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S  08 ^ `  >*  0     * Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. BWDg C H  0޽h ? 3380___PPT10.td Default Design& 0 &(     0p  JChapter 7 Statistical Intervals Based on a Single Sample (K(2 0A<H  0޽h ? 3380___PPT10.^ .&@$(  $ $ 0X4/@@: HConfidence Intervals 2( $ 08/P@@ vAn alternative to reporting a single value for the parameter being estimated is to calculate and report an entire interval of plausible values  a confidence interval (CI). A confidence level is a measure of the degree of reliability of the interval.d 2$$$$$<$H $ 0޽h ? 3380___PPT10.~UZU PU(    6E/8c"`  Y7.1 Basic Properties of Confidence Intervals 4Z(20U<<H  0޽h ? 3380___PPT10. Pۘ C;`((  ( ( 0P/  K95% Confidence Interval 2( ( 0pT/ P`  PIf after observing X1 = x1,& , Xn = xn, we compute the observed sample mean , then a 95% confidence interval for can be expressed as  2$$ $$$ $$$ $$$ $2$$$ $,g` ( c $A ?? @mf ` ( c $A ?? ` ( c $A ?? @  H ( 0޽h ? 3380___PPT10.9:N3 p 03(  0 0 00i/: NOther Levels of Confidence 2(` 0 c $A ??   x 0 <A ??d!  `  0 c $A ??  `  0 c $A ?? ` XB  0@ 0DjJ`  0 c $A ??P d XB  0 0DjJp  0` 0 c $A "?? 0  "` 0 c $A %?? K  % 0 0o/ P p  50 2H 0 0޽h ? 3380___PPT10.:NP   ,(  , , 0w/j NOther Levels of Confidence 2(` , c $A ?? PE   , 0$|/P  A confidence interval for the mean of a normal population when the value of is known is given by 2$` , c $A  ??    ` , c $A  ??0]F  `  , c $A  ??  H , 0޽h ? 3380___PPT10.:NPc 4c(  4 4 04/pj ? Sample Size 2 ( 4 0/ TThe general formula for the sample size n necessary to ensure an interval width w isLU 2($$&$$$` 4 c $A &?? p*  &H 4 0޽h ? 3380___PPT10.;A  8(  8 8 0̕/0z RDeriving a Confidence Interval 2(\ 8 0,/P`B DLet X1,& , Xn denote the sample on which the CI for the parameter is to be based. Suppose a random variable satisfying the following properties can be found:z 2$$ $$$ $$ $  8 0/PID<4___PPT9 F The variable depends functionally on both X1,& ,Xn and The probability distribution of the variable does not depend on or any other unknown parameters.l 2,$$ $$$ $q$1q8X` 8 c $A )?? 0  )` 8 c $A -?? h p  -`  8 c $A +??  p +X  8 00PPH 8 0޽h ? 3380___PPT10.<` :2 <(  < < 0/  pLet denote this random variable. In general, the form of h is usually suggested by examining the distribution of an appropriate estimator For any between 0 and 1, constants a and b can be found to satisfy f 2S$$z$$$$$ < 0/0z RDeriving a Confidence Interval 2(` < c $A 0??@  0` < c $A 1??p  1` < c $A 2??   2`  < c $A 5??.: 5`  < c $A 8?? * 8H < 0޽h ? 3380___PPT10.=4D  @(  @ @ 0/0z RDeriving a Confidence Interval 2(` @ c $A <??0  <  @ 0/@~ s?Now suppose that the inequalities can be manipulated to isolate@ 2@$`  @ c $A @??, ` @`  @ c $A C??  C  @ </jJ?"` P@  Jlower confidence limit 2  @ 0$/P 0 RFor a CI. 2$^B @ 6DjJ  @ <P/jJ?"`   Jupper confidence limit 2^B @ 6DjJ  H @ 0޽h ? 3380___PPT10.=4Dl l(  4  6/8c"`p p7.2 Large-Sample Confidence Intervals for a Population Mean and Proportion 4q(20l<<H  0޽h ? 3380___PPT10. PۘA  HA(  H H 6,/jJ" @ U!Large-Sample Confidence Interval " 2"( H 6/jJ" p 5If n is sufficiently large, the standardized variable26 2$$2$ H 6/jJ" @  xDhas approximately a standard normal distribution. This implies thatE 2E$ H 6P/jJ"   m9is a large-sample confidence interval for with level: 2:$` H c $A `??@  `  H B@IjJ" l 6  <$ `  H c $A c??@  c`  H c $A d??p d`  H c $A g?? r  gH H 0޽h ? 3380___PPT10.R  $ L(  L L 6d IjJ" @* <Confidence Interval for a Population Proportion p with level2= 20(( ( L 6IjJ" @4 n:Lower( ) and upper(+) limits: 2$ L BIjJ" l 6  <$ ` L c $A m??pj1 m`  L c $A s??` sH L 0޽h ? 3380___PPT10.R   P(  P P 6IjJ" @* X$Large-Sample Confidence Bounds for % 2%( P 6(IjJ" pp KUpper Confidence Bound: 2$ P B#IjJ" l 6  <$ `  P c $A }?? }`  P c $A {??  {  P 6%IjJ" d  KLower Confidence Bound: 2$`  P c $A ??  B  H P 0޽h ? 3380___PPT10.R ? ?(    60I8c"`  C7.3 Intervals Based on a Normal Population Distribution 4D(20?<<H  0޽h ? 3380___PPT10. Pۘm  Dm(  D D 68IjJ" p0Z GNormal Distribution 2(  D 0=IP`  The population of interest is normal, so that X1,& , Xn constitutes a random sample from a normal distribution with both zz 2.$$ $$$ $C$ $4B` D c $A D?? P T  DH D 0޽h ? 3380___PPT10.?o31M 0 TM(  T T 6lIIjJ" @j ^t Distribution4 2( ((E T 6NIjJ" 08 aWhen is the mean of a random sample of size n from a normal distribution with mean the rv2b 20$$1$_` T c $A ?? @ ` T c $A ??l ` T c $A! ??@    T 6xUIjJ" @  Bhas a probability distribution called a t distribution with n  1 degrees of freedom (df).P[ 2($$$$$VH T 0޽h ? 3380___PPT10.U@x @X(  X X 6laIjJ"  ` kProperties of t Distributions2 2(((R X 6fIjJ" `N 2Let tv denote the density function curve for v df.\3 2$$ $'$$$,)( X 6|IjJ"  P}D<4___PPT9 hz Each tv curve is bell-shaped and centered at 0. Each tv curve is spread out more than the standard normal (z) curve. { 2$$ $0$$ $4$$ $,0@8Xr X <jJ" p H X 0޽h ? 3380___PPT10.V~o P`o(  ` ` 6IjJ"  ` kProperties of t Distributions2 2((( ` 6IjJ" p 3. As v increases, the spread of the corresponding tv curve decreases. 4. As , the sequence of tv curves approaches the standard normal curve (the z curve is called a t curve with df =  2 2$$,$$ $6$$ $2$$$$$>46S 8X` ` c $A" ??  ` ` c $A ??  i  H ` 0޽h ? 3380___PPT10.V~` `d`(  d d 6IjJ" 0 c t Critical Value4 2(((l d 6hIjJ" `0  nLet = the number on the measurement axis for which the area under the t curve with v df to the right ofLo 2M$$ $$$\` d c $A$ ??`V ` d c $A# ?? 0    d 6IjJ" P p  is called a t critical value.X 2 $$$$$H d 0޽h ? 3380___PPT10.WU`  p h (  h h HA% ?jJ?" 0`   h 6IjJ" j KPictorial Definition of 2(` h c $A$ ?? P `  ^B h 6D8c @@ ` h c $A( ??@0 `  h c $A& ??0 jB  h@ BDjJ?}- ^B  h 6DjJ  h 6IjJ"   `  50 2 `  h c $A$ ??0p H h 0޽h ? 3380___PPT10.Xk" 80 l(  l l 0IP@  GConfidence Interval 2(` l c $A+ ??@ 0$e A l 0I`  Let and s be the sample mean and standard deviation computed from the results of a random sample from a normal population with mean The 2 2 $$$` l c $A ??`h ` l c $A' ??_&  ` l c $A) ?? a  H l 0޽h ? 3380___PPT10.:NP MEp(  p p 6IjJ" P@  GPrediction Interval 2(1 p 6IjJ" 8 kA prediction interval (PI) for a single observation to be selected from a normal population distribution is6l 2$$W$ p 60IjJ" 0   eThe prediction level is2 2$$$` p c $A* ??pH  ` p c $A, ??` ,  H p 0޽h ? 3380___PPT10.ds ~vt(  t t 6IjJ" `: FTolerance Interval 2( t 6IjJ" p  8Let k be a number between 0 and 100. A tolerance interval for capturing at least k% of the values in a normal population distribution with a confidence level of 95% has the formj 2$$#$$$$`$` t c $A- ?? 0  H t 0޽h ? 3380___PPT10.d b/_ _(  '  6Tc8c"`p c7.4 Confidence Intervals for the Variance and Standard Deviation of a Normal Population 4d(20_<<H  0޽h ? 3380___PPT10. Pۘ  |(  | | 0 c@z ENormal Population 2( | 0(cP` NLet X1,& , Xn be a random sample from a normal distribution with parameterszL 2$$ $$$ $?$ $ > | 0Dcp00  w Then the rv 2 $ 8X` | c $A0 ??  ` | c $A/ ?? 0  ` | c $A. ??@P0  h  | 6cjJ"   has a chi-squared probability distribution with n  1 df.2C 29$$ $?H | 0޽h ? 3380___PPT10.<` ]U(    6<*cjJ"  NChi-squared Critical Value 2(  6.cjJ" `P\  ?Let , called a chi-squared critical value, denote the number of the measurement axis such that of the area under the chi-squared curve with v df lies to the right of P 2$$g$$$`  c $A2 ??   `  c $A1 ??00[- `  c $A3 ?? &  H  0޽h ? 3380___PPT10.e@g`  x`(  x x 6;cjJ" j HNotation Illustrated 2(` x c $A1 ??}  x HA5 ?jJ?" @Yd  ` x c $A1 ?? + ^B x 6DjJ `  x c $A& ??` z ^B  x@ 6DjJ@ `  x c $A6 ??`@ F ^B  x@ 6DjJ@ H x 0޽h ? 3380___PPT10.e( } (    6,jJ" P  GConfidence Interval 2(`  c $A7 ??`    6jJ" `0 @ Kconfidence interval for 2$  6jJ" `n e1for the variance of a normal population has 2 22$`   c $A4  ??P0   `   c $A8 ??    6/jJ"   ? lower limit 2 $  6/jJ" P   ? upper limit 2 $`   c $A9 ?? x'    6/jJ"   LFor a confidence interval for , take the square root of each limit above.M 2M$`  c $A: ??  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