# Chapter 7: Confidence Interval and Sample Size

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1 Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum sample size whe determiig a cofidece iterval for the mea ad for a proportio. Level of cofidece, maximum error of Estimate (E) ad the sample size are iter-related. I. Iferece Icludes: 1. Estimatio of a populatio parameter (μ, ρ, or ) usig data from a sample. 2. Hypothesis Testig or usig sample data to test a cojecture about the populatio mea (μ), populatio proportio (ρ), or populatio stadard deviatio ( ). II. Two Kids of Estimate for Parameters 1. A poit estimate of the populatio parameter is the sample statistic, i.e., the poit estimate for the populatio mea μ is the sample mea of, the poit estimate for the populatio proportio is the sample proportio, ad the poit estimate for the populatio stadard deviatio is the sample stadard deviatio s. 2. A iterval estimate of a parameter is a rage of values determied from the poit estimate. Dr. Jaet Witer, Stat 200 Page 1

2 III. Cofidece Iterval Estimates for Populatio Parameters The cofidece level is the probability that itervals determied by these methods will cotai the parameter. A cofidece iterval is the rage of values determied from a sample statistic ad the specified cofidece level. The commo cofidece itervals use 90%, 95%, or 99% cofidece levels. IV.Cofidece Iterval Estimates for the Populatio Mea μ A. Whe to use the Normal Distributio (z) ad whe to use the t Distributio for Cofidece Iterval Estimates of the Populatio Mea Start Yes Is σ kow? No Yes Is the populatio ormally distributed? No Yes Is the populatio ormally distributed? No Yes Is > 30? No Yes Is > 30? No z Use the ormal distributio Use oparametric or bootstrappig methods. t Use the t distributio Use oparametric or bootstrappig methods. Elemetary Statistics: Usig the Graphig Calculator for the TI-83/84, Triola, Mario F. Dr. Jaet Witer, Stat 200 Page 2

3 B. Roudig Rules for all Cofidece Itervals Estimates of the Mea I. Whe usig actual data: a) fid the mea ad stadard deviatio to 2 extra places tha the data. b) roud the aswer to oe more decimal place tha the origial data. Note: This is very importat! Aswers ot rouded correctly are marked wrog o Mathzoe. II. Whe usig a mea ad stadard deviatio, work with oe more decimal place tha the data ad roud to the same umber of decimal places give for the mea. C. Meaig of ALL Cofidece Iterval Estimates Be sure to reread P 353 (6th editio) or P 361 (7th editio) i the textbook to better uderstad the meaig of the cofidece iterval. For example: a 90% cofidece iterval estimate for the populatio mea is iterpreted as 90% of the cofidece iterval estimates formed with this process iclude the value of the populatio mea. D. z Iterval Estimates for the populatio Mea I. Requiremets a) the populatio stadard deviatio ( ) is give b) the sample size 30; c) But, if the sample size < 30, the variable must be selected from a ormal distributio II. Cofidece Coefficiet Dr. Jaet Witer, Stat 200 Page 3

4 a) Meaig of the Cofidece Coefficiet z is called the cofidece coefficiet, i.e., the umber of multiples of the stadard error for a iterval estimate with a level of cofidece. Complete the rest of the table usig the cofidece level (1- ). The first 2 have bee completed for you (aswers at the ed) b) Method to fid the Cofidece Coefficiet: Fid the z value with area to its left, i.e., 1. Locate iside the Normal Probability Table (Table E) 2. Startig at, move your had to the left alog the row util you reach the Z colum. This is the iteger ad teths digits. Go back to, ext move your had to the top of its colum. This is the hudreds digits. 3. Add the iteger ad teths digits to the hudredths digits to fid the value for z. 4. Affix a sig i frot of the umber. Dr. Jaet Witer, Stat 200 Page 4

5 Usig the method described, complete the table below. The first 2 have bee completed for you (aswer at the ed). Cofidece Level 1 α α α/2 α Cofidece Coefficiet (1 α ) + 2 z(a/2) III. Developmet of the Cofidece Iterval Formula σ x z < μ < x + z σ Wheever the populatio stadard deviatio σ is kow ad either the populatio is ormally distributed or 30, the Cetral Limit Theorem guaratees the sample mea is ormally distributed or: z < x μ σ x < z z σ x < x μ < z σ x z σ x < μ < x + z σ x ( x z σ x ) < ( μ) < ( x + zσ x ) ( x z σ x ) > ( μ) > ( x + zσ x ) x + z σ x > μ > x zσ x x z σ x < μ < x + zσ x x z σ < μ < x + z σ Note: If the populatio stadard deviatio is ot kow or stated, use x t s s < μ < x + t (see sectio E page 9). Dr. Jaet Witer, Stat 200 Page 5

6 IV. Review of Cocepts ad Maximum Error of Estimate is the poit estimate ad the ceter of the cofidece iterval z is the cofidece coefficiet, the umber of multiples of the stadard error eeded to costruct a iterval estimate of the correct width to have a level of cofidece 1 α is called the maximum error of estimate. V. Example: 35 fifth-graders have a mea readig score of 82. The stadard deviatio of the populatio is 15. a) Fid the 95% cofidece iterval estimate for the mea readig scores of all fifthgraders. Sice we kow the populatio stadard deviatio ad 30, use. Use Table E backwards with the area to the left of z equal to.025. The value of z or the cofidece coefficiet is z = = This meas approximately 95% of the sample meas will fall withi 1.96 stadard errors of the populatio mea. Use z = 1.96 i the formula. 4.97, rouded to 5, is the maximum error of estimate. Be sure to list it for full credit i your aswers. Dr. Jaet Witer, Stat 200 Page 6

7 b) Fid the 99% cofidece iterval estimate of the mea readig scores of all fifthgraders. Sice approximately 99% of the sample meas will fall withi 2.58 stadard errors of the populatio mea, use z = 2.58 X = 82.1, = 35, σ = 15 Questio 1 X z σ < μ < X + z σ < μ < < μ < ± < μ < , rouded to 6.5, is the maximum error of estimate. Be sure to list it i the ext to last step. c) Is the 95% cofidece iterval or the 99% cofidece iterval larger? Explai why. 95% cofidece level: 77 < μ < 87 99% cofidece level: 75 < μ < 89 The 99% cofidece level is larger because it has a larger z value. A study of 40 Eglish compositio professors showed that they spet, o average, 12.6 miutes correctig a studet s term paper. Fid the 90% cofidece iterval of the mea time for all compositio papers whe σ = 2.5 miutes. = 40 X = 12.6 Sice the populatio stadard deviatio is give ad = 40 is greater tha 30, use the formula: X z σ < μ < X + z σ If a professor stated that he spet, o average, 30 miutes correctig a term paper, what would be your reactio? Dr. Jaet Witer, Stat 200 Page 7

8 VI. Maximum Error of Estimate for Cofidece Iterval Estimates of μ a) Defiitio The maximum error or estimate is always the largest differece betwee the poit estimate of a parameter ad the actual value of the parameter. The maximum error of estimate is ½ the width of the cofidece iterval. b) Maximum Error of Estimate for Cofidece Iterval Estimates of μ It is the term E = z σ VII. Fid the Sample Size Usig E ad the Cofidece Level a) Cocept: E is like tolerace or allowable error where: E = z σ E = zσ = zσ E = zσ E 2 b) Formula for the Miimum Sample Size for a Iterval estimate of the populatio mea = zσ E 2 where E is the maximum error of estimate. If the aswer is ot a whole umber, roud up to the ext larger whole umber to fid the sample size,. If the populatio stadard deviatio is ot available use the sample stadard deviatio. c) Example: A isurace compay is tryig to estimate the average umber of sick days that fulltime food service workers use per year. A pilot study foud the stadard deviatio to be 2.5 days. How large a sample must be selected if the compay wats to be 95% cofidet of gettig a iterval that cotais the true mea with a maximum error of 1 day? s= 2.5 cofidece level = 95% maximum error = 1 day = zσ 2 E = 1 = = 25 workers 2 = or Dr. Jaet Witer, Stat 200 Page 8

9 Questio 2 Fid the sample size ecessary to estimate a populatio mea to withi 0.5 with 95% cofidece if the stadard deviatio is 6.2 z σ 2 = E Note: Whe solvig for sample size, always roud up to the ext larger iteger. E. t Cofidece Iterval Estimates for the Populatio Mea I. Requiremets a) σ is ukow b) 30 c) But, if < 30, the variable is ormally distributed. II. Characteristics of the t Distributio Similarities with the ormal distributio: a) Bell shaped. b) Symmetrical about the mea. c) The mea, media, ad mode are equal to 0 at the ceter of the distributio. d) The curve ever touches the x axis. a) The variace is greater tha 1. Differeces from the stadard ormal: b) The t distributio is actually a family of curves based o the degrees of freedom, which is related to sample size. c) As the sample size icreases, the t distributio approaches the stadard ormal distributio. Dr. Jaet Witer, Stat 200 Page 9

10 Read textbook page 362 (6 th Editio) or page 370 (7 th Editio) for the compariso betwee Normal ad t distributios. (Triola & Triola, 2006) III. Tabled Values for the t Table F: a) Locatio 6 th Editio: Table F located o the iside cover of the text o the opposite side from Table E (stadard ormal). 7 th Editio: Table F located o the last page of the textbook or the pull-out card. b) Method to fid the cofidece coefficiet for t Use the colum for the appropriate cofidece level Use the row for the appropriate degrees of freedom. The itersectio of the appropriate colum ad appropriate row is the cofidece coefficiet. Note: If the degrees of freedom eeded are ot listed i the table, always roud dow to the earest table value. For example, if we eed degrees of freedom 44, use df=40 sice 44 is ot listed i the table. IV. Degrees of Freedom for Estimates of the Populatio Mea Degrees of freedom are the umber of values that are free to vary after a sample statistic has bee computed. For the cofidece iterval for the mea the degrees of freedom are: sample size mius 1 or d.f. = 1 Dr. Jaet Witer, Stat 200 Page 10

11 V. Example: 28 employees of XYZ Compay travel a average (mea) of 14.3 miles to work. The stadard deviatio of their travel time was 2 miles. Fid the 95% cofidece iterval of the true mea or populatio mea. Sice the populatio stadard deviatio is ot give, use the formula: X t s s < μ < X + t = 28 X = 14.3 s = 2 df: < μ < < μ < ± < μ < 15.1 VI. Example: The average yearly icome for 28 egieerig graduates i 2008 is \$56,718. The stadard deviatio was \$ Fid the 95% cofidece iterval estimate for the populatio mea. Sice the populatio stadard deviatio is ot give, use the formula: X t s s < μ < X + t = 28 X = s = 650 df: 27 \$56, < μ < \$56, < μ < \$56, 718 < μ < \$56,970 Note: Now that you are familiar with this problem, it is simpler to record the steps: ± ± ± 252 (rouded to the same umber of places as the mea) < μ < If a idividual graduate wishes to see if he or she is beig paid below average, what salary value should he or she use? Use the lower boud of the cofidece iterval: \$56,466. Dr. Jaet Witer, Stat 200 Page 11

12 Questio 3 The prices (i dollars) for a particular model of 6.0 megapixels digital camera with 3x optical zoom are listed as: \$225, \$240, \$215, \$202, \$206, \$211, \$210, \$193, \$250, \$225. Estimate the true mea usig this data with 90% cofidece. Sice the populatio stadard deviatio is ot give, use: X ± t s. Do ot use the σ from the calculator. This is a sample, so be sure to use s ad work i 2 more places tha the data ad roud the aswers to oe more place tha the data. X = s = t = df: 9 = 10 Questio 4 Joh wats to estimate the average value of the homes i his tow with a 99% cofidece iterval. Use his radom sample of 36 homes with a average value of \$251, ad stadard deviatio \$ to fid the cofidece iterval. Sice the populatio stadard deviatio is ot give, use the formula - X ± t s.. The degrees of freedom equals 35, but df = 35 is ot available i the table. Use the ext lower df or df = 34. V. Cofidece Iterval Estimates for Populatio Proportios A. Symbols Used to Estimate Proportio p = symbol for the populatio proportio p = symbol for the sample proportio; read p hat q = 1 p = symbol for the same proportio of failures. Where x = umber of sample uits that possess the characteristics of iterest ad = sample size. p + x B. Developmet of the Formula For a Biomial Probability Distributio with x = umber of successes For x: with p 5 ad (1 p) 5 X = umber of successes is approximately ormally distributed with: μ = p σ = p(1 p) Thus for proportios p = x μρ = μ = p = p Dr. Jaet Witer, Stat 200 Page 12

13 . 1. The mea of p is p 2. The stadard deviatio for p becomes: σρ = σ = p(1 p) p(1 p) σρ = 2 p(1 p) σρ = Next we use the patter for the cofidece iterval estimate of the populatio mea, poit estimate ± z stadard deviatio It becomes p ± z p (1 p ) C. Formula for the cofidece iterval estimate of the populatio proportio Note: a shorter versio of the formula to estimate the populatio proportio is: p ± z p q whe p ad q are each greater tha or equal to 5. D. Roudig Rules for proportios Always use 4 decimal places for the computatio ad roud the aswers to 3 decimal places. E. Example: studets i a radom sample of 450 erolled i summer classes. Estimate the populatio proportio of studets takig classes this summer. p = X = 55 = p z p q < p < p + z p q q = = ± ± < p < % < p < 15.2% Dr. Jaet Witer, Stat 200 Page 13

14 2. Is a estimate of 11% about right? Questio 5 Yes, 11% is about right sice it is cotaied withi the cofidece iterval estimate. A survey foud that out of 200 studets, 168 said they eeded loas or scholarships to pay their tuitio ad expeses. Fid the 90% cofidece iterval for the populatio proportio of studets eedig loas or scholarships. p = q = p q p z < p < p + z p q Questio 6 A study by the Uiversity of Michiga foud that oe i five 13 ad 14 year olds is a sometime smoker. To see how the smokig rate of the studets at a large school district compared to the atioal rate, the superitedet surveyed two hudred 13 ad 14 year old studets ad foud that 23% said they were sometime smokers. Fid the 99% cofidece iterval of the true proportio ad compare this with the Uiversity of Michiga s study. p q = 200 p = 0.23 q = = 0.77 p z < p < p + z p q F. Formula for the Miimum Sample Size to Estimate a Populatio Proportio E = z p q E z = p q E 2 z = p q z 2 E = p q z 2 E 2 = p q p q z2 = E2 = p q z 2 E Use p from a pilot study or previous estimate if it is available. Otherwise, use p =.5. must be a whole umber. If it s ot a whole umber, roud up to the ext larger whole umber. Dr. Jaet Witer, Stat 200 Page 14

15 G. Example: A medical researcher wishes to determie the percetage of drivers usig GPS systems i their car. He wishes to be 99% cofidet that the estimate is withi 2 percetage poits of the true proportio. A recet study of 180 drivers showed that 25% used GPS systems. a) How large should the sample size be? Sice a recet study showed 25% used GPS Systems, p = 0.25 ad q = = p q z E 2 = = Sice the computed is ot a whole umber, roud up ad use = b) If o estimate of the sample proportio is available, how large should the sample be? Sice there is o prior estimate of p, use p = 0.5 ad q = 0.5 = p q z E 2 = = Sice the computed is a ot a whole umber, roud up ad use = Note: the sample size eeds to be larger whe there is o prior estimate for p. VI.Cofidece Iterval Estimate for the Populatio Variace ad Populatio Stadard Deviatio A. Geeral Commets To fid cofidece itervals for variaces ad stadard deviatios, Use the chi-square distributio Samples must be selected from ormally distributed populatios. Assume the populatio variace is σ 2. The chi-square distributio is obtaied from the values of ( 1)s2 or x 2 = ( 1)s2 σ 2 σ 2 Dr. Jaet Witer, Stat 200 Page 15

16 B. Chi-Square Distributio Referece textbook page 378 (6 th Editio) or page 386 (7 th Editio). I. Characteristics Chi-Square is always positive. It is a family of distributios depedet o degrees of freedom ( 1). The mode is always slightly to the left of degrees of freedom. As icreases, Chi-Square walks off to the right. Chi-Square distributio is skewed to the right. II. Fidig Chi-Square values o the Chi-Square table Sice is ot symmetrical, two differet values are used i the cofidece iterval formula for the populatio variace. For right, use the colum. For left, use the colum i the table for. Process: 1. Use the cofidece level to fid. 2. Use the colum with the appropriate degrees of freedom to fid right. 3. Fid. 4. Use the colum with the appropriate degrees of freedom to fid left. Chi-Square Values of df: 18 Chi-Square left Left Cofidece Level Right Chi-Square right Dr. Jaet Witer, Stat 200 Page 16

17 C. Formulas 1. Cofidece Iterval Estimate for the Populatio Variace: df: - l Note: right is o the left side of the equatio but the right side of the graph ad left is o the right side of the equatio but the left side of the graph. 2. Cofidece Iterval Estimate for the Populatio Stadard Deviatio Sice the populatio stadard deviatio is the square root of the populatio variace, the cofidece iterval estimate of the populatio stadard deviatio is: D. Roudig Rules for Stadard Deviatio or Variace I. Whe usig actual data: a) fid the stadard deviatio to 2 extra places tha the data. b) roud the aswer to oe more decimal place tha the origial data. II. Whe usig sample stadard deviatio or variace, work with oe more decimal place tha the statistic ad roud to the same umber of places as the stadard deviatio or variace give. E. Example: Fid the cofidece iterval for the stadard deviatio i the time it takes to fill a car with gas. I a sample of 23 fill-ups, the stadard deviatio of the time it takes to fill the car is 3.8 miutes. Assume the variable is ormally distributed. Dr. Jaet Witer, Stat 200 Page 17

18 Note: the aswer has the same umber of decimal places as the give sample stadard deviatio sice the work is doe with statistics istead of data. Questio 7 Fid the 99% cofidece iterval for the variace ad stadard deviatio of the weights of oe-gallo cotaiers of motor oil whe the sample of 14 cotaiers has a variace of 3.2. Assume the variable is ormally distributed. F. Example: The umber of calories i a 1-ouce servig of various regular cheeses is show. Estimate the populatio variace with 90% cofidece Is the 90% cofidece iterval estimate for the populatio variace. Note: Use the tabled values for ad use s rouded to 2 more places tha the data i the computatio. Sice data is used to compute the stadard deviatio, the aswer has oe more place tha the origial data. Dr. Jaet Witer, Stat 200 Page 18

19 Questio 8 A service statio advertises a wait of o more tha 30 miutes for a oil chage. A sample of 28 oil chages has a stadard deviatio of 5.2 miutes. Fid the 95% cofidece iterval of the populatio stadard deviatio of the times spet waitig for a oil chage. ( 1)s 2 x 2 right < σ2 < ( 1)s2 x 2 left VII. Summaries A. Estimates for Populatio Parameters Estimatio is a importat aspect of iferetial statistics. A poit estimate is a sigle value with o accuracy specified. A iterval estimate is a rage of values with its accuracy specified by the cofidece level. Every questio about a cofidece iterval will have the words Fid a cofidece iterval estimate for. Pay particular attetio to the words determie whether the cofidece iterval is for the mea, proportio, or variace (or its square root, the stadard deviatio). B. Miimum Sample Sizes to Estimate Populatio Parameters You always eed to kow both the cofidece level ad the maximum error of estimate. I additio, for the: 1. Mea the populatio stadard deviatio (give or estimate) is also required. z σ 2 = E 2. Proportio a estimate of the proportio from a pilot study is preferred (or use p =.5). = p q z E 2 Dr. Jaet Witer, Stat 200 Page 19

20 C. Roudig Rules I. For Estimates of the Mea a) Whe usig actual data: (a) fid the mea ad stadard deviatio to 2 extra places tha the date. (b) roud the aswer to oe more decimal place tha the origial data. Note: This is very importat! Aswers ot rouded correctly are marked wrog o Mathzoe. b) Whe usig a mea ad stadard deviatio, work with oe more decimal place tha the data ad roud to the same umber of decimal places give for the mea. II. For Estimates of the Stadard Deviatio or Variace a) Whe usig actual data: (a) fid the stadard deviatio to 2 extra places tha the data. (b) roud the aswer to oe more decimal place tha the origial data. b) Whe usig sample stadard deviatio or variace, work with oe more decimal place tha the statistic ad roud to the same umber of places as the stadard deviatio or variace give. III. For Estimates of the Proportios a) Always use 4 decimal places for the computatio ad roud the aswers to 3 decimal places. Dr. Jaet Witer, Stat 200 Page 20

21 Aswer: Cofidece Coefficiet z is called the cofidece coefficiet, i.e., the umbers of multiples of the stadard error for a iterval estimate with a 1 level of cofidece Aswer: Method to fid the Cofidece Coefficiet 1. Use the cofidece level (1 α) to fid α/2. 2. Use the α/2 colum with the appropriate degrees of freedom to fid x 2 right. 3. Fid 1 α/2. 4. Use the 1 α/2 colum with the appropriate degrees of freedom to fid x 2 left. Cofidece Level 1 α α α/2 α Cofidece Coefficiet (1 α ) + 2 z(a/2) Dr. Jaet Witer, Stat 200 Page 21

22 Aswer: Questio 1 A study of 40 Eglish compositio professors showed that they spet, o average, 12.6 miutes correctig a studet s term paper. a) Fid the 90% cofidece iterval of the mea time for all compositio papers whe σ= 2.5 miutes. = 40 X = 12.6 Sice the populatio stadard deviatio is give ad =40 is greater tha 30, use the formula: X z σ < μ < X + z σ < μ < < μ < < μ < < μ < 13.3 b) If a professor stated that he spet, o average, 30 miutes correctig a term paper, what would be your reactio? 11.9 < μ < 13.3 It would be highly ulikely sice 30 miutes is far loger tha the upper boud of 13.3 miutes. Aswer: Questio 2 Fid the sample size ecessary to estimate a populatio mea to withi 0.5 with 95% cofidece if the stadard deviatio is 6.2. z σ 2 = E. = (1.96)(6.2) 0.5 = = [24.304] 2 = Note: Whe solvig for sample size, always roud up to the ext larger iteger (Why?) Dr. Jaet Witer, Stat 200 Page 22

23 Aswer: Questio 3 The prices (i dollars) for a particular model of 6.0 megapixels digital camera with 3x optical zoom are listed as: \$225, \$240, \$215, \$202, \$206, \$211, \$210, \$193, \$250, \$225. Estimate the true mea usig this data with 90% cofidece. Sice the populatio stadard deviatio is ot give, use: X ± t s. Do ot use the σ from the calculator. This is a sample, so be sure to use s ad work i 2 more places tha the data ad roud the aswers to oe more place tha the data. X = s = t = df: 9 = ± ± < μ < Note: X ad s are foud to two decimal places more tha the data, but the aswer is rouded back to oe more place tha the data. Aswer: Questio 4 Joh wats to estimate the average value of the homes i his tow with a 99% cofidece iterval. Use his radom sample of 36 homes with a average value of \$251, ad stadard deviatio \$ to fid the cofidece iterval. Sice the populatio stadard deviatio is ot give, use the formula X ± t s. The degrees of freedom equals 35, but df = 35 is ot available i the table. Use the ext lower df or df = ± ± < μ < Note: Sice statistics are give, work oe more place tha the statistic but roud the aswer back to the same umber of places as X. Dr. Jaet Witer, Stat 200 Page 23

24 Aswer: Questio 5 A survey foud that out of 200 studets, 168 said they eeded loas or scholarships to pay their tuitio ad expeses. Fid the 90% cofidece iterval for the populatio proportio of studets eedig loas or scholarships. p = 0.84 q = 0.16 p z p q < p < p + z p q < p < ± ( ) < p < < p < % < p < 88.3% Aswer: Questio 6 A study by the Uiversity of Michiga foud that oe i five 13 ad 14 year olds is a sometime smoker. To see how the smokig rate of the studets at a large school district compared to the atioal rate, the superitedet surveyed two hudred 13 ad 14 year old studets ad foud that 23% said they were sometime smokers. Fid the 99% cofidece iterval of the true proportio ad compare this with the Uiversity of Michiga s study.. = 200 p = 0.23 q = = 0.77 p z p q < p < p + z p q ± ± 2.58 ( ) 0.23 ± < p < % < p < 30.7% Sice 1/5 = 0.20, the Uiversity of Michiga study falls withi the cofidece iterval ad it is OK. Dr. Jaet Witer, Stat 200 Page 24

25 Aswer: Questio 7 Fid the 99% cofidece iterval for the variace ad stadard deviatio of the weights of oegallo cotaiers of motor oil whe the sample of 14 cotaiers has a variace of 3.2. Assume the variable is ormally distributed. = 14 s 2 = 3.2 ( 1)s 2 ( 1)s2 x 2 right < σ2 < x 2 left < σ2 < < σ 2 < 11.7 variace < σ < 3.4 stadard deviatio Note: The aswer has the same umber of decimal places as the give sample stadard deviatio. Aswer: Questio 8 A service statio advertises a wait of o more tha 30 miutes for a oil chage. A sample of 28 oil chages has a stadard deviatio of 5.2 miutes. Fid the 95% cofidece iterval of the populatio stadard deviatio of the times spet waitig for a oil chage. ( 1)s 2 x 2 right < ( 1)s2 σ2 < x 2 left < σ2 < < σ 2 < 50.1 variace i waitig time < σ < 7.1 stadard deviatio i waitig time. Works Cited Triola, M.D., Marc M. ad Mario F. Triola. Biostatistics for the Biologoical ad Health Scieces. New York: Pearso Educatio, Ic., Dr. Jaet Witer, Stat 200 Page 25

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