Agenda

1. Agenda Review In-Class Group Problems Review

2. 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?

3. 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)

4. 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

5. 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)

6. 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).

7. 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

8. 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

9. 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 = +/- 2.042

10. 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)