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Unformatted text preview: 318 STA 2023 c D.Wackerly  Lecture 23 STA 2023 c D.Wackerly  Lecture 23 319 Thought: Why don’t you ever see the headline ”Psychic Minitab? Wins Lottery”? ' Punch in data values Today : p. 402 – 406 ' Stat For Tuesday : Exer. 9.46, 9.52–54, 9.56, 9.59, ' Click in box labelled ”First”, double click on Paired ¡ ( ( 9.60, 9.97, 9.100, 9.107 and rest of Chpt 9 Basic Statistics Variable 1 (Before in this case). problems on syllabus! ' Click in box labelled ”Second”, double click on Last Time : PairedDifference Experiments ' Click Options, type in conﬁdence level (for CI) Assumptions: Differences appr. Normally dist. ' Choose alternative (Greater than in this case), null Wednesday : Systematic approach to material in Variable 2. (After in this case) Chpts 7, 8, 9 Method of Analysis : do a onesample “t” ON THE
DIFFERENCES value ' Test Statistic: Paired T for Before  After
N
Mean
StDev
Before
6
166.27
22.00
After
6
156.07
16.64
Difference 6
10.20
8.39
©§¥
¨¦¤ £¡
¢ 95% CI for mean difference : (1.39, 19.01)
TTest of mean difference = 0 (vs > 0): TValue = 2.98 (table value)(standard error) PValue=0.015
%#!
&$"¡ estimator SE Mean
8.98
6.79
3.43 ¤ STA 2023 c D.Wackerly  Lecture 23 320 Conﬁdence Interval: Click OK, OK. STA 2023 c D.Wackerly  Lecture 23 321 Comparing Two Population Proportions
Independent Samples (p.402)
Have: Two populations female managers who are married.
(Male = “1”, Female = “2”). from pop 2, # with attribute
# with attribute estimates
estimates
estimates (p. 402) (p. 402) is approx. normally dist’d when both are large. (p. 402) Find a 95% CI for difference in proportions of male and from pop 1, ¥D
% B C EB C '
% hiDH e¢ aUYX1UB c'
d `B WV b
D
% f % C 1fgD C
¥D
`B WV S
% C EC ¢ aUYX1UB T'
¥
¥
%C DC
%
BC D BC
% G¢ C
F %B
%C
%
D
F
D G¢ D B C of females managers married. Independent samples: DC @2
3A managers married, of the male attribute F 20
31) from Fortune 500 corporations. Pop 2 %
D female managers attribute ¢%
¢DF @958
64
754 male managers, PQI
P
QI women. Pop 1 %C ¢R
DC ¢R Ex. # 9.54, p. 407 Managerial careers of men and ’s D
¢ H
¢%
¢ % G¢
F
%
D
¢ D G¢
F D
EB C
% BC ¡ © § ¥ £ ¡
¨¦¤¢ 323 CI for (P. 403) %#
¢ &$! 6 5'
4 2
54
' ¢ 2
30 % h D
D
% Bf % B C 1Bf D B C 2 4 2
30 0 R& $
)# formula sheet ¢ 8%
7 &$
P " &# )!#
% standard errors table Conﬁdence interval: ¢1
@2
30
£ %
&# ! hD
% iH
D
% Bf % B C 1Bf D B C "
'
( & $
%# " formula sheet ! estimator STA 2023 c D.Wackerly  Lecture 23 322 1 Large Sample STA 2023 c D.Wackerly  Lecture 23 R % C ¥ D CP
B
B '
¦ female managers who are married. At the conﬁdence level , the proportion of male (Male = “1”, Female = “2”). managers who are married exceeds the proportion of ¢ female managers
of the male ¢ 2
30 of females managers married. @2
330 female managers who are married by between STA 2023 c D.Wackerly  Lecture 23 estimator ¢
T F¢
2
2 , then , estimate this with total # “S” in experiment
total sample size h
% F h D i¢ R C
DF
R f H RC W R
R % ¢ on formula sheet. h H
D ¢ RC standard error U
e% § ¥ ¥
' ' (tail area) c
dT §
6
6 W COMMON value of c ¢ If , use the individual ’s I
PI 6 ' ¢ Q
5Q H
G4 ! P
I
! P
I
or score If C
C ¢%C ¢DC
R `aYP R C
b
X
D
h H
VAU
%
d
D
% f H % C 1f H D C
§ ¥ S¥ ER C
R
%C D
6 ¥ A©
@
¢%C DC
§ C
' ' @ ¥ § '
E %
&#$! ' C
%
&# ! ' ¥ 6
6
' E
' 6 C
D%
E
D% § ' B
9
OR smaller score NULL HYPOTHESIS I
PI Q OR larger Q
5Q '
9 versus
pvalue standard error Formula Sheet a ﬁxed particular value of difference RR hypothesized value I ' Consider testing (p. 404) 325 TEST STATISTIC ¢ 324 I 958 0 STA 2023 c D.Wackerly  Lecture 23 ¥ @2
3A
2A
0
¢ % F ¢ % BC
%
D
D
¢ D F ¢ EB C 64
754 ¢ %
2301) ¢ H
D and I
PI 20
31) Find a 95% CI for difference in proportions of male and ¢ 2) @958
64
754 from Fortune 500 corporations.
managers married, d male managers, h R % C ¥ EC P
B DB '
¦ women. %
&# ! Ex. # 9.54, p. 407 Managerial careers of men and C ¥ EC
D
C ¥DC § ¢ C ¥ EC
D
F
G% STA 2023 c D.Wackerly  Lecture 23 Ex. : # 9.107, p. 431 Does inositol (found in breast milk) ' Test Statistic ' Decision : at the ¢
' c c5c
2
30 premature infants given inositol had an 2 ¢
1 of . R HR
Ue% h D
fC
c
cT
§ ¥ R C ¥ ER C
D
% reduce the risk of eye damage in premature infants? A Use 327 6 326 STA 2023 c D.Wackerly  Lecture 23 c ' eye injury to to high oxygen levels used to
compensate for poorly developed lungs
on standard diet had eye injuries. ¢
' c
5c
2 C
¢ D '
0
74 ' of true prop. of premi’s given inositol with eye injuries 1 ¥ D A©
@
% C EC
¥@
%C DC B level, there evidence to claim a lower proportion
of premature infants with eye injury due to high (2) oxygen levels for infants given inositol.
RR : ' Pvalue : Lower tail test –
pvalue ¢ ( 2
0 c ¢ % RC
¢ ¢
1 '
¢ 2 § 6 ' STA 2023 c D.Wackerly  Lecture 23 328 9 ¢
¢ D RC
h
D R
F
¢ % F h D i¢ C
% ¢
0 9
? ¢ C
¢ % '
Is 2
30 tail test, (1) A6
772 ¢ eye injuries ¢ true prop. of premi’s not given inositol with STA 2023 c D.Wackerly  Lecture 23 329 Minitab? '
'
' ”First Sample” in box labelled ”Trial”, type in # of c , claim a lower 2 22
530 1 ' proportion with breathing irregs if given inositol. Not type in # successes (14 in Ex 9.107). ' ”Second Sample” in box labelled ”Trial”, type in # of 2 22
530 E
1 ? trials (110 in Ex 9.107), in box labelled ”Successes” 2 ¢ 1
a' 2
30 – Claim type in # successes (29 in Ex 9.107). c ? ' Click ”Options” ' Choose alternative (Less than in Ex 9.107) ' Click in box ”Use pooled estimate for p for test”. ' Click OK, OK ¢ 1
a' 2
30 – Claim 2 22
530 ¢ 1
a'
– 2Proportions trials (110 in Ex 9.107), in box labelled ”Successes” c signiﬁcantly lower for any ( value larger than Click Radio button “Summarized data” Basic Statistics ( For any Stat ? Test and Confidence Interval for Two Proportions
Sample
1
2 X
14
29 N
110
110 Sample p
0.127273
0.263636 Estimate for p(1)  p(2) : 0.136364
95% CI for p(1)  p(2): (0.239604, 0.0331235)
Test for p(1)  p(2) = 0 (vs < 0) : Z = 2.55
P=0.005 330 STA 2023 c D.Wackerly  Lecture 24 STA 2023 c D.Wackerly  Lecture 24 331 Thought: Common sense is the collection of prejudices
acquired by age 18. Albert Einstein
4. How to proceed?
(a) Sample Size conﬁdence &
¢R standard error $
)# ' c 4 table formula sheet b c &$
P " %&# )!# ' "
R1 ¥ P
H (b) c (a) 2
52 with units @ ' samples from? Estimate to within b 1. How many POPULATIONS have I taken (will I take) ¡ Systematic Approach to Chapters 7, 8, 9 (c) More than 2 (STA3024)
(b) Conﬁdence Interval with conﬁdence coefﬁcient H & $
%# @
@ " & $
%# " %#!
&$X¡
%#
&$! ' ¥D
% C EC C S ¥D
% S HS 3. What is the OBJECTIVE of the exercise? P" form. sheet table value std. error $
%# ¥P estimator 0 R& c 1 . R or H (b) 2 Populations : c or 2
52 (a) 1 Population : @ 2. What is the PARAMETER if interest? form. sheet large S.S.
small S.S. (a) Find Sample Size(s)
(b) Conﬁdence Interval
(c) Hypothesis Test (reach a decision) 332 STA 2023 c D.Wackerly  Lecture 24 STA 2023 c D.Wackerly  Lecture 24 333 Large Sample(s)
(c) Hypothesis Testing
standard error ¦ ¢
' C ¨ Hypothesized Value from NULL HYPOTH. ££ ' ' £
££
££ ££
££
££
£ or ££
' ¥ %&# ! ' C
¥
%
&# ! ' '
' E
' “something” RR ££ E ¤
££
££ F¢ ££
Param E ££ OR ! “something” ££ E ££ ¤ @
££
££ F¢ B
9 B
9 ££¥ ¢ @ © £¥ ££ ££ ¡ ' ii. Independent or Paired? £
££ ££ Param 9 – How about sample(s)
i. Large ( ) or small ( )? C OR “something” Test Statistic “something” C ¢££ Param “something” ! ££ OR ££ ¨
©¨ ££
££ “something” Estimator and Stand. Error from Form. Sheet ££
' ¢££
Parameter hypothesized value “something”
OR Param ¦ Param ¥ Param estimator ¦
§¦ What am I “trying to prove”? : ' ' Formula Sheet
£ unknown ¡ “small” ¡ ¡ 'P
YI £ '
' H
G4 £
££
££ E ¤
££
££ B
9 £
££ F¢ £¥ Small Sample(s) “large” ' £
££ ' (tail area) known. b R
££ E ¢££
£ “something” 1. Use if
2. Use if pvalue ££ calc ££ C Param 'P
YI OR C “something” ' Param calc R OR £
££ ' “something” ' Param 335 ££
¢ the calculated value of STA 2023 c D.Wackerly  Lecture 24 b Pvalues : calc 334 STA 2023 c D.Wackerly  Lecture 24 3. Use for CI
. 4. Use for test 9 5. Use for CI or tests : Large Samples %% 6. Use for CI or tests : Small Samples ¡ ¡ &
R34 ¥ )$ # h D P "
¥ P % h D R ¥ D P ¢ %U
%
%c ' ¡ ' Use onesamp. proc. to analyse the differences. STA 2023 c D.Wackerly  Lecture 24 Use “system” for several examples from text. 336 9 c Calculate DIFFERENCES for all pair ¥ D D©
@
¢ % C EC R X` 8. Use to test 3. Paired Samples 2 7. Use for CI or test # d.f. P ) X` ' 2. Independent Samples (assuming . R ¡ , not © C C@ ©
¢ 1. ...
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