Mathacle. PSet Stats, Concepts In Statistics Level Number Name: Date:

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1 PSet Stats, Cocepts I Statistics 7.3. Cofidece Iterval for a Mea i Oe Sample [MATH] The Cetral Limit Theorem. Let...,,, be idepedet, idetically distributed (i.i.d.) radom variables havig mea µ ad fiite oero variace. Let =, the µ lim Pr = Φ( ) / The Cetral Limit Theorem says that whe the sample sie is large, this ubiased µ estimator ~ N( µ, ) ~ N(0,). Note that the radom variables eed ot be ormally distributed. [MATH] The correct hypergeometric model should add the correctio factor to the variace: N = N Whe the 0% rule is satisfied, the stadard deviatio is 34

2 PSet Stats, Cocepts I Statistics N N = = N N The populatio mea ca be estimated i the followig two cases: CASE #: Large Sample Sie ad Kow Suppose...,,, be idepedet, idetically distributed (i.i.d.) radom variables havig mea µ ad fiite oero variace. The is the ubiased mea estimator: = The variace is = = E[... E[ ] E i ] = µ = µ + [ ] Var i = Var[ ] = Var = = = Whe the sigificat level is give ad sample sie is relatively large, the populatio mea µ ad the sample statistic are related through the ormal distributio: µ Pr > = i i= where the sample statistic =. The cofidece iterval CI is, CI =, + 35

3 PSet Stats, Cocepts I Statistics Or, i the form of iterval otatio: < µ < + I the form of margi of error (MoE) : ± [PROCEDURE] Cofidece Iterval for a Mea with Large Sample Sie ad Kow Variace The coditios to obtai CI for the sample mea (statistic) with sample sie are.) The sample is a idepedet radom sample..) The sample sie is less tha 0% of populatio to reduce the w/o replacemet effect 3.) is kow ad > 30 for the coditio to use ormal distributio. 4.) The CI is costructed from the sample statistic for a give sigificace level : CI =, + Where is the terms of margi of error (MoE). [Ti-84] Cofidece Iterval for a Mea with Large Sample Sie ad Kow Variace.) STAT -> TESTS ->7. ZIterval, select Stats if the statistic is give..) Iput,,,. 36

4 PSet Stats, Cocepts I Statistics 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: The sample sie = 50 > 30, ad it is assumed that the school has more tha 500 studets to make the sample sie less tha 0% of the populatio so that the o replacemet effect is reduced. =, = 3., = 0. 05, = =. 96, CI =? or < µ < CI = (.6, 3.8) 50 That is, the Presidet ca say with 95% cofidece that the average age of studets is betwee.6 ad 3.8 years old. Eample From the last eample, the Presidet would like to be 99% cofidet that the estimate of average age should be accurate withi year whe the stadard deviatio of the ages is 3 years. How large a sample is ecessary? 37

5 PSet Stats, Cocepts I Statistics Solutio: = 3, = 0. 0, = =. 58, =? < The sample sie eeds to be at least < > 60 Eample [CB] 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, =.96, 00 ( ) = The least umber of household that should be surveyed is about

6 PSet Stats, Cocepts I Statistics CASE #: Small Sample Sie or/ad Ukow [MATH] For a radom variable ~ T ( ), where T ( ) is the t-distributio with the sample sie of ad the degrees of freedom k =. The mea E [ ] = 0, the variace k 6 Var[ ] = for k >, the skewess is 0 for k > 3, ad the kurtosis is for k k 4 k > 4. Whe variace for,,, is ukow, the ubiased variace estimator is: That is, the epected value of s is s = i i= ( ) E[ s ] = S N S = i where ( ) N i= hypergeometric distributio,, which is the populatio stadard error. For the s is s s N = N Whe the 0% rule is satisfied, the stadard error of is s s Note that the sample stadard deviatio estimator s = i i= ( ) is a biased estimator of the populatio stadard error S. 39

7 PSet Stats, Cocepts I Statistics Whe the sample sie is less tha 30, ad/or is ukow, similar to use ormal distributio, the mea estimator ca be studetied by the t-distributio µ ~ T ( ) s ad the cofidece level is the domai for µ that is determied by µ Pr > t s = The CI is s s CI = t, + t Or, i the iterval otatio: s t < µ < + t s I the form of margi of error t s : ± t s [PROCEDURE] Cofidece Iterval for a Mea with Small Sample Sie ad/or Ukow Variace The coditios to obtai CI for the sample mea (statistic) with sample sie are.) The sample is a idepedet radom sample..) The sample sie is less tha 0% of populatio. 3.) < 30 ad/or is ukow. 4.) The CI is costructed from the sample statistic for a give sigificace level : 40

8 PSet Stats, Cocepts I Statistics s CI = t, + t s Where t s s = i i= ( ) is the term of margi of error (MoE). [Ti-84] Cofidece Iterval for a Mea with Small Sample Sie ad/or Ukow Variace.) STAT -> TESTS ->8. TIterval, select Stats if the statistic is give..) Iput, s,,. Eample The average (mea) travel time from home to school for a sample of 8 THS teachers was 4.3 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 =, = 4. 3, = 0. 05, the umber of degrees of 7 freedom is k = 8 = 7, t = =. 05, CI =? t 4

9 PSet Stats, Cocepts I Statistics < µ < or 3.5 < µ < 5. CI = ( 3.5, 5.) Eample [MC008] A test egieer wats to estimate the mea gas mileage µ (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 gas-mileage values is roughly symmetrical ad has o outliers. The mea ad stadard deviatio of these values are 5.5 ad 3.0, respectively. Assumig that these automobiles ca be cosidered a simple radom sample of cars of this model, which of the followig is a correct statemet? 3.0 (A) A 95% cofidece iterval for µ is 5.5 ± (B) A 95% cofidece iterval for µ is 5.5 ± (C) A 95% cofidece iterval for µ is 5.5 ± (D) A 95% cofidece iterval for µ is 5.5 ±.0. 0 (E) The results caot be trusted; the sample is too small. Solutio: The aswer is A = = k = df = t ± 5.5,, 0,.8,

10 PSet Stats, Cocepts I Statistics Eample [FRQ30] 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. 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. ppm. Costruct ad iterpret a 95 percet cofidece iterval for the mea lead level of crows i the regio. Solutio: 4 a.) p ˆ = = df b.) = 4.90, s =., = 3 df = 3 =, t = t =.069 s. ± t = 4.90 ±.069 = 4.90 ± CI = (4.47, 5.383) We ca be 95% cofidet that the populatio mea lead level amog all crows i this regio is betwee 4.46 ad parts per millio

11 PSet Stats, Cocepts I Statistics 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 sie is less tha 0% of the populatio. Case # s df Cof. Iterval Margi of error Solutio: Case s df Cof. Iterval Margi of # error (4.446, 5.554) (4.065, 5.935) Case #: 30 = 0.05, * =.95, Case #: = 0.0, * =.576,, so use oe-sample Z-distributio for the CI. ± * = 5 ±.96 5 ± or CI = (4.446, 5.554) (7000).576 * = 00 = Case #3: < 30, so, use t-distributio for the CI. = 0.0, df = = 9, t = ivt (0.975,9).09, s ± t = 5 ±.09 5 ± 0.935, or CI = (4.065, 5.935) 0 Case #4: < 30, so, use t-distributio for the CI. t 00 = 30 t.6, = tcdf (.6,00,4) , so 89 44

12 PSet Stats, Cocepts I Statistics Quick-Check Cofidece Iterval for a Mea QC [MC06] 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. ouce. What would be the miimum sample sie for the umber of packages the ispector must select? (A) 8 (B) 5 (C) 5 (D) 5 (E) 60 QC [MC974] A radom sample of costs of repair jobs at a large muffler repair shop produces a mea of $7.95 ad a stadard deviatio of $4.03. If the sie 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) $7.95 ± $4.87 (B) $7.95 ± $6.5 (C) $7.95 ± $7.45 (D) $7.95 ± $30.8 (E) $7.95 ± $39.53 QC [MC037] A simple radom sample procedure produces a sample mea,, of 5. A 95 percet cofidece iterval for the correspodig populatio mea is 5 ± 3. Which of the followig statemets must be true? (A) Niety-five percet of the populatio measuremets fall betwee ad 8. (B) Niety-five percet of the sample measuremets fall betwee ad 8. (C) If 00 samples were take, 95 of the sample meas would fall betwee ad 8. (D) P( 8 ) = 0.95 (E) If µ = 9, this of 5 would be ulikely to occur. 45

13 PSet Stats, Cocepts I Statistics QC [MC033] 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 5 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 -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 -cofidece iterval should ever be used with a small sample. 46

14 PSet Stats, Cocepts I Statistics Aswers QC D. Assume that the coditios to use ormal distributio are satisfied. = 0.3, = 0.004, = QC B QC E. The populatio mea is out of CI. QC B. The variace is ukow. 47