod ay poit p 3
The samplig distributio of the mea (agai) E ( ) µ µ As icreases, the distributio approaches the ormal Kowig or assumig the populatio mea ad stadard deviatio, we ca use the table of the stadard ormal distributio to determie the probability of obtaiig sample meas withi ay iterval or beyod ay poit Thus, we ca test simple hypotheses about sample meas (called the z-test) Calculate z-score for sample mea, z obt Compare to critical z-score, z crit 0.03 Samplig Distributio - Large (Distributio is Normal) f(mea) 0.0 0.01 0 40 60 80 100 10 140 160 µ Sample mea 4
Covert sample mea to z obt Compare to z crit, which is either z a for a 1-tailed test of z a/ for a -tailed test Decide whether or ot to reject H 0 f(z) 0.50 0.40 0.30 0.0 0.10 Two-tailed test.95 0.00-5 -4-3 - -1 0 1 3 4 5 z z.05 1.96 z.05 1.96 Two reasos large sample sizes are importat Samplig distributio approaches ormal Power of the test icreases Note square root of i deomiator of stadard error To calculate power for simple z-test ( µ ) ( µ µ µ > µ, or µ < ) State H 0 ull ad H 1 real ull, real ull real µ ull Determie z crit, which depeds o 1-tailed versus -tailed test ad o alpha Determie sample size, Determie sample outcome(s) that would reject H 0 Assume a particular H 1 Covert the sample outcome(s) leadig to rejectio of H 0 to z-scores uder the assumed H 1 Use the table of the stadard ormal dist to determie the probability of the the outcome(s) uder H 1 That is the power of the test 5