106 = = = ( ) The single tail is p value = = both tails

Transcription

1 Section 4: Comparing Proportions ) In the 980's, a simple random sample of 84 job applicants revealed that 5 of them had lied on their resumes. In the 2000's, a SRS of 06 revealed that 2 had lied on their resumes. Is there evidence (choose your own reasonable level) that the proportion who lie on their resumes has changed? p proportion of population who lie on resumes in 980 s p 2 proportion of population who lie on resumes in 2000 s p p pooled p H 0 : p p 2 H a : p p SE pooled p ( p ) n + n ( 0.895) z p p SE pooled z 0.34 The single tail is p value both tails is not "small"; at any reasonable level: no evidence of H a ; do not reject H 0 Is there evidence that the proportion who lie on their resumes has changed has increased? H 0 : p p 2 H a : p < p 2 p value single tail is not "small"; at any reasonable level: no evidence of H a ; do not reject H 0 2) In the 2000 season baseball season, the New York Yankees played 80 games at home and 8 games away. They won 44 of their home games and 43 of their away games. We can consider these games as samples from potentially large populations of games played at home and away. Is there evidence that they're more likely to win at home than away? p proportion of all home games won p 2 proportion of away games won p p H 0 : p p 2 H a : p > p 2

2 pooled p SE pooled p ( p ) n + n ( ) z p p SE pooled z The single tail is is not "small"; at any reasonable level: no evidence of H a ; do not reject H 0 3) Farmers in two Indiana counties were asked whether they favor a certain government program. In Tippecanoe County, 263 favored and 252 opposed. In Benton County, 260 favored and 377 opposed. Is there evidence that opinion is different (i.e., that different proportions of all farmers favor the program) in the two counties? p proportion of all in Tippecaneo who favor p 2 proportion of all in Benton who favor p p pooled p H 0 : p p 2 H a : p p SE pooled p ( p ) n + n ( 0.895) z p p SE pooled z 3.47 The single tail is p value both tails is "small"; at any reasonable level: evidence of H a ; reject H 0 4) A new type of aspirin is being tested for effectiveness. 63 of the 78 people given the aspirin got better, and 43 of 77 people given a placebo got better. Is there evidence that the aspirin actually works (i.e., better than the placebo, which has no medical effect at all)? p proportion of all who d get better from aspirin p 2 proportion of all who d get better from placebo p p

3 pooled p H 0 : p p 2 H a : p > p SE pooled p ( p ) n + n ( ) z p p SE pooled z The single tail is is "small"; at any reasonable level: evidence of H a ; reject H 0 5) A sample of 02 men and 062 women (over the age of 65) revealed that 4 men and 535 women had arthritis. Do we have evidence that the proportions of men and women (over the age of 65) who have arthritis are different? p proportion of all older men who have arthritis p 2 proportion of all older women who have arthritis p p pooled p H 0 : p p 2 H a : p p SE pooled p ( p ) n + n ( 0.456) z p p SE pooled z 4.46 The single tail is 0 (off the chart) doubling, it s still "small"; at any reasonable level: evidence of H a ; reject H 0 6) 47 of 222 men surveyed have bought books online, and likewise 37 of 208 women. (a) Do we have evidence that men are more likely than women to buy books online at the 5%, 0%, 20% levels? p proportion of all men who buy books online p 2 proportion of women who buy books online p p H 0 : p p 2

4 pooled p H a : p > p SE pooled p ( p ) n + n ( 0.953) z p p SE pooled z The single tail is ; no evidence of H a ; do not reject H ; no evidence of H a ; do not reject H < 0.20; evidence of H a ; reject H 0 (b) Do we have evidence that the proportions of all men and of all women who've bought books online are different? H 0 : p p 2 H a : p p 2 P value is is not "small"; at any reasonable level: no evidence of H a ; do not reject H 0 7) Two ways of treating Carpal Tunnel Syndrome are with surgery and wrist splints. Of 88 people given surgery, 70 showed improvement. Of 88 given wrist splints, 48 showed improvement. Do we have evidence that surgery works better? p proportion of all surgery who d improve p 2 proportion of all wrist splints who d improve p p pooled p H 0 : p p 2 H a : p < p SE pooled p ( p ) n + n ( ) z p p SE pooled z The single tail is 0 (off the chart) "small"; at any reasonable level: evidence of H a ; reject H 0

5 8) A salesman for a new type of cell phone claims not only that his brand costs less, but that the percentage of defective phones will be no higher than those of his competitor. To test this statement, a retailer took random samples of each manufacturer's product. Of the salesman's product, he checked 50 phones and 5 were defective. Of the competitor's, he checked 50 and 6 were defective. Can we reject the salesman's claim at the 5% significance level? At the % significance level? p proportion of all salesmen s phones that are defective p 2 proportion of competitor s phones that are defective p p H 0 : p p 2 H a : p > p 2 pooled p SE pooled p ( p ) n + n ( 0.07) z p p SE pooled z The single tail is < 0.05; evidence of H a ; reject H ; no evidence of H a ; do not reject H 0 9) In a survey of working parents (both parents working), one of the questions asked was, "Have you refused a job, promotion, or transfer because it would mean less time with your family?" 200 men and 200 women were asked this question, and 29% of the men and 24% of the women responded, "yes". (a) Based on this survey, can we conclude that there is a difference in the proportion of men and women responding "yes" at the 5% significance level? p proportion of all men who d refuse a job p 2 proportion of women who d refuse a job p p pooled p H 0 : p p 2 H a : p p SE pooled p ( p ) n + n ( 0.265)

6 z p p SE pooled z +.3 The single tail is p value both tails ; no evidence of H a ; do not reject H 0 (b) Can we conclude (at the 5%, 0%, 5% levels) that men are more likely to refuse a job in this way? H 0 : p p 2 H a : p > p 2 The single tail is p value ; no evidence of H a ; do not reject H ; no evidence of H a ; do not reject H < 0.5; evidence of H a ; reject H 0 0) Does sleep improve mental performance? Is creativity and problem-solving linked to adequate sleep? A group of German scientists divided 200 volunteers into two equal-sized groups. Prior to examination, one group received eight hours of sleep while the other group stayed awake all night. The volunteers took a complicated math test that involved transforming a string of digits into a new string that fit a set of given rules, as well as a hidden rule. Of the volunteers who slept, 39 discovered the hidden rule; of the group that stayed awake, 5 discovered the hidden rule. Do we have evidence at the % level that eight hours of sleep enhanced discovery of the hidden rule? p proportion of people who sleep who would discover the rule p 2 proportion of people who don t sleep who would discover the rule p p pooled p H 0 : p p 2 H a : p > p SE pooled p ( p ) n + n ( 0.27) z p p SE pooled z The single tail is 0 (off the chart) "small"; at any reasonable level: evidence of H a ; reject H 0