Math 225 Test 3 A Name: (1 poit) SHOW YOUR WORK FOR FULL CREDIT! Problem Max. Poits Your Poits 1-10 10 11 4 12 14 13 17 14 4 15 8 16 6 17 6 Total 70 1
1. Suppose you coduct a sigificace test for a populatio proportio usig α=10% ad your p- value is.184. Which of the followig should be your coclusio? a. Reject H 0 b. Reject H a c. Do ot to reject H a d. Do ot to reject H 0 2. Whe are p-values egative? a. whe the test statistic is egative. b. whe the sample statistic is smaller tha the hypothesized value of the parameter c. whe we fail to reject the ull hypothesis d. ever 3. You have measured the systolic blood pressure of a radom sample of 25 employees of a compay. A 95% cofidece iterval for the mea systolic blood pressure for the employees is computed to be (122,138). Which of the followig statemets gives a valid iterpretatio of this iterval? a. About 95% of the sample of employees have a systolic blood pressure betwee 122 ad 138. b. About 95% of the employees i the compay have a systolic blood pressure betwee 122 ad 138. c. If the samplig procedure were repeated may times, the approximately 95% of the resultig cofidece itervals would cotai the mea systolic blood pressure for employees i the compay. d. If the samplig procedure were repeated may times, the approximately 95% of the sample meas would be betwee 122 ad 138. e. The probability that the sample mea falls betwee 122 ad 138 is equal to 0.95. 4. The average growth of a certai variety of pie tree is 10.1 iches i three years. A biologist claims that a ew variety will have a greater three-year growth. A radom sample of 45 of the ew variety has a average three-year growth of 10.8 iches ad a stadard deviatio of 2.1 iches. The appropriate ull ad alterate hypotheses to test the biologist s claim are: a. H 0 : µ = 10.8 agaist H a : µ > 10.8 b. H 0 : µ = 10.8 agaist H a : µ 10.8 c. H 0 : µ = 10.1 agaist H a : µ > 10.1 d. H 0 : µ = 10.1 agaist Ha: µ < 10.1 e. H 0 : µ = 10.1 agaist H a : µ 10.1 5. What is statistical iferece o µ? a. Drawig coclusios about a populatio mea based o iformatio cotaied i a sample. b. Drawig coclusios about a sample mea based o iformatio cotaied i a populatio. c. Drawig coclusios about a sample mea based o the measuremets i that sample. d. Drawig coclusios about the populatio proportio based o iformatio cotaied i the sample. 6. The otatio for the populatio proportio, the sample proportio, ad the claimed populatio proportio respectively are a. p, p 0, $p b. $p, p, p 0 c. p 0, p, $p d. p, $p, p 0 e. p 0, $p, p 2
7. The 90% cofidece iterval for a populatio mea is (1.2, 5.2) a. The the populatio mea is 3.2, ad the margi of error is 2. b. The the sample mea is 3.2, ad the margi of error is 2. c. The the sample mea is 2, ad the margi of error is 3.2. d. The the sample mea is 1.2, ad the margi of error is 5.2. 8. Researchers at a Uiversity coducted a study i which 67 studets were weighed i September of their freshma year ad agai i April of their freshma year. The two samples are a. idepedet b. paired 9. A appropriate 95% cofidece iterval for µ has bee calculated as ( -0.73, 1.92 ) based o =15 observatios from a populatio with a ormal distributio. Suppose we wish to test H 0 : µ = 0 versus H a : µ 0. Based o this cofidece iterval, a. we should reject H 0 at the α = 0.05 level of sigificace. b. we should ot reject H 0 at the α = 0.05 level of sigificace. c. we should reject H 0 at the α = 0.10 level of sigificace. d. we should ot reject H 0 at the α = 0.10 level of sigificace. 10. Which oe of these statemets is FALSE? a. Icreasig the sample size will decrease the margi of error. b. Icreasig the cofidece level will decrease the margi of error. c. It is usually urealistic to assume that we kow the populatio stadard deviatio. d. Cofidece itervals are ot valid if the sample is ot chose radomly. 11. Briefly explai the differece betwee the goals of a cofidece iterval ad a hypothesis test. Goal of a CI: to estimate a ukow populatio parameter Goal of a HT: to test a claimed populatio parameter. 12. The brakig distaces of a simple radom sample of 50 hybrid cars were collected, ad their mea ad stadard deviatio were calculated. Based o the sample results, a 95% cofidece iterval was calculated: (134.6ft, 139.4ft). a. I this study, what is the parameter we wat to estimate? Deote this quatity by a symbol ad explai what the symbol stads for i this problem. We wat to estimate a populatio mea, µ. I this example, we wat to estimate the mea brakig distaces of ALL hybrid cars. b. Iterpret the calculated cofidece iterval i cotext. We are 95% cofidet that the mea breakig distace of ALL hybrid cars is betwee 134.6 ft ad 139.4 ft. 3
c. What is exactly i the middle of the give cofidece iterval? Circle ALL the correct aswers: poit estimate sample proportio sample mea populatio proportio populatio mea 95% 137ft margi of error 134. 6 + 139. 4 50 d. What is the margi of error of the give cofidece iterval? E = 137 134.6 = 2.4 OR E = (139.4 134.6) / 2 = 2.4 e. What could be doe to reduce the margi of error? List two ways to achieve this goal. Either icrease the sample size, or lower the cofidece level. f. True or False? If they had take 100 samples of 50 radomly selected hybrid cars ad had calculated a 95% cofidece iterval from each sample, probably about 95 of these cofidece itervals would ot cotai the populatio parameter, ad about 5% would cotai it. 13. A magazie article titled Are You Ready for a Iterview? claims that 50% of all seior executives say that the most commo job iterview mistake is to have little or o kowledge of the compay. To check this claim, Accoutemps (a compay specialized i temporary professioal accoutig ad fiace positios) coducted a survey of 150 radomly selected seior executives, ad foud that 64 of them said just that: the most commo job iterview mistake is to have little or o kowledge of the compay. Based o these survey results, ca you coclude that the magazie s article published a false claim? (That is ot 50%?) a. Specify the ull ad alterative hypotheses for this test, usig the correct symbols ad umbers. Null: p = 0.50 Alterative: p 0.50 b. Check the coditios for a hypothesis test. Radom sample, checked. p = 150( 050. ) > 10 0 ( 1 p ) = 150( 1 05. ) > 10 0 c. Determie the value of the test statistic, ad the p-value. Test statistic: z = -1.796 p-value = 0.072 4
d. Which oe of the followig statemets is true at the 5% level of sigificace? (i) the results are sigificat, ad so we ca reject the ull hypothesis. (ii) the results are ot sigificat, ad so we ca reject the ull hypothesis. (iii)the results are sigificat, ad so we caot reject the ull hypothesis. (iv) the results are ot sigificat, ad so we caot reject the ull hypothesis. e. State your coclusio i cotext. Sice the p-value is greater tha 5%, we caot reject the ull hypothesis. That is, at the 5% sigificace level, we ca t reject the magazie s claim that 50% of all seior executives say that the most commo job iterview mistake is to have little or o kowledge of the compay. f. Calculate the 95% cofidece iterval. ( 0.348, 0.506) g. Based o the cofidece iterval you calculated i part (f), would you reject the ull hypothesis or ot? Explai your aswer. Did you arrive to the same coclusio as i part (e)? Sice the claimed value, 0.50 is iside the 95% cofidece iterval, we caot reject the ull hypothesis at the 5% level. Same coclusio as i part (e). 14. Durig routie screeig, a doctor otices that 22% of her adult patiets show higher tha ormal levels of glucose i their blood a possible warig sigal for diabetes. Hearig this, some medical researchers decide to coduct a large-scale study, plaig to estimate the proportio to withi 2% with 90% cofidece. How may radomly selected adults must they test? 2 2 z = * m p p = 1645. $( 1 $) 0. 22( 1 0. 22) = 116088. 0. 02 They eed to select at least 1161 adults. 15. Durig a agiogram, heart problems ca be examied via a small tube (catheter) threaded ito the heart from a vei i the patiet s leg. It s importat that the compay that maufactures the catheter maitai a diameter of 2.00mm. Each day, quality cotrol persoel make several measuremets to check the diameters of the catheters. If they discover a problem, they will stop the maufacturig process util it is corrected. a. State the ull ad alterative hypotheses i symbols. H 0 : µ = 2 mm H a : µ 2 mm b. A radom sample of 55 catheters yielded the followig results. T Cofidece Itervals Variable N Mea StDev 95.0 % CI DIAMETER 55 2.06 0.19 (2.0086, 2.1114) T-Test of the Mea 5
Variable N Mea StDev T P DIAMETER 55 2.06 0.19 2.34 0.0229 If you were oe of the quality cotrol persoel, what would be your recommedatio? I your discussio use both the results from the T-Test, ad from the T-Cofidece Iterval. You must explai your decisio, ad write your coclusio i cotext. Sice the p-value is less tha 5%, ad the claimed value, 2mm is ot i the 95% cofidece iterval, we have eough evidece to reject the ull hypothesis. That is, we ca reject the claim that the mea diameter of the catheters is 2mm. If I were oe of the quality cotrol persoel, I would stop the maufacturig process util the problems are corrected. c. Is the cofidece iterval valid if the distributio from which the 55 measuremets were take is ot ormally distributed? Explai briefly. Yes, the cofidece iterval is still valid because the sample size is greater tha 30. 16. A compay istitutes a exercise break for its workers to see if it will improve job satisfactio, as measured by a questioaire that assesses workers satisfactio where a higher score meas higher satisfactio. Scores for te radomly selected workers before ad after the implemetatio of the exercise program are show below. Worker Before After 1 34 33 2 28 30 3 29 32 4 45 41 5 26 37 6 27 41 7 24 25 8 15 21 9 15 20 10 27 37 Here are the summaries of two possible aalyses: Paired t-test of mu(before-after) = 0 vs. mu(before-after) < 0 t-statistic = -2.627 p = 0.014 2-Sample t-test of mu(before) mu(after) = 0 vs. mu(before) mu(after) < 0 t-statistic = -1.284 p = 0.108 a. Which of these tests is correct for these data? Explai. It s a paired t-test because the before / after scores are recorded for each worker. 6
b. Usig the test you selected, state your coclusio i cotext. p-value = 0.014. Sice the p-value is smaller tha 5%, we ca reject the ull hypothesis. That is, we ca reject the claim that there is o differece betwee the satisfactio scores before ad after the exercise program. We ca coclude that there is a differece betwee the scores. 17. A maufacturer claims that the callig rage (i feet) of its 2.4 GHz cordless phoe is greater tha that of its leadig competitor. You perform a study usig 14 radomly selected phoes from the maufacturer ad 16 radomly selected phoes from its competitor. At the 5% sigificace level, ca you support the maufacturer s claim? Assume the populatios are ormally distributed. State the claims, ad write your coclusio i cotext based o the give p-value. p-value: 0.046 Hypotheses: H : 0 maufacturere = µ H :µ µ > µ Coclusio i cotext: competitor a maufacturer competitor Sice p-value < 5%, we ca reject the ull hypothesis. We ca reject the claim that the mea callig rages (i feet) are the same for both phoes. x $p = p$ ± z * x ± z * σ x ± t * p$( 1 p$) s z = p p$ p x µ t = 0 s 0 ( 1 p ) 0 0 z = *σ 2 z = E * E 2 p$( 1 p$) z* for 90% cofidece: 1.645 z* for 95% cofidece: 1.96 z* for 99% cofidece: 2.576 7