Large Sample Hypothesis Tests for a Population Proportion

Transcription

1 Ntes-10.3a Large Sample Hypthesis Tests fr a Ppulatin Prprtin ***Cin Tss*** 1. A friend f yurs claims that when he tsses a cin he can cntrl the utcme. Yu are skeptical and want him t prve it. He tsses the cin, and yu call heads; it s tails. Yu try again and lse again. a) D tw lsses in a rw cnvince yu that he really can cntrl the tss? Explain. b) Yu try a third time, and again yu lse. Wuld three lsses in a rw cnvince yu that yur friend cheats? Explain using prbability. c) Hw many times in a rw wuld yu have t lse in rder t be pretty sure that this friend really can cntrl the tss? Justify yur answer by calculating a prbability. ***Deck f Cards*** 1. p = prprtin f red cards in a deck p = prprtin f red cards in a deck

2 2. Yu are cncerned that a majrity f smkers d nt view themselves as being at increased risk f heart disease r cancer. Because f this, the authrs call fr a public health campaign t educate smkers abut the assciated risks. In supprt f this recmmendatin, the authrs ffered the results f a study f 737 current smkers selected at randm frm U.S. husehlds with telephnes. Of the 737 smkers surveyed, 295 indicated that they believed they have a higher than average risk f cancer. Des this data suggest that p, the true prprtin f smkers wh view themselves as being at increased risk f cancer is, in fact, less that.5 as claimed by the authrs f the paper? Use a significance level f.05. Ntes Hypthesis must be abut the ppulatin, p. H : 0 p = H : p < r p > r p A Hypthesis Test SRS? Apprx Nrmal? Independence? 5. We use ur ppulatin, p, when satisfying ur cnditins and finding ur standard deviatin. statistic - parameter pˆ p z = = standard deviatin p(1 p) n 6. We either: Reject H Fail t Reject H All cnclusins are made abut the alternative hypthesis Hw d yu interpret the p-value:

3 Ntes-10.3b Large Sample Hypthesis Tests fr a Ppulatin Prprtin This is an Essential Skills Mastery Prblem. Yu must cmplete in class and turn in when it is 100% crrect. Yur instructr will crrect it befre it is turned in. 1. A preacher wuld like t establish that f peple wh pray, less than 80% pray fr wrld peace. In a randm sample f 110 persns, 77 f them said that when they pray, they pray fr wrld peace. Test at 10% level f significance. 2. The US Department f Transprtatin reprted that 77% f all fatally injured autmbile drivers were intxicated. A randm sample f 150 recrds in Kit Carsn Cunty, Clrad, shwed that 104 invlved a drunk driver. Use a 1% significance level t decide whether r nt there is evidence that indicates the ppulatin prprtin f driver fatalities related t alchl is different than 77%.

4 Ntes Type I and II Errr, Pwer f a Test Nt Making Errrs Pwer: Making Errrs Type I Errr: Type II Errr: If yu are using a 95% cnfidence level. It is bvius that 5% f the time yu will be wrng. There are tw different ways f being wrng. The null hypthesis is false The null hypthesis is true Reject H Fail t Reject H T reduce errrs: T increase pwer:

5 1. A stats prfessr has bserved that fr several years abut 13% f the students wh initially enrll in his stats curse withdraw befre the end f the semester. A salesman suggests that he try a statistics sftware prgram that gets students mre invlved with cmputers, predicting that it will cut the drput rate. The sftware is expensive, and the salesman ffers t let the prfessr use it fr a semester t see if the drput rate decreases. The prfessr will have t pay fr the sftware nly if he chses t cntinue using it. He des a hypthesis test with 5% significance level. a) Write the null and alternate hyptheses. b) In cntext, explain what wuld happen if the prfessr makes a Type I errr. c) In cntext, explain what wuld happen if the prfessr makes a Type II errr. d) What is meant by the pwer f this test? e) If the hypthesis test is tested at the 1% significance level instead f 5%, hw will this affect the pwer f the test? f) The prfessr wanted t base his decisin n 100 students, but he was nly able t use 70 students. Hw wuld this affect the pwer f the test? Explain. 2. Prductin managers n an assembly line must mnitr the utput t be sure that the level f defective prducts remains small, They peridically inspect a randm sample f the items prduced. If they find a significant increase in the prprtin f items that must be rejected, they will halt the assembly prcess until the prblem can be identified and repaired. a) In this cntext, what is a Type I errr? b) What are the cnsequences f a Type I errr? c) In this cntext, what is a Type II errr? d) What are the cnsequences f a Type II errr? e) Which type f errr wuld the factry wner cnsider mre serius? f) Which type f errr wuld the custmer cnsider mre serius?

6 Ntes-10.4a Hypthesis Test fr Large Sample Ppulatin Mean 1. A teacher s unin wuld like t establish that the average salary fr high schl teachers in a particular state is less than $ A randm sample f 100 public high schl teachers in the particular state has a mean salary f $31,578. The sample standard deviatin is $4,415. Test the unin s claim at the 5% level f significance. Ntes Hypthesis must be abut the ppulatin, µ. H : 0 µ = H : µ < r µ > r µ A Hypthesis Test SRS? Apprx Nrmal? Independence? 5. statistic - parameter µ z = = x standard deviatin s n 6. We either: Reject H Fail t Reject H All cnclusins are made abut the alternative hypthesis Hw d yu interpret the p-value:

7 2. The dean f students f a large cmmunity cllege claims that the average distance that cmmuting students travel t the campus is 32 mi. The cmmuting students feel therwise. A sample f 64 students was randmly selected and yielded a mean f 35 mi and a standard deviatin f 5 mi. Test the dean s claim at the 5% level f significance. 3. When the manufacturing prcess is wrking prperly, NeverReady batteries have lifetimes that fllw a right-skewed distributin with µ = 7 hurs. A quality cntrl supervisr selects a simple randm sample f 50 batteries every hur and measures the lifetime f each. If she is cnvinced that the mean lifetime f all batteries prduced that hur is less than 7 hurs at the 5% significance level, then all thse batteries are discarded. a) State the apprpriate hyptheses. b) Describe a Type I errr and its cnsequences in this cntext. c) Describe a Type II errr and its cnsequences in this cntext. d) In this cntext, what is meant by the pwer f the test the inspectrs inspect? e) Their test currently uses a 5% level f significance. If the supervisr tests at the 1% level f significance instead f 5%, hw will this affect the pwer f the test?

8 Ntes-10.4b Hypthesis Test fr Large Sample Ppulatin Mean 1. A grwing cncern f emplyers is time spent in activities like surfing the Internet and ing friends during wrk hurs. The San Luis Obisp Tribune summarized the findings frm a survey f a large sample f wrkers in an article that ran under the headline "Wh Gfs Off 2 Hurs a Day? Mst Wrkers, Survey Says"(August 2006) Suppse ur GBHS principal felt that it was less at GB and tk a SRS f Granite Bay High Schl Staff. Each selected staff member was asked abut daily wasted time at wrk. Assume that the ppulatin f hurs gfing ff can be cnsidered nrmal. The resulting data(in minutes) are listed belw: Test the claim at a 5% level f significance. 2. The Envirnmental Prtectins Agency (EPA) is charged with mnitring the envirnment. One aspect f this is keeping track f "acid rain," a brad term describing the fall f water thrugh an acidic atmsphere. Acidity is measured n the ph scale, where pure water has a ph f 7.0. Nrmal rain is slightly acidic because carbn dixide disslves int it, and thus has a ph f abut 5.5 ( A lwer ph indicates greater acidity.) Suppse the EPA wishes t determine whether a particular area is subject t acid rain. Let µ dente the true average f ph in this area. a. Write the apprpriate null hypthesis? H : b. Write the apprpriate alternate hypthesis? H A : c. Describe a Type I errr in cntext. d. Describe a Type II errr in cntext. e. In this cntext, what is meant by the pwer f the test?

9 Yu Try: 3. A bat manufacturer claims that a particular bat and mtr cmbinatin will burn less than 4.0 gallns f fuel per hur. Fuel cnsumptin fr a randm sample f 10 similar bats resulted in the data belw: Is there sufficient evidence t cnclude that the manufacturer's claim is crrect? 4. A new prcess fr prducing synthetic gems yielded six stnes weighting 0.43, 0.52, 0.46, 0.49, 0.60, and 0.56 carats, respectively, in this first run. Find a 90% cnfidence interval estimate fr the mean carat weight frm this prcess.