Inference for Means (C23-C25 BVD)

* Unless the standard deviation of a population is known, a normal model is not appropriate for inference about means. Instead, the appropriate model is called a t-distribution. * T-distributions are unimodal and symmetric like Normal models, but they are fatter in the tails. The smaller the sample size, the fatter the tail. * In the limit as n goes to infinity, the t-distribution goes to normal. * Degrees of freedom (n-1) are used to specify which t- distribution is used. * T-table only has t-scores for certain df, and the most common C/alphas. If using table and desired value is not shown, tell what it would be between, or err on the side of caution (choose more conservative df, etc.) * Use technology to avoid the pitfalls of the table when possible.

* X-bar +/- t* df (Sx/sqrt(n)) * Sample statistic +/- ME * ME = # standard errors reaching out from statistic. * T-interval on calculator

* Draw or imagine a normal model with C% shaded, symmetric about the center. * What percent is left in the two tails? * What percentile is the upper or lower fence at? * Look up that percentile in t-table to read off t(or use invt(.95,df) or whatever percent is appropriate)

* ME = t*(SE) Plug in desired ME (like within 5 inches means ME = 5). Plug in z* for desired level of confidence (you can’t use t* because you don’t know df). Plug in standard deviation (from a sample or a believed true value, etc. Solve equation for n.

* For inference for means check: * 1. Random sampling/assignment? * 2. Sample less than 10% of population? * 3. Nearly Normal? – sample size is >30 or sketch histogram and say could have come from a Normal population. * 4. Independent – check if comparing means or working with paired means * 5. Paired - check if data are paired if you have two lists

* Null: µ is hypothesized value * Alternate: isn’t, is greater, is less than * Hypothesized Model: centers at µ, has a standard deviation of s/sqrt(n) * Find t-score of sample value using n-1 for df * Use table or tcdf to find area of shaded region. (double for two-tail test). * T-test on calculator– report t, df and p-value.

* If data are paired, subtract higher list – lower list to create a new list, then do t-test/t-interval. * If data are not paired: * Check Nearly Normal for both groups – both must individually be over 30 or you have to sketch each group’s histogram and say could’ve come from normal population * CI: mean1-mean2 +/- ME --- use calculator because finding df (and therefore also t*) is rather complicated. * SE for unpaired means is sqrt(s1 2 /n1 + s2 2 /n2) * If calculator asks “pooled” – choose “No”. * Null for paired: µ d = 0 (usually) * Null for unpaired: µ 1 - µ 2 = 0 * Don’t forget to define variables. * Use 2-Sample T-Test and 2-Sample T-Interval in calculator for data that are not paired.

* State: name of test, hypothesis if a test, alpha level if a test, define variables * Plan: check all conditions – check marks and “yes” not good enough * Do: interval for intervals, test statistic, df (if appropriate) and p-value for tests It is good to write the sample difference if doing inference for two proportions or two means, but make sure no undefined variables are used * Conclude: Interpret Confidence Interval or Hypothesis Test – See last slide show for what to say