Stat 217 – Day 20 Comparing Two Proportions The judge asked the statistician if she promised to tell the truth, the whole truth, and nothing but the truth? The statistician replied, “Yes, 95% of the time.”

Last Time – Two-sample t procedures Observed about 7 letters Parameter:  JFK –  JFKC = difference in “population” means Populations = all potential JFK memorizers, all potential JFKC memorizes Activity 22-1 (p. 439)   men –  women = difference in population mean number of close friends

Last Time – Two-sample t procedures If the goal is to compare two population means  Random samples: Does the average number of close friends differ between men and women?  Random assignment: Does the average melting time differ between chocolate chip and butterscotch (first chip)?  Ho: 1 1 – 2 2 = 0 (1 (1 = 2 2 )  Ha: 1 1 – 2 2, ≠ 0 (1 (1 ≠ 2 2 )

Last Time – Two-sample t procedures Technical conditions  Both sample sizes are large or both populations are normally distributed  Random assignment to two groups or independent random samples from two populations  Classic counter example: paired data Everyone times both chips Appropriate analysis: one sample t test on differences   = mean difference between times in Cal Poly population

Last Time – Two-sample t procedures Test of Significance Calculator: Two means  Test statistic  p-value = 0019 About.19% of random assignments have a difference in sample means of 7.29 or larger if population means equal Confidence interval for  1 –  2  Example: (2.706, )  I’m 95% confident that the JFK population will remember 2.71 to more words, on average, than JFKC population

Example: Lab 2 Is yawning contagious? Is this difference in the conditional (sample) proportions larger than we would expect from the random assignment process alone? Yawn seed No seedTotal Yawned10414 No Yawn Total341650

Example: Lab 2 Parameter:  seed –  no seed = the difference in the probability of yawning between the two treatments (or difference in proportion of all potential yawn seed people vs. all potential no seed population) Null Ho:  seed –  no seed = 0 (  seed =  no seed ) Alt Ha:  seed –  no seed > 0 (  seed >  no seed )

Example: Lab 2 If the yawn seed has no effect, about 52% of random assignments would have a difference in sample proportions of at least.044 by chance alone No evidence against null hypothesis

Can we use the normal distribution? Yes if  Random assignment or random samples  Large sample sizes (at least 5 successes and 5 failures in each group – four numbers to check) Test statistic p-value

Informal Confidence Interval Estimate + margin of error (.136) = (-.228,.316) We would be 95% confident that the difference in population proportions is between and.316 (zero is a plausible value!) If had been (.228,.316): 95% confident that the probability of yawning with the yawn seed is.228 to.316 higher than the probability without a yawn seed

To turn in with partner: Activity 21-2 (p. 419) (a) random sampling or random assignment? (b) Null and alternative hypotheses (c) Conclusions? For Tuesday Review p. 418, 420 Activity 21-6 (self-check) HW 6

Rest of Topic 21 Activity 21-3: Same procedure with randomized experiments but can draw cause- and-effect conclusions (but may not be generalizing) Activity 21-4: Larger samples lead to smaller p-values (all else the same)  71% vs. 81% Activity 21-5: Don’t jump to cause and effect conclusions!