Statistical Intervals for a Single Sample

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1 3/5/06 Applied Statistics ad Probability for Egieers Sixth Editio Douglas C. Motgomery George C. Ruger Chapter 8 Statistical Itervals for a Sigle Sample 8 CHAPTER OUTLINE 8- Cofidece Iterval o the Mea of a Normal distributio, σ Kow 8-. Developmet of the Cofidece Iterval & Its Properties 8-. Choice of Sample Size Sided Cofidece Bouds 8-.4 Large-Sample Cofidece Iterval for μ 8- Cofidece Iterval o the Mea of a Normal distributio, σ Ukow 8-. t Distributio 8-. Cofidece Iterval o μ Statistical Itervals for a Sigle Sample 8-3 Cofidece Iterval o σ & σ of a Normal Distributio 8-4 Large-Sample Cofidece Iterval for a Populatio Proportio 8-5 Guidelies for Costructig Cofidece Itervals 8-6 Tolerace & Predictio Itervals 8-6. Predictio Iterval for a Future Observatio 8-6. Tolerace Iterval for a Normal Distributio Chapter 8 Title ad Outlie

2 3/5/06 Learig Objectives for Chapter 8 After careful study of this chapter, you should be able to do the followig:. Costruct cofidece itervals o the mea of a ormal distributio, usig ormal distributio or t distributio method.. Costruct cofidece itervals o variace ad stadard deviatio of ormal distributio. 3. Costruct cofidece itervals o a populatio proportio. 4. Costructig a approximate cofidece iterval o a parameter. 5. Predictio itervals for a future observatio. 6. Tolerace iterval for a ormal populatio. Chapter 8 Learig Objectives Cofidece Iterval ad its Properties A cofidece iterval estimate for is a iterval of the form l u, where the ed-poits l ad u are computed from the sample data. There is a probability of α of selectig a sample for which the CI will cotai the true value of. The edpoits or bouds l ad u are called lower- ad uppercofidece limits,ad α is called the cofidece coefficiet. Sec 8- Cofidece Iterval o the Mea of a Normal, σ Kow 4

3 3/5/06 Cofidece Iterval o the Mea, Variace Kow If X is the sample mea of a radom sample of size from a ormal populatio with kow variace, a 00( α)% CI o is give by x z x z / / / / (8-) where z α/ is the upper 00α/ percetage poit of the stadard ormal distributio. Sec 8- Cofidece Iterval o the Mea of a Normal, σ Kow 5 EXAMPLE 8- Metallic Material Trasitio Te measuremets of impact eergy (J) o specimes of A38 steel cut at 60 C are as follows: 64., 64.7, 64.5, 64.6, 64.5, 64.3, 64.6, 64.8, 64., ad The impact eergy is ormally distributed with = J. Fid a 95% CI for, the mea impact eergy. The required quatities are z α/ = z 0.05 =.96, = 0, = l, ad x The resultig 95% CI is foud from Equatio 8- as follows: x z x z / Iterpretatio: Based o the sample data, a rage of highly plausible values for mea impact eergy for A38 steel at 60 C is 63.84J 65.08J 0 Sec 8- Cofidece Iterval o the Mea of a Normal, σ Kow 6 3

4 3/5/ Sample Size for Specified Error o the Mea, Variace Kow If x is used as a estimate of, we ca be 00( α)% cofidet that the error x will ot exceed a specified amout E whe the sample size is z E (8-) Sec 8- Cofidece Iterval o the Mea of a Normal, σ Kow 7 EXAMPLE 8- Metallic Material Trasitio Cosider the CVN test described i Example 8-. Determie how may specimes must be tested to esure that the 95% CI o for A38 steel cut at 60 C has a legth of at most.0j. The boud o error i estimatio E is oe-half of the legth of the CI. Use Equatio 8- to determie with E = 0.5, =, ad z α/ =.96. z E / Sice, must be a iteger, the required sample size is = 6. Sec 8- Cofidece Iterval o the Mea of a Normal, σ Kow 8 4

5 3/5/ Oe-Sided Cofidece Bouds A 00( α)% upper-cofidece boud for is x z / (8-3) ad a 00( α)% lower-cofidece boud for is x z / l (8-4) Sec 8- Cofidece Iterval o the Mea of a Normal, σ Kow 9 Example 8-3 Oe-Sided Cofidece Boud The same data for impact testig from Example 8- are used to costruct a lower, oe-sided 95% cofidece iterval for the mea impact eergy. Recall that z α =.64, = 0, = l, ad x A 00( α)% lower-cofidece boud for is x z 0 Sec 8- Cofidece Iterval o the Mea of a Normal, σ Kow 0 5

6 3/5/ A Large-Sample Cofidece Iterval for Whe is large, the quatity X S / has a approximate stadard ormal distributio. Cosequetly, s s x z/ x z/ (8-5) is a large sample cofidece iterval for, with cofidece level of approximately 00( ). Sec 8- Cofidece Iterval o the Mea of a Normal, σ Kow Example 8-5 Mercury Cotamiatio A sample of fish was selected from 53 Florida lakes, ad mercury cocetratio i the muscle tissue was measured (ppm). The mercury cocetratio values were Fid a approximate 95% CI o. Sec 8- Cofidece Iterval o the Mea of a Normal, σ Kow 6

7 3/5/06 Example 8-5 Mercury Cotamiatio (cotiued) The summary statistics for the data are as follows: Variable N Mea Media StDev Miimum Maximum Q Q3 Cocetratio Because > 40, the assumptio of ormality is ot ecessary to use i Equatio 8-5. The required values are 53, x 0.550, s , ad z 0.05 =.96. The approximate 95 CI o is s s x z0.05 μ x z μ μ Iterpretatio: This iterval is fairly wide because there is variability i the mercury cocetratio measuremets. A larger sample size would have produced a shorter iterval. Sec 8- Cofidece Iterval o the Mea of a Normal, σ Kow 3 Large-Sample Approximate Cofidece Iterval Suppose that θ is a parameter of a probability distributio, ad let ˆ be a estimator of θ. The a large-sample approximate CI for θ is give by ˆ z / ˆ ˆ z / ˆ Sec 8- Cofidece Iterval o the Mea of a Normal, σ Kow 4 7

8 3/5/ The t distributio Let X, X,, X be a radom sample from a ormal distributio with ukow mea ad ukow variace. The radom variable T X S/ (8-6) has a t distributio with degrees of freedom. Sec 8- Cofidece Iterval o the Mea of a Normal, σ Ukow The Cofidece Iterval o Mea, Variace Ukow x If ad s are the mea ad stadard deviatio of a radom sample from a ormal distributio with ukow variace, a 00( ) cofidece iterval o is give by x t s/ x t/, s/ /, (8-7) where t, the upper 00 percetage poit of the t distributio with degrees of freedom. Oe-sided cofidece bouds o the mea are foud by replacig t /,- i Equatio 8-7 with t,-. Sec 8- Cofidece Iterval o the Mea of a Normal, σ Ukow 6 8

9 3/5/06 Example 8-6 Alloy Adhesio Costruct a 95% CI o to the followig data. The sample mea is ad sample stadard deviatio is s = Sice =, we have = degrees of freedom for t, so t 0.05, =.080. The resultig CI is x 3.7 x t/, s/ x t/, s/ (3.55)/ / Iterpretatio: The CI is fairly wide because there is a lot of variability i the measuremets. A larger sample size would have led to a shorter iterval. Sec 8- Cofidece Iterval o the Mea of a Normal, σ Ukow 7 Distributio Let X, X,, X be a radom sample from a ormal distributio with mea ad variace, ad let S be the sample variace. The the radom variable X S (8-8) has a chi-square ( ) distributio with degrees of freedom. Sec 8-3 Cofidece Iterval o σ & σ of a Normal Distributio 8 9

10 3/5/06 Cofidece Iterval o the Variace ad Stadard Deviatio If s is the sample variace from a radom sample of observatios from a ormal distributio with ukow variace, the a 00( )% cofidece iterval o is ( ) s ( ) s (8-9) where ad are the upper ad lower 00/ percetage poits of the chi-square distributio with degrees of freedom, respectively. A cofidece iterval for has lower ad upper limits that are the square roots of the correspodig limits i Equatio 8 9. Sec 8-3 Cofidece Iterval o σ & σ of a Normal Distributio 9 Oe-Sided Cofidece Bouds The 00( )% lower ad upper cofidece bouds o are ( ) s ( ) s ad (8-0) Sec 8-3 Cofidece Iterval o σ & σ of a Normal Distributio 0 0

11 3/5/06 Example 8-7 Deterget Fillig A automatic fillig machie is used to fill bottles with liquid deterget. A radom sample of 0 bottles results i a sample variace of fill volume of s = Assume that the fill volume is approximately ormal. Compute a 95% upper cofidece boud. A 95% upper cofidece boud is foud from Equatio 8-0 as follows: s A cofidece iterval o the stadard deviatio ca be obtaied by takig the square root o both sides, resultig i 0.7 Sec 8-3 Cofidece Iterval o σ & σ of a Normal Distributio 8-4 A Large-Sample Cofidece Iterval For a Populatio Proportio Normal Approximatio for Biomial Proportio If is large, the distributio of Z X p Pˆ p p ( p) p ( p) is approximately stadard ormal. The quatity p ( p) / is called the stadard error of the poit estimator Pˆ. Sec 8-4 Large-Sample Cofidece Iterval for a Populatio Proportio

12 3/5/06 Approximate Cofidece Iterval o a Biomial Proportio If ˆp is the proportio of observatios i a radom sample of size, a approximate 00( )% cofidece iterval o the proportio p of the populatio is z ( ) p z ( ) (8-) where z / is the upper / percetage poit of the stadard ormal distributio. Sec 8-4 Large-Sample Cofidece Iterval for a Populatio Proportio 3 Example 8-8 Crakshaft Bearigs I a radom sample of 85 automobile egie crakshaft bearigs, 0 have a surface fiish that is rougher tha the specificatios allow. Costruct a 95% two-sided cofidece iterval for p. A poit estimate of the proportio of bearigs i the populatio that exceeds the roughess specificatio is x / 0/ A 95% two-sided cofidece iterval for p is computed from Equatio 8- as z ˆ 0.05 p p z p p 0.43 Iterpretatio: This is a wide CI. Although the sample size does ot appear to be small ( = 85), the value of is fairly small, which leads to a large stadard error for cotributig to the wide CI. 85 Sec 8-4 Large-Sample Cofidece Iterval for a Populatio Proportio 4

13 3/5/06 Choice of Sample Size Sample size for a specified error o a biomial proportio : If we set E z p p / ad solve for, the appropriate sample size is z E p The sample size from Equatio 8- will always be a maximum for p = 0.5 [that is, p( p) 0.5 with equality for p = 0.5], ad ca be used to obtai a upper boud o. p (8-) z E 0.5 (8-3) Sec 8-4 Large-Sample Cofidece Iterval for a Populatio Proportio 5 Example 8-9 Crakshaft Bearigs Cosider the situatio i Example 8-8. How large a sample is required if we wat to be 95% cofidet that the error i usig to estimate p is less tha 0.05? Usig 0. as a iitial estimate of p, we fid from Equatio 8- that the required sample size is z E If we wated to be at least 95% cofidet that our estimate of the true proportio p was withi 0.05 regardless of the value of p, we would use Equatio 8-3 to fid the sample size z E 0.05 Iterpretatio: If we have iformatio cocerig the value of p, either from a prelimiary sample or from past experiece, we could use a smaller sample while maitaiig both the desired precisio of estimatio ad the level of cofidece. Sec 8-4 Large-Sample Cofidece Iterval for a Populatio Proportio 6 3

14 3/5/06 Approximate Oe-Sided Cofidece Bouds o a Biomial Proportio The approximate 00( )% lower ad upper cofidece bouds are z p ad p z (8-4) respectively. Sec 8-4 Large-Sample Cofidece Iterval for a Populatio Proportio 7 Example 8-0 The Agresti-Coull CI o a Proportio Recosider the crakshaft bearig data itroduced i Example 8-8. I that example we reported that 0. ad 85. The 95% CI was p. Costruct the ew Agresti-Coull CI z z UCL 4 z z (85) 85 4(85) 0.04 (.96 / 85) z z LCL 4 z z (85) 85 4(85) (.96 / 85) Iterpretatio: The two CIs would agree more closely if the sample size were larger. Sec 8-4 Large-Sample Cofidece Iterval for a Populatio Proportio 8 4

15 3/5/ Guidelies for Costructig Cofidece Itervals Table 8- provides a simple road map for appropriate calculatio of a cofidece iterval. Sec 8-5 Guidelies for Costructig Cofidece Itervals Tolerace ad Predictio Itervals 8-6. Predictio Iterval for Future Observatio A 00 ( )% predictio iterval (PI) o a sigle future observatio from a ormal distributio is give by x t s X x t s (8-5) The predictio iterval for X + will always be loger tha the cofidece iterval for. Sec 8-6 Tolerace & Predictio Itervals 30 5

16 3/5/06 Example 8- Alloy Adhesio The load at failure for = specimes was observed, ad foud that x 3.7 ad s The 95% cofidece iterval o was Pla to test a 3rd specime. A 95% predictio iterval o the load at failure for this specime is 3.7 x t s X 6.6 X X x t.6 s 3.55 Iterpretatio: The predictio iterval is cosiderably loger tha the CI. This is because the CI is a estimate of a parameter, but the PI is a iterval estimate of a sigle future observatio. Sec 8-6 Tolerace & Predictio Itervals Tolerace Iterval for a Normal Distributio A tolerace iterval for capturig at least % of the values i a ormal distributio with cofidece level 00( )% is x ks, x ks where k is a tolerace iterval factor foud i Appedix Table XII. Values are give for = 90%, 95%, ad 99% ad for 90%, 95%, ad 99% cofidece. Sec 8-6 Tolerace & Predictio Itervals 3 6

17 3/5/06 Example 8- Alloy Adhesio The load at failure for = specimes was observed, ad foud that x 3.7 ad s = Fid a tolerace iterval for the load at failure that icludes 90% of the values i the populatio with 95% cofidece. From Appedix Table XII, the tolerace factor k for =, = 0.90, ad 95% cofidece is k =.64. The desired tolerace iterval is x ks, x ks [ , ] (5.67,.74) Iterpretatio: We ca be 95% cofidet that at least 90% of the values of load at failure for this particular alloy lie betwee 5.67 ad.74. Sec 8-6 Tolerace & Predictio Itervals 33 Importat Terms & Cocepts of Chapter 8 Chi-squared distributio Cofidece coefficiet Cofidece iterval Cofidece iterval for a: Populatio proportio Mea of a ormal distributio Variace of a ormal distributio Cofidece level Error i estimatio Large sample cofidece iterval -sided cofidece bouds Precisio of parameter estimatio Predictio iterval Tolerace iterval -sided cofidece iterval t distributio Chapter 8 Summary 34 7