Samples from Normal Populations with Known Variances

Transcription

1 Samples from Normal Populatios with Kow Variaces If the populatio variaces are kow to be σ 2 1 adσ2, the the 2 two-sided cofidece iterval for the differece of the populatio meas µ 1 µ 2 with cofidece level 1 α is ( ( X Ȳ ) zα/2 σ 2 1 m + σ2 2, ( X Ȳ ) σ z α/2 m + σ2 2 )

2 Samples from Normal Populatios with Kow Variaces I case of kow populatio variaces, the procedures for hypothesis testig for the differece of the populatio meas µ 1 µ 2 is similar to the oe sample test for the populatio mea: Null hypothesis H 0 : µ 1 µ 2 = 0 Test statistic value z = ( X Ȳ ) 0 σ1 2 m + σ2 2 Alterative Hypothesis Rejectio Regio for Level α Test H a : µ 1 µ 2 > 0 z z α (upper-tailed) H a : µ 1 µ 2 < 0 z z α (lower-tailed) H a : µ 1 µ 2 0 z z α/2 or z z α/2 (two-tailed)

3 Large Size Samples Whe the sample size is large, both X ad Ȳ are approximately ormally distributed, ad Z = ( X Ȳ ) (µ µ ) 1 2 S1 2 m + S2 2 is approximately a stadard ormal rv.

4 Large Size Samples I case both m ad are large (m, > 30), the procedure for costructig cofidece iterval ad testig hypotheses for the differece of two populatio meas are similar to the oe sample case. The two-sided cofidece iterval for the differece of the populatio meas µ 1 µ 2 with cofidece level 1 α is ( X Ȳ ) z α/2 S 2 1 m + S 2 2, ( X Ȳ ) + z α/2 S 2 1 m + S 2 2

5 Large Size Samples I case both m ad are large (m, > 30), the procedures for hypothesis testig for the differece of the populatio meas µ 1 µ 2 is : Null hypothesis H 0 : µ 1 µ 2 = 0 Test statistic value z = ( X Ȳ ) 0 S1 2 m + S2 2 Alterative Hypothesis Rejectio Regio for Level α Test H a : µ 1 µ 2 > 0 z z α (upper-tailed) H a : µ 1 µ 2 < 0 z z α (lower-tailed) H a : µ 1 µ 2 0 z z α/2 or z z α/2 (two-tailed)

6 Two-Sample t Test ad C.I. Theorem Whe the populatio distributios are both ormal, the stadardized variable T = ( X Ȳ ) (µ X µ Y ) S 2 X m + S2 Y has approximately a t distributio with df ν estimated from the data by ν = ( s 2 X m + s2 Y ) 2 (s 2 X /m)2 m 1 + (s2 Y /)2 1 (roud ν dow to the earest iteger.)

7 Two-Sample t Test ad C.I. The two-sample t cofidece iterval for µ X µ Y cofidece level 100(1 α)% is give by ( x ȳ) t α/2,ν s 2 X m + s2 Y, ( x ȳ) + t α/2,ν with s 2 X m + s2 Y A oe-sided cofidece boud ca be obtaied by replacig t α/2,ν with t α,ν.

8 Two-Sample t Test ad C.I. The two-sample t test for testig H 0 : µ X µ Y = 0 is as follows: Test statistic value: t = ( x ȳ) 0 s 2 X m + s2 Y Alterative Hypothesis Rejectio Regio for Approximate Level α Test H a : µ X µ Y > 0 t t α,ν (upper-tailed) H a : µ X µ Y < 0 t t α,ν (lower-tailed) H a : µ X µ Y 0 t t α/2,ν or t t α/2,ν (two-tailed)

9 Aalysis of Paired Data Assumptios: The data cosists of idepedetly selected pairs (X 1, Y 1 ), (X 2, Y 2 ),..., (X, Y ), with E[X i ] = µ 1 ad E[Y i ] = µ 2.

10 Aalysis of Paired Data Assumptios: The data cosists of idepedetly selected pairs (X 1, Y 1 ), (X 2, Y 2 ),..., (X, Y ), with E[X i ] = µ 1 ad E[Y i ] = µ 2. Let D 1 = X 1 Y 1, D 2 = X 2 Y 2,..., D = X Y, so the D i s are the differeces withi pairs.

11 Aalysis of Paired Data Assumptios: The data cosists of idepedetly selected pairs (X 1, Y 1 ), (X 2, Y 2 ),..., (X, Y ), with E[X i ] = µ 1 ad E[Y i ] = µ 2. Let D 1 = X 1 Y 1, D 2 = X 2 Y 2,..., D = X Y, so the D i s are the differeces withi pairs. The the D i s are assumed to be ormally distributed with mea value µ D ad variace σ 2 D (this is usually a cosequece of the X is ad Y i s themselves beig ormally distributed.)

12 Paired t Test The paired t test for testig H 0 : µ D = 0 (where D = X Y is the differece betwee the first ad secod observatios withi a pair, ad µ D = µ 1 µ 2 ) is as follows: Test statistic value: t = d 0 s D / (where d ad s D are the sample mea ad stadard deviatio, respectively, of the d i s) Alterative Hypothesis Rejectio Regio for Approximate Level α Test H a : µ D > 0 t t α, 1 (upper-tailed) H a : µ D < 0 t t α, 1 (lower-tailed) H a : µ D 0 t t α/2, 1 or t t α/2, 1 (two-tailed)

13 Cofidece Iterval for µ D The Paired t CI for µ D with cofidece level 100(1 α)% is ( ) d t α/2, 1 sd, d + t α/2, 1 sd A oe-sided cofidece boud results from retaiig the relevat sig ad replacig t α/2, 1 by t α, 1.

14 Aalysis of Paired Data Example: (Problem 40) Lactatio promotes a temporary loss of boe mass to provide adequate amouts of calcium for milk productio. The paper Boe Mass Is Recovered from Lactatio to Postweaig i Adolescet Mothers with Low Calcium Itakes (Amer. J. Cliical Nutr., 2004: ) gave the followig data o total body boe mieral cotet (TBBMC) (g) for a sample both durig lactatio (L) ad i the postweaig period (P) L P

15 Aalysis of Paired Data Example: (Problem 40) Lactatio promotes a temporary loss of boe mass to provide adequate amouts of calcium for milk productio. The paper Boe Mass Is Recovered from Lactatio to Postweaig i Adolescet Mothers with Low Calcium Itakes (Amer. J. Cliical Nutr., 2004: ) gave the followig data o total body boe mieral cotet (TBBMC) (g) for a sample both durig lactatio (L) ad i the postweaig period (P) L P Does the data suggest that true average total body boe mieral cotet durig postweaig exceeds that durig lactatio by more tha 25g?

16 Aalysis of Paired Data The differeces for the sample are: with mea d = ad stadard deviatio s D = 103.8

17 Aalysis of Paired Data The differeces for the sample are: with mea d = ad stadard deviatio s D = The Quatile-Quatile for this sample differeces is

18 Differece Betwee Populatio Proportios Propositio Let X Bi(m, p 1 ) ad Y Bi(, p 2 ) with X ad Y idepedet. The E[ˆp 1 ˆp 2 ] = p 1 p 2 so ˆp 1 ˆp 2 is a ubiased estimator of p 1 p 2. Here ˆp 1 = X m ad ˆp 2 = Y. Furthermore, V [ˆp 1 ˆp 2 ] = p 1q 1 m + p 2q 2 where q i = 1 p i.

19 Differece Betwee Populatio Proportios Large-Sample Test Procedure Null hypothesis: H 0 : p 1 p 2 = 0 Test statistic value (large samples): z = ˆp 1 ˆp 2 ˆpˆq ( 1 m + 1 ) where ˆp = X +Y m+ = m m+ ˆp 1 + m+ ˆp 2. Alterative Hypothesis Rejectio Regio for Approximate Level α Test H a : p 1 p 2 > 0 z z α (upper-tailed) H a : p 1 p 2 < 0 z z α (lower-tailed) H a : p 1 p 2 0 z z α/2 or z z α/2 (two-tailed)

20 Differece Betwee Populatio Proportios Large-Sample Cofidece Iterval for p 1 p 2 The 100(1 α)% cofidece iterval for p 1 p 2 is give by ( (ˆp 1 ˆp 2 ) z α/2 ˆp1ˆq 1 m + ˆp 2ˆq 2, (ˆp 1 ˆp 2 ) + z α/2 ˆp1ˆq 1 m + ˆp 2ˆq 2 )

21 Differece Betwee Populatio Proportios Example: (Problem 50) Do teachers fid their work rewardig ad satisfyig? The article Work-Related Attitudes (Psychological Reports, 1991: ) reports the results of a survey of 395 elemetary school teachers ad 266 high school teachers. Of the elemetary school teachers, 224 said they were very satisfied with their jobs, whereas, 126 of the high school teachers were very satisfied with their work. Is there ay differece betwee the proportio of all elemetary school teachers who are satisfied ad all high school teachers who are satisfied?