1. Suppose X is a variable that follows the normal distribution with known standard deviation σ = 0.3 but unknown mean µ.

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1 Chapter 9 Exercises Suppose X is a variable that follows the ormal distributio with kow stadard deviatio σ = 03 but ukow mea µ (a) Costruct a 95% cofidece iterval for µ if a radom sample of = 6 observatios of X has sample mea x = 5 (b) Suppose that we wat the etire width of the cofidece iterval to be equal to 004 Fid the sample size eeded A sample of size = 00 of a variable Y is take The sample mea of these 00 observatios is foud to be ȳ = 450 Assume that the populatio stadard deviatio is σ = 50 (a) Costruct a 95% cofidece iterval for µ, the populatio mea of Y (b) What sample size is eeded so that the legth of the iterval is 0 with 95% cofidece? 3 Five observatios of a variable W are take: 680, 705, 690, 783, ad 70 Costruct a 95% cofidece iterval for µ, the populatio mea of W State ay assumptios eeded for this cofidece iterval to be valid 4 I a rural area of a developig coutry, a survey is coducted to estimate the proportio, p, of households that have access to clea water Out of the 000 households survey, 650 reported they have clea water (a) Costruct a 95% cofidece iterval for p ad state all assumptios (b) Fid the sample size eeded so that the margi of error will be ±00 with cofidece level 95% 5 A researcher wats to estimate the prevalece of a disease i a coutry What sample size should be used if she desires to be 95% cofidet that the fial estimate is withi 005 of the true prevalece? 6 A studet recorded the duratio (T i miutes) o 0 occasios whe the course website was dow: 59, 5, 57, 88, 0, 83, 35, 9, 85, 73 96, 75, 03,, 0, 05, 09, 59, 04, 05

(a) The studet wats to study the populatio mea duratio of time that the website is dow She assumes that the data is ormally distributed Based o this assumptio, fid a 95% cofidece iterval of the mea

2 (a) The studet wats to study the populatio mea duratio of time that the website is dow She assumes that the data is ormally distributed Based o this assumptio, fid a 95% cofidece iterval of the mea duratio (b) Suppose she is subsequetly told that T actually follows a Exp(λ) distributio Based o this ew piece of iformatio, fid aother 95% cofidece iterval (c) Compare the aswers i (a) ad (b) ad discuss their differeces 7 The Farøes is a group of islads situated about half way betwee Norway ad Icelad The islads have bee a depedecy of the Kigdom of Demark sice the 300s However, over the past few decades, there have bee icreasig desire from ihabitats o the islads to seek idepedece from Demark A radom sample of 00 ihabitats are used i a survey, where each perso gives their opiio (X) o whether Farøe Islads should become a idepedet coutry The survey results are as follows: (x,, x 00 ) = ( 0, 0,, 0,,,, ), where x }{{}}{{} i = if the i-th perso supports 636 observatios 564 observatios idepedece ad x i = 0, otherwise Suppose the observatios are IID Beroulli(p), where p represets the proportio of all ihabitats who wat the islads to be idepedet (a) How may observatios of X are there? (b) Fid the MLE, ˆp, of p, based o the give data? (c) Use the CLT to fid a 95% cofidece iterval for p (d) What is the margi of error i your estimate? (e) A local politicia claims that there is eough evidece i the results to suggest that 50% of all ihabitats of Farøe Islads wat idepedece Does your aalysis shed some light o her commets?

3 8 The ame of Farøe Islads is derived from the word Faøroyar, meaig sheep Sice Vikigs time, wool products have bee of major importace for the subsistece of the islads A particular farm ows a herd of

rescue Suppose o 6 out of 0 days, a sheep would require rescue (a) Use the CLT to fid a 95% cofidece iterval for p, the proportio of days whe a sheep requires rescue You may use some of the results

3 3 8 The ame of Farøe Islads is derived from the word Faøroyar, meaig sheep Sice Vikigs time, wool products have bee of major importace for the subsistece of the islads A particular farm ows a herd of sheep The sheep are free to roam aroud the moutais surroudig the farm The sheep are atural climbers ad they ca scale the steepest of all slopes However, occasioally, a sheep may be trapped ad require rescue Suppose o 6 out of 0 days, a sheep would require rescue (a) Use the CLT to fid a 95% cofidece iterval for p, the proportio of days whe a sheep requires rescue You may use some of the results you foud i Questio to aswer this questio (b) What is the margi of error i your estimate? (c) If it is desired to reduce the margi of error by a factor of /, how much the sample size eeds to be icreased? (ie, if E is the margi of error uder the curret sample, the we wat E = E uder the ew sample size) 9 Tourism accouts for a substatial part of the islads ecoomy Apart from the spectacular sceery ad ladscape, may visitors to the islads wat to see the orther lights, or Aurora Borealis Norther lights are display of lights formed from the collisio of solar clouds ad the Earth s magetic field ad are best observed at ight i the orther hemisphere Let X be the duratio (i miutes o the log-scale) of the display of lights o ay particular occasio ad suppose X N(µ, σ ), with (µ, σ ) ukow Suppose the duratios (i miutes o the log-scale) of a radom sample of 30 displays are recorded ad they are: 53, 68, 5, 69, 6, 4, 57, 59, 37, 56, 60, 77, 34, 45, 59, 4, 45, 63, 58, 50, 7, 4, 54, 56, 53, 54, 79, 50, 49, 58 (a) Fid the MLE, (ˆµ, ˆσ ), of (µ, σ ), based o the give data? (b) Use the CLT to fid a 95% cofidece iterval for µ (c) Express your 95% cofidece iterval from (b) i terms of miutes i its origial scale (d) What should be the sample size if we wat to reduce the margi of error (i log-scale) to ± 0?

4 0 A group of scietists studyig global warmig has arrived the islads They took 60 observatios of the time (i days) betwee days whe the temperature o the islads exceeded 0 degrees Celsius Suppose the

4 4 0 A group of scietists studyig global warmig has arrived the islads They took 60 observatios of the time (i days) betwee days whe the temperature o the islads exceeded 0 degrees Celsius Suppose the data are IID Exp(λ), where represets the mea time λ betwee days with temperature exceedig 0 degrees Celsius Suppose x = i= x i = 98 (i days) (a) Fid the MLE of λ, based o the give data? (b) Use the CLT to fid a 95% cofidece iterval for λ You may use the fact that var(/ x) λ / for reasoably large values of Note that i geeral var(/ x) does NOT equal /var( x) (c) Fifty years ago, λ = 004 (or mea time= 5 days) Does your aalysis give evidece that λ has chaged from 50 years ago? Arguably the biggest idustry o the islads is fishig ad fish farmig Recetly, fisherme have bee complaiig that their icome are droppig due to competitio ad over fishig i the waters surroudig the islads Data are collected to determie whether there is eough evidece to support the fisherme s claims The data cosist of records of the catch X (i 000 kg, same below) from m = 80 fishig trawlers five years ago ad the catch (Y ) from = 70 fishig trawlers this year Summary statistics of the data are give below: x ȳ = m = m x i = 753, i= y i = 558, i= m m (x i x) = 8004, i= (y i ȳ) = 88 i= Assume X N(µ X, σx ) ad Y N(µ Y, σy ), where (µ X, σx ) ad (µ Y, σy ) are ukow Furthermore, assume all data are idepedet of each other (a) Fid the MLE of µ X ad µ Y ad hece, fid the MLE for µ X µ Y, usig the give data You may use established results from Questio 3 (b) Let ˆµ X Y be the MLE of µ X µ Y Show that var(ˆµ X Y ) = σ X m + σ Y (Hit: Use the fact that var(ˆµ X ) = σ X m, var(ˆµ Y ) = σ Y 5) ad recall the rules of var(x + Y ) i Chapter (c) Use the CLT to fid a 95% cofidece iterval for µ X µ Y Does your aalysis give evidece that the amout of catch has depleted compared to five years ago?

5 5 ANSWERS (a) A 95% cofidece iterval is x ± 96 σ = 5 ± = 5 ± 047 We are 95% cofidet that µ is betwee 4853 ad 547 (b) The width of the cofidece iterval is If we wat the width to be o more tha 004, the we fid such that 004 = ( = ) (a) A 95% cofidece iterval is x ± 96 σ = 450 ± = 450 ± 98 We are 95% cofidet that µ is betwee 440 ad 4598 (b) The width of the cofidece iterval is If we wat the width to be o more tha 0, the we fid such that 0 = ( = ) (3) We may be able to cosider a 95% cofidece iterval x ± 96 σ if the sample comes from a ormal distributio ad σ is kow However, i this case, σ is ukow ad sice the sample size is small, we estimate σ usig a sample estimate ˆσ ad we replace 96 by a umber from the t-table Sice = 5, df = = 4, the umber we use is 776, hece a 95% cofidece iterval is x ± 776 ˆσ 7 ± ± 508

6 where ˆσ is the sample stadard deviatio, i= (x i x) (this estimate is better tha the alterative estimate i= (x i x), sice = 5 is quite small) Therefore, we are 95% cofidet that µ is betwee 66 ad 768 (b) The width of the cofidece iterval is If we wat the width to be o more tha 0, the we fid such that 0 = ( = ) (4a) We assume there is a probability p that a household has access to clea water ad the chace households have access to clea water are idepedet A 95% cofidece iterval is p( p) ˆp ± 96 Sice p is ukow, we estimate the margi of error usig ˆp, givig ( ˆp( ˆp) ˆp ± 96 = ) ( 350 ) 000 ± = 065 ± We are 95% cofidet that p is betwee 06 ad 068 (b) The margi of error is p( p) 96 Sice p is ukow, ad the largest margi of error, for a particular value of is whe p = 05, the we fid such that 00 = 96 05( 05) = ( ) 96 (05) = Sice a prevalece is a proportio, 0 < p < which is ukow, the a cofidece iterval estimate has the form: p( p) ˆp ± 96 meaig that we are 95% certai that p is from ˆp by the margi of error, 96 p( p)

7 7 Usig the same argumet as i Questio 4, we replace the ukow p by the value 05 that would lead to the largest margi of error, the we fid such that 005 = 96 05( 05) = ( ) 96 (05) = (6a) Based o the iformatio, we may cosider a 95% cofidece iterval x ± 96 σ if the sample comes from a ormal distributio ad σ is kow However, i this case, σ is ukow ad sice the sample size is small, we estimate σ usig a sample estimate ˆσ ad we replace 96 by a umber from the t-table For = 0, df = = 9, the umber we use is 093, hece a 95% cofidece iterval is x ± 093 ˆσ 56 ± ± 0 where ˆσ is a sample estimate of the populatio stadard deviatio We use ˆσ = i= (x i x) here; alteratively, we could have used i= (x i x) but for small, the former is better We are 95% cofidet that the mea is betwee 34 ad 78 (b) Assumig the observatios follow a Exp(λ) distributio, the the mea /λ ca be estimated by /ˆλ = x However for a expoetial distributio, the stadard deviatio is also /λ hece, we also use x to estimate the stadard deviatio So as log as the sample size is assumed to be big, a approximate 95% cofidece iterval is x ± 96 x 56 ± ± 46 We are 95% cofidet that the mea is betwee 35 ad 806 (c) Comparig (a) to (b), the mai differece is the way the margi of error, 96ˆσ/, is estimated We aim to estimate that as well as possible The estimate usig ˆσ = x is the MLE whe the data follow a expoetial distributio ad hece (b)is better tha (a) uder that assumptio Alteratively, ˆσ estimated by i= (x i x) is a simple sample stadard deviatio without ay assumptios; furthermore, whe the ormality assumptio holds, (a) gives a cofidece iterval with correct level of cofidece ad (a) is better tha (b) because i that case ˆσ = x is biased for σ To coclude, we choose a cofidece iterval that utilizes the iformatio that is give (7a) Each x i, i =,, 00 is a observatio of X Therefore, there are = 00 observatios

8 8 (b) Let ˆp be the MLE, the ˆp = x = i= x i = (c) Accordig to the CLT, i a radom sample of size, as log as is large, ( ) p( p) ˆp N p, var(ˆp) = Therefore, usig the CLT, a 95% cofidece iterval for p is p( p) ˆp( ˆp) ˆp ± 96 ˆp ± 96 = ± ( ) 00 = 047 ± 008 (d) The margi of error is 008 (e) Accordig to the 95% cofidece iterval, the level of support is betwee ( , ) = (044, 0498) Sice the upper limit is less tha 05, we ca say that we are 95% certai that the politicia is wrog (8a) The sample size is = 0 Usig the CLT, a 95% cofidece iterval for p is ˆp ± 96 p( p) ˆp ± 96 ˆp( ˆp) = 6 0 ± 96 6 ( 6 ) = 005 ± (b) The margi of error is (c) Let m be the ew sample size, so we wat ( p( p) 96 = ) p( p) 96 }{{ m } }{{} ew margi of error old margi of error = ( ) m m = ( ) 4 m = 4 The aswer shows that the ew sample size should be 480 = 4 0 Therefore, we eed 4 times the origial sample size to reduce the margi of error by a factor of / The

9 9 geeral rule is, for a reductio of every factor of / i the margi of error, we require a 4-fold icrease i the sample size For example, if we wat to reduce the margi of error by a factor of /6, the sice 6 =, we eed to icrease the sample size by = 56 times (9a) Let x,, x be iid N(µ,σ ) The MLE (ˆµ, ˆσ) are: ˆµ = x = , ˆσ = (x i x) = 003 so i terms of miutes, the mea duratio is exp( ) or about 47 miutes (b) Usig the CLT, ˆµ N(µ, var(ˆµ) = σ ) Therefore, usig the CLT, if we use value from the t-table based df = = 30 = 9, a 95% cofidece iterval for µ is i= σ ˆσ 003 ˆµ ± 045 ˆµ ± 045 = ± = ± (c) From (b), the 95% cofidece iterval o log-scale ca be writte as ( , ) which, i terms of miutes, is [exp( ), exp( )] (636, 3709) (d) The expressio for the margi of error (o log-scale) is σ , usig the sample size of = 30 ad estimatig σ by ˆσ = 003 i the expressio for margi of error To reduce the margi of error to ± 0, we ca approximate the ew sample size by: σ 045 = 0,

10 0 ad solve for The above equatio gives: σ = ( σ 0 = 045 (0a) The MLE of λ is: = ( σ ) = ( σ 0 ) 045 ( ˆσ ˆλ = i= x i ) ( ) = 003 ) = 56 6 = x = 98 = (b) Usig the CLT, ˆλ N(λ, var(ˆλ)) Assumig is large eough ad sice we are usig a MLE, therefore, var(ˆλ) λ But λ is ukow, so we estimate it by ˆλ Therefore, usig the CLT, a 95% cofidece iterval for λ is λ ˆλ ± 96 ˆλ ˆλ ± 96 = 98 ± = ± (c) Accordig to the 95% cofidece iterval, λ is betwee ( , ) = (00377, 00637) Sice the iterval icludes 004, we caot say that the rate is differet from 50 years ago (a) From Questio 9, we kow the MLE of (µ X, σx ) based o (x,, x m ) are ˆµ X = x = 753, ˆσ X = m m (x i x) = 8004 i= Similarly, the MLE of (µ Y, σ Y ) based o (y,, y ) are ˆµ Y = ȳ = 558, ˆσ Y = (y i ȳ) = 88 i=

11 Therefore, a estimate for µ X µ Y is ˆµ X ˆµ Y = = 95 (b) Recall i Chapter 5, we leared that, for idepedet radom variables X ad Y, var(x Y ) = var(x) + var(y ) var(ˆµ X Y ) = var(ˆµ X ˆµ Y ) = var(ˆµ X ) + var(ˆµ Y ) }{{} X s ad Y s are idepedet samples = var( x) + var(ȳ) = σ X m + σ Y }{{ } From Questio 3 (c) Usig the CLT for MLE, ˆµ X Y N(µ X µ Y, var(ˆµ X Y )) = N where the last result comes from (b) ( ) µ X µ Y, σ X m + σ Y, Therefore, usig the CLT, a 95% cofidece iterval for µ X µ Y is σ ˆµ X ˆµ Y ± 96 X m + σ Y ˆσ ˆµ X ˆµ Y ± 96 X m + ˆσ Y 8004 = 95 ± = 95 ± 30 Accordig to the 95% cofidece iterval, the mea differece is betwee (95 3, ) = (7, 38) Sice the lower limit of the iterval is above zero, we are 95% certai that the average catch has decreased by more tha 700 kg from five years ago So the claims from the fisherme are supported

Estimating Proportions with Confidence

Estimating Proportions with Confidence Aoucemets: Discussio today is review for midterm, o credit. You may atted more tha oe discussio sectio. Brig sheets of otes ad calculator to midterm. We will provide Scatro form. Homework: (Due Wed Chapter