AGENDA Review In-Class Group Problems Review

Homework #3 Due on Thursday Do the first problem correctly Difference between what should happen over the long run and what happens in your one random sample. Estimation: figure out what the “estimate” is (mean vs. proportion). You may have to calculate the proportion based on the data. Significance tests One tailed or two? Large (z) or small (t) sample size?

Sampling Distributions Based on probability theory, what would happen if we did an infinite number of random samples and plotted some outcome For large samples, many “outcomes” are normally distributed (z- distribution) For smaller samples, the distribution is a bit flatter (t-distribution)

Dispersion for Sampling Distributions Because sampling distributions are based on repeated samples from some population… The dispersion of the distribution depends on the size of the samples (N) being drawn The STANDARD ERROR is the measure of dispersion used for a sampling distribution N is used to calculate the standard error

The use of Sampling Distributions Estimation: the sampling distribution for means and proportions (assuming N>100) is normal Using the z-distribution, we know that 95% of sample outcomes will be within 1.96 standard errors of the population parameter (P or μ) Therefore, any single sample p or x has a 95% chance of being within 1.96 standard errors of the population parameter (P or μ)

The use of Sampling Distributions Significance Testing: Assuming the null hypothesis is true, and we did an infinite number or random samples… Sampling distributions for “test statistics” Test statistics (obtained z, obtained t) are just another sample “outcome” Single sample z test With large sample (N>100), assuming the null is true, there is a 95% chance of getting an obtained z score within 1.96 standard errors of zero (null).

What z and t “obtained” tell us For single sample significance tests They indicate the number of standard errors between the population mean (μ) and the sample mean (x) If the null was true, this number should be small—95% of all sample “obtained z’s” should be within 1.96 standard errors of zero In other words, if the null is true, there is only a 5% chance of getting an obtained z outside of +/- 1.96

Critical values and critical regions From sampling distributions based on the assumption that null is true Critical region Defined by alpha (choice of researcher) Area under sampling distribution where null is rejected Finding so rare, if null were true, that we reject null Critical value The test statistic associated with the critical region (defines critical region) For alpha of.05, two tailed test, z-critical = +/- 1.96

t-distribution Necessary because the sampling distributions that result from smaller N’s are not perfectly normal (they are flatter) Area under the curve slightly different Critical t-values depend on sample size (technically, degrees of freedom) Same logic for “critical values” From chart, for alpha of.05, two tailed test, and N of 31, t-critical = +/

t vs. z distributions For significance testing, both are sampling distributions of “obtained” stats under the assumption that the null is true As sample size gets larger, t morphs into z Look at the “infinity” row in the t chart (with an infinitely large sample size, t = z)