Unformatted text preview: 264 STA 2023 c B.Presnell & D.Wackerly  Lecture 22 STA 2023 c B.Presnell & D.Wackerly  Lecture 22 265 Last Time : Tests and Conﬁdence Intervals for a ¡ Population Mean, based on SMALL samples. ¢ Assumption : POPULATION approx. NORMALLY Thought: Why is “abbreviation” such a long word? dist.
: ¨¦£ ¤
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¢ Hypothesis Tests ¡ Conﬁdence Interval for Small Sample (p. 292): RR pvalue ©§ 1 § 5
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For Monday : P. 389 – 396 OR 266 smaller score score (tail area) STA 2023 c B.Presnell & D.Wackerly  Lecture 22 or larger G
HF Wednesday : P. 378 – 383 (rest of Sec. 9.1) 7
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0! For Tuesday: Exer. 9.1, 9.13, 9.19, 9.22, 9.24, 9.25 STA 2023 c B.Presnell & D.Wackerly  Lecture 22 267 ¢ Test statistic : (like before) AND #d.f. pvalue Can’t be any more precise using these tables! bp X
WWibYU ¢
¢
¢ s X $ 9bp X
wvUrgIWWubYU 7 @ § 64
¢
5
t X $ 9pq XU 5 $ 9pq X
eisrgdeca`Yh1 § 64 gIWca`YU 7 @ § 64
5
From the table, From the table, and ) to bX $
Wusr
T
tX $
eusr
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pX $
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X $
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sX $
vUr
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tX $
wvUr
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Fb
$ 9Fb X
5
9Fb X
eeca`WU 7 @ § !4 $
5 are Thus, in this case, best we can say is that @t
HeusX Closest values in table ( with d.f. 1.732 sX
vU@ Fb X
dca`YU 7 $ §
Look at table, , ¢ UU
WV$ pvalue 9dpWcf`YU
qX
x§
y' yA7 7 5 §
© S¢ In Everglades example, lower tail test, d.f. 0 9IbWWubYU
pX
' uBA7 7 5
x§ ) depends on T
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P
Q
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( ¡ !£ ¤ $ § (new) (looks just like !!) R (and 9 versus
Today : P. 374 – 378 (Sec. 9.1, “Large Sample”) 7
8§ Assignments '¡ $ "
(&%¡ #! IMMEDIATELY!!! (Exam Conﬂict) 9 If you are taking BUL 4310 – contact me Ho .
Ho . Ho .
Ho . Ex. : New case, lower tail test, d.f. ¢ Closest values to is are t$ ¢ bX
WuF 7 $ § U $
s 271 skills tests. Lefthanded (1) ¢
¢ $ 9qp
gIWvUX b 7 @ §
$ pX
g9 ca`F 7 @ § 64
sU 5
V$ 64 ¢
5 Find a 41 97.5 17.5 Righthanded (2) and s
eq bX
WuF
9b X
dWuF 7 @ § !4
5
then competence in preschool children. Scores on motor sU
V$ 9b X
dWiF 1 § 64
5 . Ex. Investigation between “handedness” and motor ¢ If d.f. STA 2023 c B.Presnell & D.Wackerly  Lecture 22 , that are in table for d.f. and are . 41 98.1 19.2 ¡
¢ ¢ Closest values to So and £¤ 9X
Is YU 1 § 64 ¢
5
$9p X
gdY` YU 1 § 64 ¢
5 sWsvUXr$ 9dpY` Yh1 § !4
XU 5
t$ tail area 270 then ¢ pvalue = . $ 9ps X
gIWWubF 7 @ §
$ 9sp X
gdWeuYU 7 @ § !4
5
E$ 64 ¢
5 s YU
X ¢ STA 2023 c B.Presnell & D.Wackerly  Lecture 22 Ex. : New case, twotailed test, d.f. that are in table for d.f. and If d.f. if d.f. So pvalue sX
vUF that is in table for d.f. pvalue = , conﬁdence interval for the true difference in mean motor skills scores for left and right handed
preschoolers. ¢
@
@ pvalue .
. Objective is to compare two means ¢ @
@ And tail area Samples from two populations ¢ ¢ So
E$ 5
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9 ¢ Closest value to 269 , ¢ $ pvalue = STA 2023 c B.Presnell & D.Wackerly  Lecture 22 sX
vUF 7 $ § $ d.f. s YV$ § E$
XU
p Ex. : New case, upper tail test, 268 STA 2023 c B.Presnell & D.Wackerly  Lecture 22 HOW???? ¢ STA 2023 c B.Presnell & D.Wackerly  Lecture 22 Large Sample Inferences about Differences 273 £¤ 7 £¤
£¤ 7 £¤ 272 STA 2023 c B.Presnell & D.Wackerly  Lecture 22 ¢ If both populations normal, then ¢ Independent Samples (Section 9.1) If and Between Two Population Means, “large”, then normal. approx normal, regardless of shapes of sampled populations. Independent samples from two pop’s. (P. 375) 3
1 5 62 1
3
5 © R ¥ 5 42 1
estimator formula sheet [formula sheet, P. 375] STA 2023 c B.Presnell & D.Wackerly  Lecture 22 ¡ ¢ skills tests. 97.5 £¤ 17.5 41 98.1 19.2 ¡ ¢ What does it mean if ¡ ¢ conﬁdence interval for the true difference in is than by ? equals ? equals by ? How can we interpret the 90% conﬁdence interval, 9` p7 ` X
WdusX 8eiF` 7 5 preschoolers. , for the difference in mean motor skills R
$ y' x E$ I© R scores for left and right handed preschoolers? ¡ ¢ (the true mean dexterity score for lefthanders) $
2T R 9 £ £
I© ¥ ¤ 7 ¤ 5 could be as much as larger than (the true mean dexterity score for righthanders) . (the true mean dexterity score for righthanders) ¡ ¢ ¥ could be as much as larger than true mean dexterity score for lefthanders).
or anything in between. ¡ s
eq ¡
¢ mean motor skills scores for left and right handed is equals ¡
41 90% CI is than What does it mean if Righthanded (2) ¢ Find a is 275 `X
diFI` 7 What does it mean if competence in preschool children. Scores on motor Lefthanded (1) ’s s Ex. Investigation between “handedness” and motor ’s, use them, otherwise use ¡
¡ 7¡
¡
¡ 7¡
¡
¡ 7¡ $ ©¡ £ §¥¡ £ ¨ ¦¤
¡ ¨ ¦ ¤ ¡£
¡ 7 &$ ©¡ £ §¥¢¡ 274 formula sheet table `
IusX p ¢ STA 2023 c B.Presnell & D.Wackerly  Lecture 22 Know [formula sheet, P. 375] standard errors
© R ¥ 9 £ ¤ 7 £ ¤ 5
£¤ 7 £¤ ¡ 7¡ (p. 375) large. 3
62 – Use the difference in sample means, and X 95 .
£¤
Want: Estimate for
£¤
Sample variance Both Sample mean (p. 376) ¡ ¡ Sample size $"
%#! Pop. variance Large Sample CI for '
(& Pop. mean Pop. 2 )
0& ¢ Pop. 1 (the $ dupX p ¥
` 276 STA 2023 c B.Presnell & D.Wackerly  Lecture 22 STA 2023 c B.Presnell & D.Wackerly  Lecture 22 277 Consider testing (p. 376) G
HF R1
I© 36R
@
© R 7 ¦R ' @¡ 7¡ C$ ¡ 7 ¡ (tail area) hypothesized value standard error ¡
W NULL HYPOTHESIS
6 7 ¤ 7 ¤ £ ¡ $R ¢ Formula Sheet ¢
$R
¦ 7 ¤ 7 ¤ preschoolers by 7 points? 7 estimator ¡
¢¡ ¢ handed preschoolers is larger that the mean for RH 278 or score TEST STATISTIC Is it plausible that the mean dexterity score for left STA 2023 c B.Presnell & D.Wackerly  Lecture 22 ' RH preschoolers by 5 points? @
© R 7 ¦R ¢ handed preschoolers is smaller that the mean for 5
!4 OR Is it plausible that the mean dexterity score for left smaller score 7R OR larger STA 2023 c B.Presnell & D.Wackerly  Lecture 22 Ex. Investigation between “handedness” and motor 279 competence in preschool children. Scores on motor
skills tests. £¤ 17.5 Righthanded (2) 41 98.1 19.2 ¢ Conclusion : motor skills scores for left and right handed Signiﬁcant difference between mean motor skills scores sX $
vUV2T preschoolers at the
pvalue = $ FT 7
$
2T
¡ 7 ¡ 0!
")
¡ 7 ¡ #!
" Test Statistic level! ? Cannot claim a difference in mean motor skills scores for left and right handed preschoolers for
any value of `t
eeusX $ $' Rejection Region : level? ¢ for left and right handed preschoolers claim a difference in mean T 97.5 tX $
eusr
T 41 sX $
U r2T
Lefthanded (1) that is less than !! 9 ' preschoolers by 5 points? pvalue R1
© 3¦R s
Wq ¡
¢ handed preschoolers is larger that the mean for RH RR 5
64 ¢ Is it plausible that the mean dexterity score for left 1¡ 7¡
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0! conﬁdence level. 7R versus 9 $ ¡ 7 ¡ " ! rejected at the be left and right handers. This claim a ﬁxed particular value of difference ' Claim : no difference in mean dexterity scores for ¤ $R ¢ STA 2023 c B.Presnell & D.Wackerly  Lecture 23 STA 2023 c B.Presnell & D.Wackerly  Lecture 23 280 281 Last Time:
Large Sample Inferences about Differences Between
Two Population Means : Independent Samples (Section Thought: Diplomacy is the art of saying “Nice doggie” – 9.1) 3
1 5 42 1
3
5 I© R ¥ 5 42 1
estimator IMMEDIATELY!!! (Exam Conﬂict) formula sheet For Monday : P. 389 – 396 Know ’s STA 2023 c B.Presnell & D.Wackerly  Lecture 23 282 Hypothesis testing (p. 376) ' $ ¡ w ¡ "
7 versus ’s, use them, otherwise use STA 2023 c B.Presnell & D.Wackerly  Lecture 23 formula sheet
I© R ¥ 9 £ ¤ 7 £ ¤ 5
For Thursday : Exer. 9.7, 9.15, 9.16, 9.18, 9.20 table standard errors 3
42 If you are taking BUL 4310 – contact me X 95 9 T 7 U 5 WU ¢
ss
(p. 376) Today : P. 378 – 383 (rest of Sec. 9.1) Large Sample CI for ¡ Assignments : ¡ 7¡ until you can ﬁnd a rock. 283 Ex. : #9.23, p. 388 Manufacturing plant discharges a ﬁxed particular value of difference “puriﬁed” liquid waste into a river. EPA inspector
collected water specimens at the point of discharge and 7R 5 times, average bacteria count for each specimen score reported. Six specimens at each location At Discharge(1) G
HF ' R1
I© 36R
@
© R 7 ¦R ' ¢ @¡ 7¡ ' C$ ¡ 7 ¡ ¡
¢¡ 33.4 29.7 30.3 26.4 29.8 34.9 27.3 31.7 32.2 Can it be concluded that the mean count at the
discharge location exceeds that for the upstream
location? ¡ " !
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W NULL HYPOTHESIS ¢ 7
7¤ 7¤
£ ¡ $R Formula Sheet 36.2 28.2 hypothesized value standard error (2) 30.1 (tail area) TEST STATISTIC estimator Upstream HOW??? "
#! @
© R 7 ¦R 5
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or smaller score 9 R1
© 3¦R 5
64 OR larger 7R OR also upstream from the plant. Each specimen analysed pvalue 9 1¡ 7¡
") RR
$R
7¤ 7¤ ¢ £¡
¤¢ Assumptions: (p. 346)
1.
Both pop.’s have the Same variance, ¢ Could use ¢ , standard error of d.f.
d.f. Will combine or “pool” these values, using BOTH ¢ The “pooled” estimator will have (p. 379) 95 U
9 U 7 5
7 5
F 7
3
62
1
9 U 7 5
$ ©
9
QU 7 5
$g9 U 7 5 9 U 7 5 $
§
¨
U U ¦ ¥
$ STA 2023 c B.Presnell & D.Wackerly  Lecture 23 286 is a “weighted average” of the individual STA 2023 c B.Presnell & D.Wackerly  Lecture 23 287 ’s ¢ MORE weight to the estimator based on the
LARGER samp size. $ ¥#
"
Small sample (P. 379)
§¨
§
U U ¥ © I© ¨¥ ¤ 7 ¤ § 9 U 7 5 9 U 7 5
§
¨
U
9 ¡ 7 ¡5 gives the d.f. for this statistic. Large sample with ¢ 5 42 1
3
ibX E$
b
aFX $ E$ E$
p
b
deg of freedom ¢ has a dist with
§
U U ¦ © R ¥ ¤ 7 ¤
¥
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$ ©
pU
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U © ¤ $
7 9 £¤ 7 £¤ 5 § Note : divisor in 3 and 4.2, closer to '
(& Note: 3.3 is # d.f. C.I. for © Ex. : ’s )& Always between the two d.f. # d.f. ¡ is ¢ Since assuming , respectively.
and ? Could use Both populations Normally distributed with (unknown) means used in Ch. 78 9 U 7 5
9 U 7 5 variance, Samples are Independent. 3. score : New Problem : how do I estimate this common
$ ©¡ £ ¦ ¥¡ £ ¨¤
£¤ 7 £¤
$ $
¡
9 £5
9 ¡5
$$
9 5 (unknown). is a R '& Useful when one or both sample size(s) less than 30. 2. 285 Since (1) and (2) are valid
U ¦ ¥
$R
7 9 £¤ 7 £¤ 5 (Section 9.1, last part) )
0& Small sample inferences about STA 2023 c B.Presnell & D.Wackerly  Lecture 23 §
¨
U
9 ¡ 7 ¡5 284 STA 2023 c B.Presnell & D.Wackerly  Lecture 23 STA 2023 c B.Presnell & D.Wackerly  Lecture 23 STA 2023 c B.Presnell & D.Wackerly  Lecture 23 288 Ex. : #9.23, p. 388 Manufacturing plant discharges Hypothesis Testing, P. 380 “puriﬁed” liquid waste into a river. EPA inspector $ ¡ w ¡ "
7 a ﬁxed particular value of difference collected water specimens at the point of discharge and
also upstream from the plant. Each specimen analysed versus 5 times, average bacteria count for each specimen
pvalue score Upstream At Discharge(1) (2) 30.1 (tail area) 36.2 33.4 29.7 30.3 26.4 28.2 ¢ 7
8§ smaller score 9 G
HF ©§ 1 § 5
!4 §
© A7 @ § 5
!4 29.8 34.9 27.3 31.7 32.2 Why might the recorded values tend to be
approximately normally distributed?
Each is the average of ﬁve actual measurements. ¢ Can it be concluded that the mean count at the
location? pE$
p
E$ STA 2023 c B.Presnell & D.Wackerly  Lecture 23 9
eF
9U 7
7 5
¡ 7 ¡"
¢ Rejection Region : $
T
Rejection Region : ¢ Conclusion : At the tX $
eusr
T $§
F 7 $ $ ©
5
9 U 7 5
¡ 7 ¡" ) ¢
$ level, there is evidence to conclude that the mean
bacteria count is greater at the discharge point than
it is upstream. ¢ 3
$ 5 eit62 c` 1 $
` X
Test Statistic d.f. $ ¢ d.f. 291 $ B' x §
290 FeupX Y$ ¤
qF
s XF
vUeb $ ¤ C$ ¡ 7 ¡ STA 2023 c B.Presnell & D.Wackerly  Lecture 23 discharge location exceeds that for the upstream p itt $
X
p sU
W XU V$
© § 1 §
§
I© A7 @ § §¨ ©
U U¥
6 7 ¤ 7 ¤ @¡ 7¡ Test Statistic : or 7
B§ OR larger reported. Six specimens at each location 9 ¦ 1 ¡ 7 ¡
") RR OR d.f. 289 Pvalue
¨ ©
U U
¤$§
¦ 7 9 £ ¤ 7 £ ¤ 5 $
X
$ wU upYU $ ¢ STA 2023 c B.Presnell & D.Wackerly  Lecture 23 292 95% CI :
§¨
U U ¥ © I© § ¥ ¤ 7 ¤
¥
ts
dWupX b ¥ or In terms of this example, what are the assumptions
necessary for the above test and CI to be valid? £ ¢ ¡
– : the POPULATION variances of
are approximately the ¡ – for the two locations. : the samples were taken at the two £ – : the measurements (remember, they are averages of ﬁve readings) are
approximately
locations. distributed for both ¢ ...
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This note was uploaded on 12/15/2011 for the course STA 2023 taught by Professor Ripol during the Spring '08 term at University of Florida.
 Spring '08
 Ripol
 Statistics

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