Chapter 7 Estimating Population Values
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1 Chapter 7 Studet Lecture Notes 7-1 Busiess Statistics: A Decisio-Makig Approach 6 th Editio Chapter 7 Estimatig Populatio Values Fall 006 Fudametals of Busiess Statistics 1 Cofidece Itervals Cotet of this chapter Cofidece Itervals for the Populatio Mea, whe Populatio Stadard Deviatio is Kow whe Populatio Stadard Deviatio is Ukow Determiig the Required Sample Size Fall 006 Fudametals of Busiess Statistics Fudametals of Busiess Statistics Murali Shaker
2 Chapter 7 Studet Lecture Notes 7- Cofidece Iterval Estimatio for Suppose you are iterested i estimatig the average amout of moey a Ket State Studet (populatio) carries. How would you fid out? Fall 006 Fudametals of Busiess Statistics 3 Poit ad Iterval Estimates A poit estimate is a sigle umber, a cofidece iterval provides additioal iformatio about variability Lower Cofidece Limit Poit Estimate Width of cofidece iterval Upper Cofidece Limit Fall 006 Fudametals of Busiess Statistics 4 Fudametals of Busiess Statistics Murali Shaker
3 Chapter 7 Studet Lecture Notes 7-3 Estimatio Methods Poit Estimatio Provides sigle value Based o observatios from 1 sample Gives o iformatio o how close value is to the populatio parameter Iterval Estimatio Provides rage of values Based o observatios from 1 sample Gives iformatio about closeess to ukow populatio parameter Stated i terms of level of cofidece. To determie exactly requires what iformatio? Fall 006 Fudametals of Busiess Statistics 5 Estimatio Process Populatio (mea,, is ukow) Radom Sample Mea x 50 I am 95% cofidet that is betwee 40 & 60. Sample Fall 006 Fudametals of Busiess Statistics 6 Fudametals of Busiess Statistics Murali Shaker
4 Chapter 7 Studet Lecture Notes 7-4 Geeral Formula The geeral formula for all cofidece itervals is: Poit Estimate ± (Critical Value)(Stadard Error) Fall 006 Fudametals of Busiess Statistics 7 Cofidece Itervals Cofidece Itervals Populatio Mea Kow Ukow Fall 006 Fudametals of Busiess Statistics 8 Fudametals of Busiess Statistics Murali Shaker
5 Fudametals of Busiess Statistics Murali Shaker Chapter 7 Studet Lecture Notes 7-5 (1-α)x100% Cofidece Iterval for 1 α α α Lower Limit Upper Limit Half Width H Half Width H Fall 006 Fudametals of Busiess Statistics 10 CI Derivatio Cotiued 1. Parameter Statistic ± Error (Half Width) Z Z H H Z H H / / / / or ± + ±
6 Chapter 7 Studet Lecture Notes 7-6 Cofidece Iterval for ( Kow) Assumptios Populatio stadard deviatio is kow Populatio is ormally distributed If populatio is ot ormal, use large sample Cofidece iterval estimate x ± z α/ (.5- ) Fall 006 Fudametals of Busiess Statistics 11 (1-α)x100% CI α 1 α α Z (α/) 0 Z (1-α/) Z Cof. Level (1-α) α (.5-α/) Z (.5-α/) Fudametals of Busiess Statistics Murali Shaker
7 Chapter 7 Studet Lecture Notes 7-7 Iterpretatio Samplig Distributio of the Mea α / 1 α α/ x x x x 1 100(1-α)% of itervals costructed cotai ; 100α% do ot. Cofidece Itervals Fall 006 Fudametals of Busiess Statistics 13 Factors Affectig Half Width H z (. 5 α/) Data variatio, : H as Sample size, : H as Level of cofidece, 1 - α : H if 1 - α Fall 006 Fudametals of Busiess Statistics 14 Fudametals of Busiess Statistics Murali Shaker
8 Chapter 7 Studet Lecture Notes 7-8 Example A sample of 11 circuits from a large ormal populatio has a mea resistace of.0 ohms. We kow from past testig that the populatio stadard deviatio is.35 ohms. Determie a 95% cofidece iterval for the true mea resistace of the populatio. Fall 006 Fudametals of Busiess Statistics 15 Cofidece Itervals Cofidece Itervals Populatio Mea Populatio Proportio Kow Ukow Fall 006 Fudametals of Busiess Statistics 16 Fudametals of Busiess Statistics Murali Shaker
9 Chapter 7 Studet Lecture Notes 7-9 Cofidece Iterval for ( Ukow) If the populatio stadard deviatio is ukow, we ca substitute the sample stadard deviatio, s This itroduces extra ucertaity, sice s is variable from sample to sample So we use the t distributio istead of the stadard ormal distributio Fall 006 Fudametals of Busiess Statistics 17 Cofidece Iterval for ( Ukow) Assumptios Populatio stadard deviatio is ukow Populatio is ormally distributed If populatio is ot ormal, use large sample Use Studet s t Distributio Cofidece Iterval Estimate ( 1) ( 1 /) ± t α s (cotiued) Fall 006 Fudametals of Busiess Statistics 18 Fudametals of Busiess Statistics Murali Shaker
10 Chapter 7 Studet Lecture Notes 7-10 Studet s t Distributio The t is a family of distributios The t value depeds o degrees of freedom (d.f.) Number of observatios that are free to vary after sample mea has bee calculated d.f. - 1 Fall 006 Fudametals of Busiess Statistics 19 Studet s t Distributio Note: t z as icreases Stadard Normal (t with df ) t-distributios are bellshaped ad symmetric, but have fatter tails tha the ormal t (df 13) t (df 5) 0 Fall 006 Fudametals of Busiess Statistics 0 t Fudametals of Busiess Statistics Murali Shaker
11 Chapter 7 Studet Lecture Notes 7-11 Studet s t Table Upper Tail Area df Let: 3 df -1 α.10 α/ α/.05 The body of the table cotais t values, ot probabilities 0.90 Fall 006 Fudametals of Busiess Statistics 1 t t distributio values With compariso to the z value Cofidece t t t z Level (10 d.f.) (0 d.f.) (30 d.f.) Note: t z as icreases Fall 006 Fudametals of Busiess Statistics Fudametals of Busiess Statistics Murali Shaker
12 Chapter 7 Studet Lecture Notes 7-1 Example A radom sample of 5 has x 50 ad s 8. Form a 95% cofidece iterval for Fall 006 Fudametals of Busiess Statistics 3 Approximatio for Large Samples Sice t approaches z as the sample size icreases, a approximatio is sometimes used whe 30: Correct formula ( 1) ± t α ± z( 0.5 α/) ( 1 / ) s Approximatio for large s Fall 006 Fudametals of Busiess Statistics 4 Fudametals of Busiess Statistics Murali Shaker
13 Chapter 7 Studet Lecture Notes 7-13 Determiig Sample Size The required sample size ca be foud to reach a desired half width (H) ad level of cofidece (1 - α) Required sample size, kow: z 0 z (.5 α/) ( 0.5 α/) H H Fall 006 Fudametals of Busiess Statistics 5 Determiig Sample Size The required sample size ca be foud to reach a desired half width (H) ad level of cofidece (1 - α) Required sample size, ukow: z 0 s z s (.5 α/) ( 0.5 α/) H H Fall 006 Fudametals of Busiess Statistics 6 Fudametals of Busiess Statistics Murali Shaker
14 Chapter 7 Studet Lecture Notes 7-14 Required Sample Size Example If 45, what sample size is eeded to be 90% cofidet of beig correct withi ± 5? Fall 006 Fudametals of Busiess Statistics 7 Cofidece Iterval Estimates Yes Is ~ N? No Sample Size? Small Large Yes Is kow? No 1. Use Z~N(0,1). Use T~t (-1) Fall 006 Fudametals of Busiess Statistics 8 Fudametals of Busiess Statistics Murali Shaker
15 Chapter 7 Studet Lecture Notes 7-15 Cofidece Itervals (1-α)% 1. Stadard Normal. T distributio Two -sided : ± Z( 0.5 α / ) Oe -sided Upper : Oe -sided Lower : Two -sided : ± t Oe -sided Upper : + Z( 0.5 α ) Z( 0.5 α ) ( 1) ( 1 α / ) + t Oe -sided Lower : t s ( 1) ( 1 α ) ( 1) ( 1 α ) s s Fall 006 Fudametals of Busiess Statistics 9 YDI A beverage dispesig machie is calibrated so that the amout of beverage dispesed is approximately ormally distributed with a populatio stadard deviatio of 0.15 deciliters (dl). Compute a 95% cofidece iterval for the mea amout of beverage dispesed by this machie based o a radom sample of 36 driks dispesig a average of.5 dl. Would a 90% cofidece iterval be wider or arrower tha the iterval above. How large of a sample would you eed if you wat the width of the 95% cofidece iterval to be 0.04? Fall 006 Fudametals of Busiess Statistics 30 Fudametals of Busiess Statistics Murali Shaker
16 Chapter 7 Studet Lecture Notes 7-16 YDI A restaurat ower believed that customer spedig was below the usual spedig level. The ower takes a simple radom sample of 6 receipts from the previous weeks receipts. The amout spet per customer served (i dollars) was recorded ad some summary measures are provided: 6, , s Assumig that customer spedig is approximately ormally distributed, compute a 90% cofidece iterval for the mea amout of moey spet per customer served. Iterpret what the 90% cofidece iterval meas. Fall 006 Fudametals of Busiess Statistics 31 Fudametals of Busiess Statistics Murali Shaker
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