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 8-.3 -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
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 3 8-. 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
3/5/06 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 64.3. 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 64.46. The resultig 95% CI is foud from Equatio 8- as follows: x z x z / 64.46.96 64.46.96 0 63.84 65.08 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
3/5/06 8.. 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 /.96 0.5 5.37 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
3/5/06 8-.3 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 64.46. 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
3/5/06 8-.4 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.30.330 0.040 0.044.00 0.70 0.490 0.90 0.830 0.80 0.70 0.500 0.490.60 0.050 0.50 0.90 0.770.080 0.980 0.630 0.560 0.40 0.730 0.590 0.340 0.340 0.840 0.500 0.340 0.80 0.340 0.750 0.870 0.560 0.70 0.80 0.90 0.040 0.490.00 0.60 0.00 0.0 0.860 0.50 0.650 0.70 0.940 0.400 0.430 0.50 0.70 Fid a approximate 95% CI o. Sec 8- Cofidece Iterval o the Mea of a Normal, σ Kow 6
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 53 0.550 0.4900 0.3486 0.0400.3300 0.300 0.7900 Because > 40, the assumptio of ormality is ot ecessary to use i Equatio 8-5. The required values are 53, x 0.550, s 0.3486, ad z 0.05 =.96. The approximate 95 CI o is s s x z0.05 μ x z0. 05 0.3486 0.3486 0.550.96 μ 0.550.96 53 53 0.43 μ 0.689 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
3/5/06 8-. 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 5 8-. 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
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 = 3.55. Sice =, we have = degrees of freedom for t, so t 0.05, =.080. The resultig CI is 9.8 0. 4.9 7.5 5.4 5.4 5.4 8.5 7.9.7.9.4.4 4. 7.6 6.7 5.8 9.5 8.8 3.6.9.4 x 3.7 x t/, s/ x t/, s/ 3.7.080(3.55)/ 3.7.080 3.55 / 3.7.57 3.7.57.4 5.8 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
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
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 = 0.053. 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 0 0.053 0.7 0.087 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
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 / 0/85 0.. A 95% two-sided cofidece iterval for p is computed from Equatio 8- as z ˆ 0.05 p p z0.05 0.0.88 0. 0.88 0..96 p 0..96 85 0.0509 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
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 0.05.96 0.05 0.0.88 63 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 z0.05.96 0.5 0.5 385 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
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. 0.0509 0.43 z z UCL 4 z z.96 0. 0.88.96 0..96 (85) 85 4(85) 0.04 (.96 / 85) z z LCL 4 z z.96 0. 0.88.96 0..96 (85) 85 4(85) 0.0654 (.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
3/5/06 8-5 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 9 8-6 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
3/5/06 Example 8- Alloy Adhesio The load at failure for = specimes was observed, ad foud that x 3.7 ad s 3.55. The 95% cofidece iterval o was.4 5.8. 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.0803.55 X 3.7.080 3 3 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 3 8-6. 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
3/5/06 Example 8- Alloy Adhesio The load at failure for = specimes was observed, ad foud that x 3.7 ad s = 3.55. 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 [3.7.64 3.55, 3.7.64 3.55] (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