Mathacle PSet Stats, Confidence Interval Level Number Name: Date: Confidence Interval Guesswork with Confidence

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1 PSet Stats, Cofidece Iterval Cofidece Iterval Guesswork with Cofidece VII. CONFIDENCE INTERVAL 7.1. Sigificace Level ad Cofidece Iterval (CI) The Sigificace Level The sigificace level, ofte deoted by, is a pre-specified value that is used to compare with the probability of a evet uder a propositio kow as the ull hypothesis. The probability of the evet is regarded as the P value ad is cosidered as the coditioal probability of makig a error (Type I error ) of rejectig the ull hypothesis while the propositio is true. The sigificace level is associated with the critical z-scores critical z i N(0,1 ) distributio or t i t distributio, depedig o which distributio i the applicatios is used. Three cases associated with are ofte practiced: the left-tailed, the right-tailed ad twotailed. The two-tailed case is used i determiig the cofidece iterval i ormal ad t distributios. df is the degree of freedom i t distributio. 1

2 PSet Stats, Cofidece Iterval [MATH] Assume the ull hypothesis or propositio H 0, the sigificace level is the probability of rejectig the ull hypothesis while the ull hypothesis is true: Pr reject H H is true 0 0 The critical z * value of Z-distributio or the critical value t * of t-distributio are defied mathematically as follows: 1.) For left-sided: Pr Z z*, Pr *.) For right-sided: Pr Z z*, Pr * 3.) For two-sided: Pr Z z*, Pr * T t T t T t [Ti-84] The critical z * or t * values ca be calculated as follows: 1.) Left-tailed: z ivnorm, t ivt, df ;.) Right-tailed: z ivnorm 1, t ivt 1, df 3.) Two-tailed: z ivnorm, t ivt, df

3 PSet Stats, Cofidece Iterval Eample For the give, df, the types of tails, ad ull hypothesis, fid as idicated. z or t Type df z or 0.1 Z - Distributio, right - tailed - t Graph 0.1 Z - Distributio, left - tailed Z - Distributio, two - tailed Z - Distributio, two - tailed Z - Distributio, two - tailed t - Distributio, two - tailed t - Distributio, two - tailed t - Distributio, two - tailed t - Distributio, two - tailed t - Distributio, two - tailed t - Distributio, two - tailed t - Distributio, two - tailed t - Distributio, two - tailed t - Distributio, two - tailed 30 3

4 PSet Stats, Cofidece Iterval Solutio: Type df z or t 0.1 Z - Distributio, right - tailed - z 1.8 Graph 0.1 Z - Distributio, left - tailed - z Z - Distributio, two - tailed - z Z - Distributio, two - tailed - z Z - Distributio, two - tailed - z t - Distributio, two - tailed 5 t t - Distributio, two - tailed 0 t t - Distributio, two - tailed 30 t t - Distributio, two - tailed 60 t t - Distributio, two - tailed 5 t t - Distributio, two - tailed 0 t t - Distributio, two - tailed 30 t t - Distributio, two - tailed 5 t t - Distributio, two - tailed 30 t.750 4

5 PSet Stats, Cofidece Iterval The Cofidece Iterval The cofidece level (CL) is a probability value that is defied as 1. I the two-tailed case whe the is give, the Cofidece Iterval (CI) is defied as the iterval cotaiig the parameter for some obtaied statistic such that Statistic Parameter Pr c * 1 Variability Where c * is the critical value for either z * or t *. The CI is the domai for the probability i terms of the parameter, ad CI is oly defied for the symmetric distributios: or Statistic c* Variability Parameter Statstic c* Variability CI Statistic c * Variability, Statstic c * Variability [MATH] A estimator ˆ is called a ubiased estimator for parameter, if the epected value of ˆ is : E[ ˆ ] The differece of E[ ˆ ] is called the bias of ˆ. The ituitive meaig of a ubiased estimator is oe that does ot systematically overestimate or uderestimate the. 5

6 PSet Stats, Cofidece Iterval 7.. Cofidece Iterval for a Proportio i Oe Sample PROBLEM: to estimate the populatio parameter from the statistic obtaied from the sample data. Uder certai coditios, the problem ca be solved by usig some ormal approimatio. Pickig skittles from a jar ca be viewed as a Beroulli process whe the sample size is small compared with the populatio size. I the followig eample, assume 4 red skittles i the small bag ad 40 red i the large jar, ad you wat to select 10 skittles radomly i each cotaier. Let N deote the total umber of uits or items i the populatio, ad a sample of size is collected, where N. Let X be the sum of those idepedet Beroulli radom variables X, X, 1, X (i.i.d. Beroulli trials) with: X i 1 0 P( X X i ) p 1 p 6

7 PSet Stats, Cofidece Iterval X X X... 1 X The mea ad variace of X are E[ X ] p ad Var ( X ) p1 p. Let the estimator of 1 p be defied as a radom variable ˆ X X X X P. The, pˆ i1 where ad each i takes the value of oe whe it is a success ad zero whe it i1 is a failure. i i [MATH] The proportio problem should really be modeled as a hypergeometric problem. That is, the k successes i draws without replacemet. The stadard deviatio for the hypergeometric model is N p(1 p) pˆ N 1 N Where N 1 is the modificatio factor. Whe the samplig fractio is small, say N 10% N, the stadard deviatio ca be approimated as pˆ 1 N 1 1 N p(1 p) p(1 p) This is the 10% rule, ad i this case the biomial approimatio is satisfactory. For a give ad from the characteristic property of quadratic fuctio, the stadard p( 1 p) deviatio p reaches maimum whe p 0. 5: ma (0.5) 1 7

8 PSet Stats, Cofidece Iterval Whe p is ukow, ˆp ca be used to approimate the 10% rule is satisfied. pˆ pˆ(1 pˆ), this is provided that [MATH] Whe the coditios p 10ad 1 p 10satisfied, the biomial distributio of X ca be approimated by the ormal distributios, ad Pr P pˆ Pr X, where Pr 1 X e p(1 p) 1 p p(1 p) 1 e p(1 p) p 1 p(1 p) Where p p(1 p) (1 ) P N p, p p. 1 pˆ p p 1 pˆ p p(1 p) 1 1 e p(1 p) 1 1 e pˆ p p p ad p 1 p. So, the distributio p follows the Z-distributio p For a give, the cofidece Iterval (CI) is the domai of the probability that cotais the parameter p : ˆ ˆ p p p p Pr z Pr z p(1 p) pˆ(1 pˆ) 8

9 PSet Stats, Cofidece Iterval That is, ˆ(1 ˆ) ˆ(1 ˆ) ˆ p p, ˆ p CI p z p z p Or, i the iterval otatio: pˆ (1 pˆ ) pˆ (1 pˆ ) pˆ z p pˆ z i the form of margi of error z pˆ(1 pˆ) : pˆ z pˆ(1 pˆ) [PROCEDURE] Cofidece Iterval for a Proportio i Oe Sample The steps to obtai CI for the populatio proportio p from the sample proportio (statistic) pˆ with sample size are 1.) The sample is a idepedet radom sample (the variables are i.i.d.)..) The sample size is less tha 10% of populatio. 3.) pˆ 10, (1 pˆ ) 10 to approimate the biomial distributio by the ormal distributio. 4.) The CI is costructed from the sample statistic ˆp for a give sigificace level : Where z ˆ(1 ˆ) ˆ(1 ˆ) ˆ p p, ˆ p CI p z p z p pˆ(1 pˆ) is the margi of error (MoE). [Ti-84] Cofidece Iterval for a Proportio i Oe Sample 1.) STAT -> TESTS ->A. 1-PropZit.) Iput,,1. Note that you eed to put i,, ot ˆp! 9

10 PSet Stats, Cofidece Iterval Eample THS admiistrators wated to kow how may 10 th graders ad 11 th graders did either iterships or commuity services i the past summer. A radom sample of 75 studets idicated that 60 studets did oe of the two. Fid the 95% cofidece iterval for the school proportio. Assume that the school populatio of these two classes is 800 ad all two grades are equally likely to do summer iterships or commuity services. Use the calculator to verify your aswers. Solutio: p ˆ 60/ , 0. 05, z ( z ivnorm(0.05) ) z 75 10% , pˆ satisfied., 1 p ˆ. The coditios are or ˆ p z pˆ(1 pˆ) CI 0.709, 0.8(1 0.8) That is, the school is 95% cofidet that the true proportio of those two grades who did the summer iterships or commuity services is betwee ad Eample 7... A previous study has suggested that about 19.3% of tees (aged 1-19) are obese. How large of a sample will be eeded i order to estimate the true proportio of obese tees with 95% cofidece ad a margi of error of o more tha 1%? Solutio: p 0.193, 0. 05, z (use z ivnorm(0.05) i Ti 84) z z z p 1 p p(1 p) ( ) It is assumed that %N, ad 10 p, p

11 PSet Stats, Cofidece Iterval Eample I wat to costruct a 99% cofidece iterval for the proportio of Americas who thik that the govermet has placed too may regulatio o busiesses, ad I wat a margi of error of o more tha 3%. Assume the populatio proportio is 0.5. How large of a sample will this require? Solutio: 0.01, p 0.5, z z , z p(1 p) 3%? 0.5(0.5).575(0.5).576 3% Eample A study of 530 people aged 60 or older i US foud 14 with rheumatoid arthritis. Costruct 90% cofidece iterval for the actual proportio of all people aged 60 ad older who have rheumatoid arthritis. Use the calculator to verify your aswers. Solutio: or 0.1, 14 pˆ 0.033, z z , 530, CI? 0.033( ) CI , Eample [APSTATSMC01-09] Based o a survey of a radom sample of 900 adults i the Uited States, a jouralist reports that 60 percet of adults i the Uited States are i favor of icreasig the miimum hourly wage. If the reported percet has a margi error of.7 percetage poits, what is the level of cofidece? Solutio: z p(1 p) 0.6(1 0.6).7% z.7% 900 z The CI is 90.0%. It is assumed that all the coditios are satisfied. 11

12 PSet Stats, Cofidece Iterval Eample [APSTATSFRQ016-05] A pollig agecy showed the followig two statemets to a radom sample of 1048 adults i the Uited States. The order i which the statemets were show was radomly selected for each perso i the sample. After readig the statemets, each perso was asked to choose the statemet that was most cosistet with his or her opiio. The results are show i the table. (a) Assume the coditios for iferece have bee met. Costruct ad iterpret a 95 percet cofidece iterval for the proportio of all adults i the Uited States who would have chose i the ecoomy statemet. (b) Oe of the coditios for iferece that was met is that the umber who chose the ecoomy statemet ad the umber who did ot choose the ecoomy statemet are both greater tha 10. Eplai why it is ecessary to satisfy that coditio. (c) A suggestio was made to use a two-sample z-iterval for a differece betwee proportios to ivestigate whether the differece i proportios betwee adults i the Uited States who would have chose the eviromet statemet ad the adults i the Uited States who would have chose the ecoomy statemet is statistically sigificat. Is the two-sample z-iterval for a differece betwee proportios a appropriate procedure to ivestigate the differece? Justify your aswer. Solutio: pq ˆˆ 0.37(1 0.37) a.) 1048, pˆ 0.37, 0.05, z 1.96, pˆ CI (0.0149) CI (0.34, 0.40) 95% of chace that the iterval cotais the populatio proportio of selectig Ecoomy Statemet. b.) Oe of the coditios is p 10, q 10. Sice p or q are less tha oe, the sample size should be at least 10. c.) No, the two sampligs should be idepedet. 1

13 PSet Stats, Cofidece Iterval Eample For each of the followig problems of fidig cofidece iterval of populatio proportio from oe-sample proportio, fid the idicated variable. Assume that the sample is idepedet ad the sample size is less tha 10% of the populatio. Case # p p ( 1 p) Cof. Iterval Margi of error , Solutio: Case # p p ( 1 p) Cof. Iterval Margi of error ( 0.311, ) , , , Case #1: p 0.5(1 0.5) 30(0.5) 15, ( 1 p) 30(1 0.5) 15, Case #: p 0.8(1 0.) 0.8, MoE , Case #3: 0.8(1 0.) , CI Case #4: 0.6(1 0.6) z * 0.03 z* 1.83, (0.336)

14 PSet Stats, Cofidece Iterval Eample Give 95% cofidece level ad sample size, prove that the margi of 1 error (MoE) of CI is bouded by. Assume that the coditios for costructig the CI are satisfied. Proof: The MoE is z p(1 p) ad z p(1 p) (0.5)(0.5) (0.5). Note that ma{ pˆ (1 pˆ )} pˆ (1 pˆ ) 0.5. p ˆ 0.5 Eample The data below show the geographic distributio of 00 people. Costruct the cofidece iterval for the white proportio by usig the followig methods: a.) SRS b.) Stratified Samplig c.) Clustered Samplig Describe the way you obtai the sample data, ad verify the coditios for each case. Assume 95% cofidece level. 14

15 PSet Stats, Cofidece Iterval a.) Cesus: N,, p b.) SRS:,, p, Coditios: c.) Stratified:,, p, Coditios: d.) Clustered:,, p, Coditios: Solutio: the aswers vary. 81 a.) Cesus: N 00, 81, p b.) SRS: 10% of the populatio is 0. Step 1.) Assig a umber to each idividual from left to right, ad from top to bottom. Step.) Radomly select 0 umbers from 1 to 00: X {4,18,3,4,5,46,48,49,51,55,7,9,106,108,143,161,166,170,17,188} Step 3.) Cout the umber of white ad calculate the sample proportio: 0, 10, 10 p 0 Step 4.) Check Coditios: p 0(0.5) 10, ( 1 p) 0(0.5) 10 Step 5.) Calculate the CI: ( 0.809, ). That is, there is 95% of chace that the populatio white proportio is betwee ad

16 PSet Stats, Cofidece Iterval c.) Stratified Method: 10% of the populatio is 0.,, p, Coditios: d.) Clustered Method: 10% of the populatio is 0. Step 1.) divide the area ito 0 plots (colums), Step.) geerate oe radom umber from I our case, 3 is selected. Step 3.) select the 3 rd row. Step 4.) 0, 6, p 0.3, Coditios: Step 5.) CI (0.099, 0.501) 16

17 PSet Stats, Cofidece Iterval Quick-Check 7.. Cofidece Iterval for a Proportio i Oe Sample QC [CBAPStatsPracticeQuestio] Courtey has costructed a cricket out of paper ad rubber bads. Accordig to the istructios for makig the cricket, whe it jumps it will lad o its feet half of the time ad o its back the other half of the time. I the 50 jumps, Courtey s cricket laded o its feet 35 times. I the et 10 jumps, it laded o its feet oly twice. Based o this eperiece, Courtey ca coclude that (A) the cricket was due to lad o its feet less tha half the time durig the fial 10 jumps, sice it had haded too ofte o its feet durig the first 50 jumps. (B) a cofidece iterval for estimatig the cricket s true probability of ladig o its feet is wider after the fial 10 jumps tha it was before the fial 10 jumps. (C) a cofidece iterval for estimatig the cricket s true probability of ladig o its feet after the fial 10 jumps is eactly the same as it was before the fial 10 jumps. (D) a cofidece iterval for estimatig the cricket s true probability of ladig o its feet is more arrow after the fial 10 jumps tha it was before the fial 10 jumps. (E) a cofidece iterval for estimatig the cricket s true probability of ladig o its feet based o the iitial 50 jumps does ot iclude 0., so there must be a defect i the cricket s costructio to accout for the poor showig i the fial 10 jumps. QC 7... [APSTATSMC01-4] A radom sample of 43 voters revealed that 100 are i favor of a certai bod issue. A 95 percet cofidece iterval for the proportio of the populatio of voters who are i favor of the bod issue is (A) (B) (C) (D) (E) (0.5) (0.5) (0.769) (0.769) (0.769) 43 17

18 PSet Stats, Cofidece Iterval QC [APSTATSMC014-13] The maager of a car compay will select a radom sample of its customers to create a 90 percet cofidece iterval to estimate the proportio of its customers who have childre. What is the smallest sample size that will result i a margi of error of o more tha 6 percetage poits? QC [APSTATSMC01-17] A large-sample 98 percet cofidece iterval for the proportio of hotel reservatio that is caceled o the iteded arrival day is (0.048, 0.11). What is the poit estimate for the proportio of hotel reservatios that are caceled o the iteded arrival day from which this iterval was costructed? (A) 0.03 (B) (C) (D) (E) It caot be determied form the iformatio give. QC [APSTATSMC01-6] I 009 a survey of Iteret usage foud that 79 percet of adults age 18 years ad older i the Uited States use the Iteret. A broadbad compay believes that the percet is greater ow tha it was i 009 ad will coduct a survey. The compay plas to costruct a 98 percet cofidece iterval to estimate the curret percet ad wats to the margi of error to be o more tha.5 percetage poits. Assumig that at least 79 percet of adults use the Iteret, which of the followig should be used to fid the sample size () eeded? (A) (0.5) (B) (C) (D) (E) 0.5(0.5) (0.1) (0.1)

19 PSet Stats, Cofidece Iterval QC [APSTATSMC007-1] A city is iterested i buildig a waste maagemet facility i a certai area. Oe hudred radomly selected residets from this area were asked, Do you support the city s decisio to build a waste maagemet facility i your area? Of the 100 residets iterviewed, 54 said o, 4 said yes, ad 4 had o opiio. A ˆ(1 ˆ) large sample z-cofidece iterval, ˆ * p p z p, was costructed from these data to estimate the proportio of this area s residets who support buildig a waste maagemet facility i their area. Which of the followig statemets is correct for this cofidece iterval? (A) The cofidece iterval is valid because a sample size of more tha 30 was used. (B) The cofidece iterval is valid because each area residet was asked the same questio. (C) The cofidece iterval is valid because o coditios are required for costructig a large sample cofidece iterval for a proportio. (D) The cofidece iterval is ot valid because the quatity pˆ is too small. (E) The cofidece iterval is ot valid because o opiio was icluded as a respose category for the questio. QC [1997APSTATSMC ] A 95 percet cofidece iterval of the form ˆp Ewill be used to obtai a estimate for a ukow populatio proportio p. If ˆp is the sample proportio ad E is the margi of error, which of the followig the smallest size that will guaratee a margi of error of at most 0.08? (A) 5 (B) 100 (C) 175 (D) 50 (E) 65 19

20 PSet Stats, Cofidece Iterval QC [007APSTATSMC007-34] A plaig board i Elm Couty is iterested i estimatig the proportio of its residets that are i favor of offerig icetives to hightech idustries to build plats i that couty. A radom sample of Elm Couty residets was selected. All of the selected residets were asked, Are you i favor of offerig icetives to high-tech idustries to build plats i your couty? A 95 percet cofidece iterval for the proportio of residets i favor of offerig icetives was calculated to be Which of the followig statemets is correct? (A) At 95% cofidece level, the estimate of 0.54 is withi 0.05 of the true proportio of couty residets i favor of offerig icetives to high-tech idustries to build plats i the couty. (B) At 95% cofidece level, the majority of area residets are i favor of offerig icetives to high-tech idustries to build plats i the couty. (C) I repeated samplig, 95% of sample proportios will fall i the iterval (0.49, 0.59) (D) I repeated samplig, the true proportio of couty residets i favor of offerig icetives to high-tech idustries to build plats i the couty will fall i the iterval (0.49, 0.59). (E) I repeated samplig, 95% of the time the true proportio of couty residets i favor of offerig icetives to high-tech idustries to build plats i the couty will be equal to QC [APSTATSMC ] A survey was coducted to determie what percetage of college seiors would have chose to atted a differet college if they had kow the what they kow ow. I a radom sample of 100 seiors, 34 percet idicated that they would have atteded a differet college. A 90 percet cofidece iterval for the percetage of all seiors who would have atteded a differet college is (A) 4.7% to 43.3% (B) 5.8% to 4.% (C) 6.% to 41.8% (D) 30.6% to 37.4% (E) 31.% to 36.8% 0

21 PSet Stats, Cofidece Iterval QC [APSTATSMC ] QC [AP STATSFRQ011-06] Every year, each studet i a atioally represetative sample is give tests i various subjects. Recetly, a radom sample of 9,600 1 th -grade studets from US were admiistered a multiple-choice US history eam. Oe of the multiple-choice questios is below. (The correct aswer is C.) Of the 9,600 studets, 8 percet aswered the multiple-choice questio correctly. a.). Let p be the proportio of all Uited States twelfth-grade studets who would aswer the questio correctly. Costruct ad iterpret a 99 percet cofidece iterval for p. Assume that studets who actually kow the correct aswer have a 100 percet chace of aswerig the questio correctly, ad studets who do ot kow the correct aswer to the questio guess completely at radom from amog the four optios. Let k represet the proportio of all Uited States twelfth-grade studets who actually kow the correct aswer to the questio. 1

22 PSet Stats, Cofidece Iterval b.) A tree diagram of the possible outcomes for a radomly selected twelfth-grade studet is provided below. Write the correct probability i each of the five empty boes. Some of the probabilities may be epressios i terms of k. c.) Based o the completed tree diagram, epress the probability, i terms of k, that a radomly selected twelfth-grade studet would correctly aswer the history questio. d.) Usig your iterval from part (a) ad your aswer to part (c), calculate ad iterpret a 99 percet cofidece iterval for k, the proportio of all Uited States twelfth-grade studets who actually kow the aswer to the history questio. You may assume that the coditios for iferece for the cofidece iterval have bee checked ad verified.

23 PSet Stats, Cofidece Iterval QC [APSTATSFRQ015-0] To icrease busiess, the ower of a restaurat is ruig a promotio i which a customer s bill ca be radomly selected to receive a discout. Whe a customer s bill is prited, a program i the cash register radomly determies whether the customer will receive a discout o the bill. The program was writte to geerate a discout with a probability of 0., that is, givig 0 percet of the bills a discout i the log ru. However, the ower is cocered that the program has a mistake that results i the program ot geeratig the iteded log-ru proportio of 0.. The ower selected a radom sample of bills ad foud that oly 15 percet of them received discouts. A cofidece iterval for p, the proportio of bills that will receive a discout i the log ru, is All coditios for iferece were met. a.). Cosider the cofidece iterval i. Does the cofidece iterval provide covicig statistical evidece that the program is ot workig as iteded? Justify your aswer. ii. Does the cofidece iterval provide covicig statistical evidece that the program geerates the discout with a probability of 0.? Justify your aswer. A secod radom sample of bills was take that was four times the size of the origial sample. I the secod sample, 15 percet of the bills received the discout. b.) Determie the value of the margi of error based o the secod sample of bills that would be used to compute a iterval for p with the same cofidece level as that of the origial iterval. c) Based o the margi of error i part (b) that was obtaied from the secod sample, what do you coclude about whether the program is workig as iteded? Justify your aswer. 3

24 PSet Stats, Cofidece Iterval Aswers QC D. The proportio is asumed to be p 0.5. The error term (width) of the cofidece iterval is calculated by p(1 p) z. So, whe the sample size is icreasig, the error term will be decreasig. That is, the CI is arrowig whe sample size is icreasig. QC 7... D. 100 pˆ QC Sice the proportio is ormalized to 100%, so the questio idicates that p(1 p) % without give what p is. The quadratic fuctio f ( p) p(1 p) reaches maimum whe p 0.5, therefore, for all 0 p 1, f (0.5) 0.5(1 0.5) 0.5 is maimum value. That is QC C. pˆ QC D. pˆ 0.79, 0.0, z ivorm.33 4 QC D. p is too small. 100 QC C. Assume p 0.5, QC A % (1 0.5) QC C. 0.34(1 0.34) pˆ 0.34, z 1.64, QC E QC a.) pˆ 0.80, pˆ 9600(0.80) , (1 pˆ ) 9600(1 0.80) , 4

25 PSet Stats, Cofidece Iterval 0.01, z z , pˆ(1 pˆ). pˆ z pˆ(1 pˆ) (0.68, 0.9) 0.80(1 0.80) 9600 The CI idicates that 99% cofidece that the populatio proportio is betwee 0.68 ad 0.9. That is, we are 99 percet cofidet that the iterval from 0.68 to 0.9 cotais the populatio proportio of all Uited States twelfth-grade studets who would aswer this questio correctly. b.) c.) P( Aswer _ Correctly) P( Kow _ the _ Aswers) P( Guess _ the _ Aswers) k 0.5(1 k) k d.) Sice p k, the 0.68 p k k We are 99 percet cofidet that the iterval from 0.04 to cotais the proportio of all Uited States twelfth-grade studets who actually kow the aswer to the history questio. 5

26 PSet Stats, Cofidece Iterval QC a.) i. No. The assumed proportio is 0., ad it is withi the CI. So, there is o statistical evidece to claim that the program is ot workig. ii. No. Ay umber withi CI could be the probability b.) c.) Now the CI is , so 0. is ot withi the CI. So, there is covicig evidece that the program is ot workig. 6

27 PSet Stats, Cofidece Iterval 7.3. Cofidece Iterval for a Mea i Oe Sample The populatio mea ca be estimated i the followig cases: CASE #1: Large Sample Size ad Kow Suppose X...,, X, 1 X be idepedet, idetically distributed (i.i.d.) radom variables havig mea ad fiite ozero variace. The is the ubiased mea estimator: X X 1 X... X The variace is X E[ X ] E 1 X... X E[ X i ] X [ ] 1 X... X EX Var[ X ] Var i [MATH] The Cetral Limit Theorem. Let X, X,..., X 1 be idepedet, idetically distributed (i.i.d.) radom variables havig mea ad fiite ozero variace. Let X 1 X... X X, the X lim Pr z ( z) / The Cetral Limit Theorem says that whe the sample size is large, this ubiased estimator X ~ N(, ) ~ N(0,1). 7

28 PSet Stats, Cofidece Iterval 8 [MATH] The correct hypergeometric model should add the correctio factor to the variace: 1 N N Whe the 10% rule is satisfied, the stadard deviatio is N N N N Whe the sigificat level is give, the parameters ad statistic are related by Pr z where 1 i i. The CI is, z z CI, Or, i the form of iterval otatio: z z I the form of margi of error z : z

29 PSet Stats, Cofidece Iterval [PROCEDURE] Cofidece Iterval for a Mea with Large Sample Size ad Kow Variace The coditios to obtai CI for the sample mea (statistic) with sample size are 1.) The sample is a idepedet radom sample..) The sample size is less tha 10% of populatio. 3.) is kow ad 30for the coditio of usig ormal distributio. 4.) The CI is costructed from the sample statistic for a give sigificace level : Where z CI z, is the term of margi of error. z [Ti-84] Cofidece Iterval for a Mea with Large Sample Size ad Kow Variace 1.) STAT -> TESTS ->7. ZIterval, select Stats if the statistic is give..) Iput,,,1. Eample The Presidet of a large uiversity wishes to estimate the average age of the studets presetly erolled. For the past studies, the stadard deviatio is kow to be years. A sample of 50 studets is selected radomly, ad sample mea is foud to be 3. years. Fid the 95% cofidece iterval of the school s populatio mea. Use the calculator to verify your aswers. Solutio: 50 30, ad it is assumed that the school has more tha 500 studets., 3., 0. 05, z 1. 96, CI? z or CI.6, 3.8 That is, the Presidet ca say with 95% cofidece that the average age of studets is betwee.6 ad 3.8 years old. 9

30 PSet Stats, Cofidece Iterval Eample From the last eample, the Presidet would like to be 99% cofidet that the estimate of average age should be accurate withi 1 year whe the stadard deviatio of the ages is 3 years. How large a sample is ecessary? Solutio: 3, 0. 01, z. 58,? z 1 The sample size eeds to be at least z Eample [CBAPStatsPracticeProblem] A large compay is cosiderig opeig a frachise i St. Louis ad wats to estimate the mea household icome for the area usig a simple radom sample of the households. Based o iformatio from a pilot study, the compay assumes that the stadard deviatio of household icomes is $7, 00. What is the least umber of households that should be surveyed to obtai a estimate that is withi $00 of the true mea houshold icome with 95 percet cofidece? Solutio: The variace is kow. 7, 00, 0.05, 1.96, z z z

31 PSet Stats, Cofidece Iterval CASE #: Small Sample Size or Ukow [MATH] For a radom variable X ~ T( 1), where T ( 1) is the t-distributio with the degrees of freedom k 1 ad with sample size of, the mea E [ X ] 0, the variace k 6 Var[ X ] for k, the skewess is 0 for k 3, ad the kurtosis is for k k 4 k 4. Whe variace That is, E[ s ] is ukow for X, X, 1, i1 X 1 s i 1, the ubiased variace estimator is: N 1 S, where S i, the populatio stadard error. For the N 1 hypergeometric distributio, s is s i1 s N N 1 Whe the 10% rule is satisfied, the stadard error of is s s Note that the sample stadard deviatio estimator 1 s i 1 i1 is ot a ubiased estimator of the populatio stadard error S. 31

32 PSet Stats, Cofidece Iterval Whe the sample size is less tha 30, ad/or is ukow, similar to use ormal distributio, the mea estimator X ca be studetized by the t-distributio ~ T ( 1) s ad the cofidece level is the domai for that is determied by 1 Pr t s. The CI is 1 s 1 s CI t, t Or, i the iterval otatio: 1 s 1 t t s I the form of margi of error t 1 s : t 1 s [PROCEDURE] Cofidece Iterval for a Mea with Small Sample Size ad/or Ukow Variace The coditios to obtai CI for the sample mea (statistic) with sample size are 1.) The sample is a idepedet radom sample..) The sample size is less tha 10% of populatio. 3.) 30ad/or is ukow. 4.) The CI is costructed from the sample statistic for a give sigificace level : 3

33 PSet Stats, Cofidece Iterval 1 s 1 CI t, t s Where t 1 s 1 s i 1 i1 is the term of margi of error (MoE). [Ti-84] Cofidece Iterval for a Mea with Small Sample Size ad/or Ukow Variace 1.) STAT -> TESTS ->8. TIterval, select Stats if the statistic is give..) Iput, s,,1. Eample The average (mea) travel time from home to school for a sample of 8 THS teachers was 14.3 miles. The stadard deviatio of their travel time was miles. Fid the 95% cofidece iterval of true mea or populatio mea. Use the calculator to verify the results. Solutio: 8 30 ad the is ukow. s, 14. 3, 0. 05, the umber of degrees of 7 freedom is k 8 1 7, t. 05, CI? t or CI 13.5,

34 PSet Stats, Cofidece Iterval Eample [APSTATSMC00-08] A test egieer wats to estimate the mea gas mileage (i miles per gallo) for a particular model of automobile. Eleve of these cars are subjected to a road test, ad the gas mileage is computed for each car. A dotplot of the 11 gas-mileage values is roughly symmetrical ad has o outliers. The mea ad stadard deviatio of these values are 5.5 ad 3.01, respectively. Assumig that these 11 automobiles ca be cosidered a simple radom sample of cars of this model, which of the followig is a correct statemet? 3.01 (A) A 95% cofidece iterval for is (B) A 95% cofidece iterval for is (C) A 95% cofidece iterval for is (D) A 95% cofidece iterval for is (E) The results caot be trusted; the sample is too small. Solutio: A k df t 5.5, 11, 10,.8, Eample [013APSTATSFRQ013, #1] A evirometal group coducted a study to determie whether crows i a certai regio were igestig food cotaiig uhealthy levels of lead. A biologist classified lead levels greater tha 6.0 parts per millio (ppm) as uhealthy. The lead levels of a radom sample of 3 crows i the regio were measured ad recorded. The data are show i the stemplot below. 34

35 PSet Stats, Cofidece Iterval a.) What proportio of crows i the sample had lead levels that are classified by the biologist as uhealthy? b.) The mea lead level of the 3 crows i the sample was 4.90 ppm ad the stadard deviatio was 1.1 ppm. Costruct ad iterpret a 95 percet cofidece iterval for the mea lead level of crows i the regio. Solutio: 4 a.) pˆ df b.) 4.90, s 1.1, 3 df 31, t t s 1.1 t CI (4.417, 5.383) We ca be 95% cofidet that the populatio mea lead level amog all crows i this regio is betwee ad parts per millio. 35

36 PSet Stats, Cofidece Iterval Eample For each of the followig problems of fidig cofidece iterval of populatio mea from oe-sample mea, fid the idicated variable. Assume that the sample is idepedet ad the sample size is less tha 10% of the populatio. Case # s df Cof. Iterval Margi of error Solutio: Case s df Cof. Iterval Margi of # error (4.446, 5.554) (14.065,15.935) Case #1: , z* 1.95, Case #: 0.01, z*.576,, so use oe-sample Z-distributio for the CI. z* or CI (4.446, 5.554) (7000).576* Case #3: 30, so, use t-distributio for the CI. 0.01, df 1 19, t ivt(0.975,19).09, s t , or CI (14.065,15.935) 0 Case #4: 30, so, use t-distributio for the CI. t t 1.16, tcdf (1.16,100,14) , so

37 PSet Stats, Cofidece Iterval Quick-Check 7.3. Cofidece Iterval for a Mea QC [APSTATSMC00-6] A quality cotrol ispector must verify whether a machie that packages sack foods is workig correctly. The ispector will radomly select a sample of packages ad weigh the amout of sack food i each. Assume that the weights of food i packages filled by the machie have a stadard deviatio of 0.30 ouce. A estimate of the mea amout of sack food i each package must be reported with 99.6 percet cofidece ad a margi of error of o more tha 0.1 ouce. What would be the miimum sample size for the umber of packages the ispector must select? (A) 8 (B) 15 (C) 5 (D) 5 (E) 60 QC [APSTATSMC1997-4] A radom sample of costs of repair jobs at a large muffler repair shop produces a mea of $17.95 ad a stadard deviatio of $4.03. If the size of this sample is 40, which of the followig is a approimate 90 percet cofidece iterval for the average cost of a repair at this repair shop? (A) $17.95 $4.87 (B) $17.95 $6.5 (C) $17.95 $7.45 (D) $17.95 $30.81 (E) $17.95 $39.53 QC [APSTATSMC00-37] A simple radom sample procedure produces a sample mea,, of 15. A 95 percet cofidece iterval for the correspodig populatio mea is Which of the followig statemets must be true? (A) Niety-five percet of the populatio measuremets fall betwee 1 ad 18. (B) Niety-five percet of the sample measuremets fall betwee 1 ad 18. (C) If 100 samples were take, 95 of the sample meas would fall betwee 1 ad 18. (D) P( 1 18 ) = 0.95 (E) If = 19, this of 15 would be ulikely to occur. 37

38 PSet Stats, Cofidece Iterval QC [APSTATS00-33] A egieer for the Allied Steel Compay has the resposibility of estimatig the mea carbo cotet of a particular day's steel output, usig a radom sample of 15 rods from that day's output. The actual populatio distributio of carbo cotet is ot kow to be ormal, but graphic displays of the egieer's sample results idicate that the assumptio of ormality is ot ureasoable. The process is ewly developed, ad there are o historical data o the variability of the process. I estimatig this day's mea carbo cotet, the primary reaso the egieer should use a t-cofidece iterval rather tha a z-cofidece iterval is because the egieer (A) is estimatig the populatio mea usig the sample mea. (B) is usig the sample variace as a estimate of the populatio variace. (C) is usig data, rather tha theory, to judge that the carbo cotet is ormal. (D) is usig data from a specific day oly. (E) has a small sample, ad a z-cofidece iterval should ever be used with a small sample. QC [APSTATSMC00-13] A radom sample has bee take from a populatio. A statisticia, usig this sample, eeds to decide whether to costruct a 90 percet cofidece iterval for the populatio mea or a 95 percet cofidece iterval for the populatio mea. How will these itervals differ? (A) The 90% cofidece iterval will ot be as wide as the 95 percet cof. Iterval. (B) The 90% cofidece iterval will be wider tha the 95 percet cof. iterval. (C) Which iterval is wider will deped o how large the sample is. (D) Which iterval is wider will deped o whether the sample is ubiased. (E) Which iterval is wider will deped o whether a z-statistic or a t-statistic is used. QC [APSTATSMC00-30] The populatio {, 3, 5, 7} has mea = 4.5 ad stadard deviatio = 1.9. Whe samplig with replacemet, there are 16 differet possible ordered samples of size that ca be selected from this populatio. The mea of each of these 16 samples is computed. For eample, 1 of the 16 samples is (, 5), which has a mea of 3.5. The distributio of the 16 sample meas has its ow mea ad its ow stadard deviatio. Which of the followig statemets is true? (A) (B) (C) (D) (E) = 4.5 ad = 1.9 = 4.5 ad > 1.9 = 4.5 ad > 4.5 < 4.5 <

39 PSet Stats, Cofidece Iterval QC [APSTATSMC00-40] A studet workig o a history project decided to fid a 95 percet cofidece iterval for the differece i mea age at the time of electio to office for former America Presidets versus former British Prime Miisters. The studet foud the ages at the time of electio to office for the members of both groups, which icluded all of the America Presidets ad all of the British Prime Miisters, ad used a calculator to fid the 95 percet cofidece iterval based o the t-distributio. This procedure is ot appropriate i this cotet because (A) the sample sizes for the two groups are ot equal. (B) the etire populatio was measured i both cases, so the actual differece i meas ca be computed ad a cofidece iterval should ot be used. (C) electio to office take place at differet itervals i the two coutries, so the distributio of ages caot be the same. (D) ages at the time of electio to office are likely to be skewed rather tha bell-shaped, so the assumptios for usig this cofidece iterval formula are ot valid. (E) ages at the time of electio to office are likely to have a few large outliers, so the assumptios for usig this cofidece iterval formula are ot valid. QC [APSTATSMC01-] A radom sample of 50 studets at a large high school resulted i a 95 percet cofidece iterval for the mea umber of hours of sleep per day of (6.73, 7.67). Which of the followig statemets best summarizes the meaig of this cofidece iterval? (A) About 95% of all radom samples of 50 studets from this populatio would result i a 95% cofidece iterval (6.73, 7.67). (B) About 95% of all radom samples of 50 studets from this populatio would result i a 95% cofidece iterval that covered the populatio mea umber of hours of sleep per day. (C) 95% of the studets i the survey reported sleepig betwee 6.73 ad 7.67 hours per day. (D) 95% of the studets i this school sleep betwee 6.73 ad 7.67 hours per day. (E) A studet selected at radom from this populatio sleeps betwee 6.73 ad 7.67 hours per day for 95% of the time. 39

40 PSet Stats, Cofidece Iterval Aswers QC D. Assume that the coditios to use ormal distributio are satisfied. 0.3, 0.004, z QC B QC E. The populatio mea is out of CI. QC B. The variace is ukow. QC A. The lesser cofidece, the arrower the CI. QC C. The sample size is larger, so the sample variace is smaller. QC B. The meas calculated were the populatio meas -- o eed to estimate the mea. QC B. 40

41 PSet Stats, Cofidece Iterval 7.4. Cofidece Iterval for the Differece of Two Meas or Two Proportios The graph below shows the results of two treatmets i a eperimet. The two treatmets were idepedetly carried out so that the two distributios of two outcomes X 1ad X could be viewed as some idepedet radom variables with respective probability distributios. If the two distributios are cosidered approimately ormal distributios N( 1, 1 ) ad N (, ), the distributio of the differece of the two radom variables X X 1 X is N, or the pdf pis ( ) amazigly also ormally distributed with ( ) p( ) e e 1 1( ) Where 1 ad. 1 41

42 PSet Stats, Cofidece Iterval CASE #1 : The Differece of Two Proportios For two biomial distributios b( 1, p1, 1) ad b(, p, ), whe 1 ad are less tha 10% of populatio ad 1 pˆ 1, 1 (1 pˆ ˆ ˆ 1), p, (1 p) 10, the differece of the two proportios pˆ ˆ 1 pto estimate the differece of populatio meas p1 p ca be approimated as the ormal distributio ˆ1(1 ˆ1) ˆ (1 ˆ ) ˆ ˆ 1, p p p N p p p 1 The cofidece iterval CI is pˆ 1(1 pˆ 1) pˆ (1 pˆ ) pˆ 1(1 pˆ 1) pˆ ˆ (1 p) CI pˆ ˆ 1 p z, pˆ ˆ 1 p z 1 1 or 1 1 pˆ pˆ z 1 pˆ (1 pˆ ) pˆ (1 pˆ ) 1 [Ti-84] Estimatio of the Differece of Two Proportios. 1.) Stats -> TESTS ->B. -PropZit.) Iput 1, 1,,, 1. Eample Idepedet radom samples of 100 luury cars ad 50 o-luury cars i a certai city are eamied to see if they have bumper stickers. Of the 50 o-luury cars, 15 have bumper stickers ad of the 100 luury cars, 30 have bumper stickers. What is a 90% cofidece iterval for the differece i the proportio of o-luury cars with bumper stickers ad the proportio of luury cars with bumper stickers for the populatio of cars represeted by these samples? Verify your results by usig the calculator. 4

43 PSet Stats, Cofidece Iterval Solutio: , pˆ 1 0.5, 100, pˆ 0.3, Assume that 1 ad are less tha 10% of total umbers of luury cars ad o-luury cars, respectively. z z , 1 pˆ 1, 1 (1 pˆ ˆ ˆ 1), p, (1 p) 10. pˆ 1(1 pˆ 1) pˆ ˆ (1 p ) pˆ ˆ 1 p z 1 0.5(1 0.5) 0.3(1 0.3) or (0.108, 0.9) CASE #: Kow Uequal Variaces ad Large Sample Size For two ormal distributios N1, 1 ad, 1 ad N, whe 1, 30 ad variaces are kow ad uequal, the differece of the two meas 1 differece of populatio meas 1 has the distributio N cofidece iterval is 1 to estimate the,. The 1 1 or CI ( ) z, ( ) z ( 1 ) z 1 1 [Ti-84] Cofidece Iterval for the Differece of Two Meas with Kow Uequal Variaces ad Large Sample Size. 1.) Stats -> TESTS ->9. -SampZIt.) Iput 1, 1,,, 1. 43

44 PSet Stats, Cofidece Iterval Eample 7.4. A research team is iterested i the differece betwee serum uric acid levels i patiets with ad without Dow's sydrome. A sample of 4 idividuals with Dow's sydrome yielded a mea of mg/100 ml. A sample of 45 ormal idividuals of the same age ad se were foud to have a mea value of 3.4 mg/100 ml. If it is reasoable to assume that the two populatios of values are ormally distributed with variaces equal to 1 ad 1.5, fid the 95 percet cofidece iterval for. 1 Solutio: 4 30, 4.5, , 3.4, ( 1 ) z (0.390) , or CI (0.631,1.569) 1 That is, the differece betwee the two populatio meas is 1.1 ad we are 95% cofidet that the true differece betwee the meas lies betwee 0.6 ad CASE #3: Small Sample Size ad/or with Ukow Equal Variaces For sample size 1, 30 ad/or with ukow equal variaces, the pooled stadard error is used for estimatig differece of populatio meas 1 from the differece of the two meas 1 : ad s p 1 1 p ( 1) s ( 1) s ( ) s T( ). 1 44

45 PSet Stats, Cofidece Iterval The cofidece iterval is or CI ( ) t s, ( ) t s p 1 p ( ) 1 1 t s p 1 [Ti-84] Cofidece Iterval for the Differece of Two Meas with Ukow Equal Variace ad Small Sample Size. 1.) Stats -> TESTS ->10. -SampTIt.) Iput 1, 1,,, 1. Eample A eperimet was doe to compare the mea umber of tapeworms i the stomachs of sheep that had bee treated for worms versus those ot treated. There were 7 sheep i the treatmet group ad 7 i the cotrol group. The meas ad stadard deviatio are Treatmet Cotrol s What is the cofidece iterval for the differece of two meas at sigificat level 0.1? Solutio: The sample size is small ad it is reasoable to assume the variaces are equal. So the pooled estimate will be used. 8.57, 40.0, T C s 198.6, T s 15.33, 1 7, 7, 0.1, C t t t

46 PSet Stats, Cofidece Iterval The pooled stadard error is s p ( 1 1) s1 ( 1) s (7 1)(198.6) (7 1)(15.33) ( ) t s 1 T C p ( ) 1.78(14.387)

47 PSet Stats, Cofidece Iterval Quick-Check 7.4. Cofidece Iterval for Two Meas or Two Proportios QC [007APStats, #4] QC [APSTATSFRQ01-03] Idepedet radom samples of 500 households were take from a large metropolita area i the Uited States for the years 1950 ad 000. Histograms of household size (umber of people i a household) for the years are show below. A researcher wats to use these data to costruct a cofidece iterval to estimate the chage i mea household size i the metropolita area from the year 1950 to the year 000. State the coditios for usig a two-sample t-procedure, ad eplai whether the coditios for iferece are met. 47

48 PSet Stats, Cofidece Iterval Aswers QC A. It is a two-sample proportio problem with pˆ 15 ˆ , p 0.3, , 100, z z QC There are three coditios: 1. the data come from idepedet radom samples;. both sample sizes are large; 3. The populatio sizes are at least 10 (or 0) times the sample sizes. 48

49 PSet Stats, Cofidece Iterval 7.5. Cofidece Iterval for the Coefficiets of Regressio For the CI of coefficiet of the b 1 regressio lie ŷ b0 b1 with sample size of ad oe predicator, the equatio is give by CI b1 t sb, b 1 1 t s b 1 Where sb 1 is the stadard error of b 1 calculated by s b1 i ( y yˆ ) ( ) ( ) Eample [APSTATSMC014-30] A statistics teacher wats to determie whether there is a liear relatioship betwee high school studets heights, i iches (i), ad the legths of their feet, i cetimeters (cm). The teacher obtais height ad foot-legth measuremets for a radom sample of 3 studets at the high school ad geerates the followig graph ad computer output. Provided that the assumptios for regressio iferece are satisfied, What is a 95 percet cofidece iterval estimate of the slope of the populatio regressio lie for predictig foot legth from height? i i Solutio: 3, df 31, 0.05, t.07, SE slope (0.138) (0.96, 0.87) 49

50 PSet Stats, Cofidece Iterval Quick-Check Cofidece Iterval for the Coefficiet of Regressio QC [APSTATSMC013] As part of a class project at a large uiversity, Amber selected a radom sample of 1 studets i her major field of study. All studets i the sample were asked to report their umber of hours spet studyig for the fial eam ad their score o the fial eam. A regressio aalysis o the data produced the followig partial computer output. Amber wats to compute a 95 percet cofidece iterval for the slope of the least squares regressio lie i the populatio of all studets i her major field of study. Assumig that coditios for iferece are satisfied, which of the followig gives the margi of error for the cofidece iterval? QC [00APStats, #1] I a study of the performace of a computer priter, the size (i kilobytes) ad the pritig time (i secods) for each of small tet files were recorded. A regressio lie was a satisfactory descriptio of the relatioship betwee size ad pritig time. The results of the regressio aalysis are show below. Depedet variable: Pritig Time Source Sum of Squares df Mea Square F-ratio Regressio Residual Variable Coefficiet s.e. of Coeff t-ratio prob Costat Size R-squared = 87.5% R-squared (adjusted) = 86.9% s = with = 0 degrees of freedom 50

51 PSet Stats, Cofidece Iterval Which of the followig should be used to compare a 95 percet cofidece iterval for the slope of the regressio lie? (A) (B) (C) (D) (E)

52 PSet Stats, Cofidece Iterval Aswers QC A. s At 95% CI, b1 t t.8, margi of error is t s b 1 (.8)(0.745) QC A. b , s 0.94, df 0, 0.05, t.086. Without 0 1 b calculatig the critical t value, we should kow that it is greater tha

53 PSet Stats, Cofidece Iterval Coditios Idepedet radom sample 1.) Biomial b(, p, ).) 10% of populatio 3.) pˆ 10, ( 1 pˆ) 10 1.). Normal N,.) 10% of populatio 3.) is kow, 30 1.). t-distributio ~ T ( 1) s.) k 1deg. of freedom 3.) 5 30 or is ukow 1.) Biomial b(, p, ).) 1, 10% of populatio 3.) 1 pˆ ˆ 1, p 10, ˆ 1(1 p1) 10, (1 pˆ ) ) Normal N 1, 1.) 1 are kow, 1, 30 1.). Normal ( ) 1 1 s p T( ) 1.) k 1 is deg. of freedom 3.) 5 1, 30, ad/or 1, are equal ad ukow. 1.) t-dist, Coeff. of Regressio.) SE s b1 i ( y yˆ ) ( ) ( ) 3.) k, use z whe 30 i i Sample Statistic ˆp Cofidece Iterval TI-84: STAT->TESTS-> pˆ z z 1 t pˆ 1 pˆ pˆ pˆ 1 z pˆ(1 pˆ) s pˆ 1(1 pˆ ˆ ˆ 1) p(1 p) ( 1 ) z ( ) 1 1 t s p 1 1, are the sample meas ad s 1, s are the sample deviatios. s p ( 1) s ( 1) s CI b t s, b t s 1 b1 1 b1 b 1 or b1 t s b 1 Commets Oe-sample proportio, 1-PropZit Oe-sample mea, ZIterval Oe-sample mea, TIterval Two-sample proportio -PropZIt Two-sample mea with kow variaces -SampZIt Two-sample mea with ukow variaces -SampTIt ŷ b0 b1 53

54 PSet Stats, Cofidece Iterval 54