AP Statistics Review

1. AP Statistics Review Inference for Means (C23-C25 BVD)

2. 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. T-distributions

3. X-bar +/- t*df(Sx/sqrt(n)) • Sample statistic +/- ME • ME = # standard errors reaching out from statistic. • T-interval on calculator Confidence Interval for 1 Mean

4. 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) Finding the critical value (t star)

5. 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. Finding Sample Size

6. 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 Conditions/ Assumptions to Check

7. 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. Hypothesis Test for 1 mean

8. 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(s12/n1 + s22/n2) • If calculator asks “pooled” – choose “No”. • Null for paired: µd = 0 (usually) • Null for unpaired: µ1 - µ2 = 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. Inference for 2 Means

9. 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 What to Write