MATH/STAT 352: Lecture 15

Transcription

1 MATH/STAT 352: Lecture 15 Sectios 5.2 ad 5.3. Large sample CI for a proportio ad small sample CI for a mea. 1

2 5.2: Cofidece Iterval for a Proportio Estimatig proportio of successes i a biomial experimet Biomial experimet, X = umber of successes, =umber of trials, p=probability of success is ukow. Take pˆ X umber of successes = = = sample proportio of successes. umber of trials ˆp The, is a ubiased poit estimator of p. How to get a iterval estimate of p? We start with a large sample ad use the Cetral Limit Theorem. 2

3 95% large sample CI for p Whe is large, the probability is 0.95 that the sample proportio is withi 1.96 stadard deviatios of the true proportio (usig ormal approximatio): (1 ) (1 ) 1.96 p p p pˆ p 1.96 p < < + p It is the also true that for 95% of all possible samples above iequality works. So, the above iterval is a great cadidate for the 95% CI for p. 3

4 95% large sample CI for p, cotd. Problem: Cosider the formula for the iterval we got: This expressio is ot a practical cofidece iterval, because it cotais the ukow populatio proportio p i the margi of error. The traditioal approach is to replace p with. Recet research shows that a slight modificatio of ad the followig estimate of p provide a good cofidece iterval: Defie (1 ) (1 ) ˆ 1.96 p p p p pˆ 1.96 p < < + p ~ = + 4 ad ~ X + p = ~ 2 ˆp 4

5 Cofidece Iterval for p Let X be the umber of successes i idepedet Beroulli trials with success probability p, so that X ~ Bi(, p). The a 100(1 α)% cofidece iterval for p is Where ~ = + 4 ad If the lower limit is less tha 0, replace it with 0. ~ p ~ X + p = ~ 2 ± z α / 2 ~ p(1 ~ p ~ ). If the upper limit is greater tha 1, replace it with 1. 5

6 Example It was reported that, i a sample of 507 adult Americas, oly 142 correctly described the Bill of Rights as the first te amedmets to the U.S. Costitutio. Calculate a 99% CI for the proportio of all U. S. adults that could give a correct descriptio of the Bill of Rights. 6

7 Sample size for give margi of error Suppose we wat to estimate proportio to withi margi of error m. How large a sample do we eed? = z 2 α/2 p (1 p ) m 2 4 Example: What sample size is eeded to obtai a 99% cofidece iterval for the of all U. S. adults that could give a correct descriptio of the Bill of Rights with width (margi of error) ±0.01? Example: What if we did ot have ay prior ifo o p? What sample size do we eed the? NOTE: p 1 p is maximized for p = To get (coservative) sample size, use p = 0. 5 is the above formula for to get: = z 2 α/2 4m2 4 7

8 The Traditioal Method for CI for p ˆp Let be the proportio of successes i a large umber of idepedet Beroulli trials with success probability p. The the traditioal level 100(1 α)% cofidece iterval for p is pˆ ± z α / 2 pˆ(1 pˆ). The method should ot be used uless the sample cotais at least 19 successes ad 10 failures. To obtai a (1- α)100% CI for proportio usig this method, that has margi of error =m, we eed sample size = z 2 α/2 p (1 p ) If we do ot have ay prior estimate of p, use =0.5 i the formula for the sample size, you get: = z 2 α/2 4m 2 ˆp m 2 8

9 Traditioal method of estimatig p. Example: use the data o the Bill of Rights study. Use the traditioal method to fid 99% CI for p, ad fid the sample size eeded to obtai a 99% cofidece iterval for p with width (margi of error) ±0.01? Also, assumig that we do ot have ay estimates of p available, estimate the sample size eeded to obtai a 99% cofidece iterval for p with width (margi of error) ±0.01? Compare the results to those obtaied usig the previous method. 9

10 5.3: Small Sample CIs for a Populatio Mea The methods that we have discussed for a populatio mea previously require that the sample size be large. Whe the sample size is small, there are o good geeral methods for fidig CIs. However, whe the populatio is approximately ormal, a probability distributio called the Studet s t distributio ca be used to compute cofidece itervals for a populatio mea. 10

11 Small-Sample Cofidece Itervals for the Mea X What ca we do if is the mea of a small sample? If the sample size is small, s may ot be close to σ, ad may ot be approximately ormal. If we kow othig about the populatio from which the small sample was draw, there are o easy methods for computig CIs. If the populatio is approximately ormal, it will be approximately ormal eve whe the sample size is small. It turs out that we ca use the quatity ( X µ ) /( s / ) but sice s may ot be close to σ, this quatity has a Studet s t distributio. X 11

12 Studet s t Distributio Let X 1,,X be a small ( < 30) radom sample from a ormal populatio with mea µ. The the quatity ( X µ ) s/ has a Studet s t distributio with 1 degrees of freedom (deoted by t 1 ). Whe is large, the distributio of the above quatity is very close to ormal, so the ormal curve ca be used, rather tha the Studet s t. 12

13 More o Studet s t The probability desity of the Studet s t distributio is differet for differet degrees of freedom. The t curves are more spread out tha the stadard ormal distributio. Table A.3, called a t table, provides probabilities associated with the Studet s t distributio. 13

14 Example: usig t-table Fid the value for the t 14 distributio whose lower-tail probability is Sol: Look dow the colum headed with 0.01 to the row correspodig to 14 degrees of freedom. The value for t = This value cuts off a area, or probability, of 1% i the upper tail. The value whose lower-tail probability is 1% is

15 Studet s t CI for the mea: ormal populatio, σ ot kow Let X 1,, X be a small radom sample from a ormal populatio with mea µ. The a level 100(1 α)% CI for µ is X ± t 1, α / 2 s. The sample must come from a populatio that it approximately ormal. Note: Normal or approximately ormal samples are roughly symmetric ad (practically) do ot cotai outliers. Other CIs: ormal populatio, σ kow If a small sample is take from a ormal populatio with stadard deviatio σ kow, the we use the CI that is usig the z value. 15

16 Example 8 A radom sample of = 8 E-glass fiber test specimes of a certai type yielded a sample mea iterfacial shear yield stress of 30.5 ad a sample stadard deviatio of 3.0. Assumig that iterfacial shear yield stress is ormally distributed, compute a 95% CI for true average stress? 16

17 Example The article Direct Strut-ad-Tie Model for Prestressed Deep Beams presets measuremets of the omial shear stregth (i kn) for a sample of 15 prestressed cocrete beams. The results are Assume that o the basis of a very large umber of previous measuremets of other beams, the populatio of shear stregths i kow to be approximately ormal, with stadard deviatio σ = kn. Fid a 99% cofidece iterval for the mea shear stregth. MINITAB exercise. Oe-Sample Z: stregth The assumed stadard deviatio = 180 Variable N Mea StDev SE Mea 99% CI stregth (548.6, 788.0) What is SE Mea i MINITAB output? It is σ X =StDev/ N What if we did ot have the populatio stadard deviatio σ? Oe-Sample T: stregth Variable N Mea StDev SE Mea 99% CI stregth (520.6, 815.9) 17

18 MINITAB: computig CI for p. It was reported that, i a sample of 507 adult Americas, oly 142 correctly described the Bill of Rights as the first te amedmets to the U.S. Costitutio. Calculate a 99% CI for the proportio of all U. S. adults that could give a correct descriptio of the Bill of Rights. Test ad CI for Oe Proportio Sample X N Sample p 99% CI ( , ) 18