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Unformatted text preview: STA 2023 c D.Wackerly  Lecture 21 STA 2023 c D.Wackerly  Lecture 21 289 290 Thought: If you pick up a starving dog and make him
prosperous, he will not bite you; that is the principal Last Time: Large Sample Inferences about difference between a dog and a man. Mark Twain Differences Between Two Population Means :
Independent Samples (Section 9.1) § ¤¤¢
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(p. 376) ' &$ #
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estimator For Tuesday: Exer. 9.7(a)(c), 9.26, 9.114(a)(b)(d) formula sheet
Wednesday : P. 378 – 383 (rest of Sec. 9.1), table standard errors %
&$ Monday : P. 378 – 383 (rest of Sec. 9.1), 5 ' Large Sample CI for !
" Assignments formula sheet P. 389 – 396 9.29, 9.33, 9.35, 9.38, 9.39, 9.42, 9.94, 9.96, 9.101,
9.110 Know For Monday : p. 402 – 406 ’s, use them, otherwise use ’s STA 2023 c D.Wackerly  Lecture 21 !D
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!! A C ! A 41 @( 9 786 § For Thursday : 9.7, 9.16, 9.17, 9.95, 9.105, 9.114, STA 2023 c D.Wackerly  Lecture 21 291 292 Consider testing (p. 376) Ex. Investigation between “handedness” and motor TS Q! HF
U9R" P IGE competence in preschool children. Scores on motor a ﬁxed particular value of difference skills tests. 17.5 41 98.1 19.2 Signiﬁcant difference between mean motor skills scores q r Q)
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TEST STATISTIC or Lefthanded (1) ¤
¥¢ 5 Q ) smaller score larger B OR pvalue 68 OR RR D versus 293 STA 2023 c D.Wackerly  Lecture 21 STA 2023 c D.Wackerly  Lecture 21 294 Ex. : #9.23, p. 388 Manufacturing plant discharges
“puriﬁed” liquid waste into a river. EPA inspector
collected water specimens at the point of discharge and
claim a difference in mean also upstream from the plant. Each specimen analysed motor skills scores for left and right handed 5 times, average bacteria count for each specimen level! reported. Six specimens at each location that is less than 31.7 32.2 Since (1) and (2) are valid
is a score : used for L.S. inf. ¥ £¡
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©§ New Problem : how do I estimate this common
variance, ¡ ? Could use , respectively. ¡ Both populations Normally distributed with Could use ¡ , standard error of d.f.
d.f. Will combine or “pool” these values, using BOTH
The “pooled” estimator will have (p. 379) ' ¢ ! §B
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!! D ¢ ! §B C ! D 2¢
d ! B C B Q ¢ ! #
§B §B Q !$ D
§B C ¢ §B §
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§§ Samples are Independent. !! D
! D (unknown). 296 ¢ ! §B
¢ §B h ¡ STA 2023 c D.Wackerly  Lecture 21 Assumptions: (p. 346)
1.
Both pop.’s have the Same variance, ¡ !B B
!A C !A ! A Q ! ! A Q ! A
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Q is 27.3 HOW??? Useful when one or both sample size(s) less than 30. Since assuming 34.9 H GE
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295 and 29.8 location? (Section 9.1, last part) (unknown) means 26.4 .8808!! Small sample inferences about 3. 30.3 discharge location exceeds that for the upstream scores for left and right handed preschoolers for STA 2023 c D.Wackerly  Lecture 21 2. 29.7 Can it be concluded that the mean count at the Can only claim a difference in mean motor skills that are 33.4 28.2 !! 36.2 !A any value of (2) 30.1 scores for left and right handed preschoolers for "
#! B
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! § h Cannot claim a difference in mean motor skills values of Upstream At Discharge(1) ) t
u¤ 5 Q h ? HF
pGE ¡ pvalue = ¤
¥¢ 5 Q preschoolers at the ¡ ¡ Conclusion : # d.f. d.f. 297 !D !$ D is a “weighted average” of the individual STA 2023 c D.Wackerly  Lecture 21 STA 2023 c D.Wackerly  Lecture 21 298 ’s ¡ MORE weight to the estimator based on the Small sample (P. 379) ¡ £ has a dist with ¥
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" P § ¢ ! §B C ¢ §B deg of freedom £ gives the d.f. for this statistic. !$ D
B $D
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! 6 8 786 § £ 299 Ex. : #9.23, p. 388 Manufacturing plant discharges
“puriﬁed” liquid waste into a river. EPA inspector STA 2023 c D.Wackerly  Lecture 21 300 95% CI : collected water specimens at the point of discharge and reported. Six specimens at each location Upstream At Discharge(1) (2) 30.1 36.2 33.4 29.7 30.3 26.4 28.2 29.8 34.9 27.3 31.7 32.3 ¡ Why might the recorded values tend to be
approximately normally distributed? %
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v t5 v
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! C D ¢ §B §B Q !$ D 5 times, average bacteria count for each specimen ! 41 £ ( ! 8 8
3 also upstream from the plant. Each specimen analysed d.f. Each is the average of ﬁve actual measurements.
Want : 95% a CI for the difference in mean bacteria ( counts at the discharge location and the upstream
location? or d !B
!! D 2¢ ! §B C
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#! B B $ D
¢ C ¢!! Q !$ D STA 2023 c D.Wackerly  Lecture 21 ¢ Q B 3 and 4.2, closer to Note : divisor in Large sample with "
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Q R! B ¡ STA 2023 c D.Wackerly  Lecture 21 ¡ 301 STA 2023 c D.Wackerly  Lecture 21 Test Statistic ¡ Rejection Region : Q
¥
¦! B B $ D
¢ C ¢ ¤! r Q
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F S ! 9 786 § £ Ex. : #9.23, p. 388 : Can it be concluded that the mean
count at the discharge location exceeds that for the
upstream location? Upstream At Discharge(1) (2)
26.4 27.3 31.7 32.3 s t5 t
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! ¢ 5 ¥¢ Conclusion : At the Rej.Reg : level, there is evidence to conclude that the mean d
v t 5 v Q t 5 §t u §t C C ¢ 5 ¥©¢&u §t Q !$ D
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u¤ 5 Q 34.9 Q 29.8 Q 28.2 30.3 ¡ 29.7 Q T£ 33.4 ¢ 5 ¢ Q 36.2 Q 30.1 302 STA 2023 c D.Wackerly  Lecture 22 304 Thought: Diplomacy is the art of saying “Nice doggie” –
until you can ﬁnd a rock. Assignments :
Today : P. 378 – 383 (rest of Sec. 9.1), P. 389 – 396
In terms of this example, what are the assumptions Tomorrow : Exer. 9.7, 9.16, 9.17, 9.95, 9.105, 9.114, necessary for the previous CI hypothesis test to be 9.29, 9.33, 9.35, 9.38, 9.39, 9.42, 9.94, 9.96, 9.101, valid? 9.110 §
©§
¨
– Monday : p. 402 – 406
: the POPULATION variances of the for the two locations. : the samples were ¨ – Tuesday : Exer. 9.46, 9.52–54, 9.56, 9.59, are approximately problems on syllabus!
taken at the two – : the 9.60, 9.97, 9.100, 9.107 and rest of Chpt 9 measurements (remember, For Wednesday : Print out a copy of the ANNOTATED FORMULA
SHEET from the course web page. The link is they are averages of ﬁve readings) are right under the “Formula Sheet” link near the top of approximately the page. locations. distributed for both ¡ 305 STA 2023 c D.Wackerly  Lecture 22 STA 2023 c D.Wackerly  Lecture 22 Last Time: Small sample inferences about S
F 9Q ! H F E Useful when one or both sample size(s) less than 30. !
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d.f. STA 2023 c D.Wackerly  Lecture 22 or score Test Statistic : (P. 379) Small sample CI for §Y degrees of freedom. OR smaller score £ , respectively. larger STA 2023 c D.Wackerly  Lecture 22 308 Minitab?
Ex. : #9.101, P. 430 Study to assess the effect of Basic Statistics 2Sample £ ¡ Stat biofeedback exercises on blood pressure. Six subjects
were taught biofeedback. Blood pressure ¡ Select variable and click Radio button “Samples in measurements (millimeters of mercury) were taken
before and after the training. Mean blood pressure after different columns” learning biofeedback less than before? Use ¡ Variable 2. ¢ ¡ Choose alternative (Greater than in this case) Subject Mean
32.10
29.62 StDev
3.19
2.35 SE Mean
1.3
0.96 95% CI for mu AtDisc  mu Upstream: ( 1.1, 6.09)
TTest mu AtDisc = mu Upstream (vs >): T = 1.53
P=0.078
Both use Pooled StDev = 2.80 N
6
6 DF = 10 t ¡ AtDisc
Upstream ¡s Two sample T for AtDisc vs Upstream d Click (Check) in box ”Assume equal variances”. Before After ¢5 ¤ ¢ ¡5 " ¢ 5 d ¢ d5 ¡ v ¢
d5 ¡ t ¢ ¤5 ¤ t ¢ 5 " s ¢ 5 ! ¢ v5 ¤ ¢ s5 ¢ ¤ d
d5 ¤ ¡ ¢ "5 ¡ ¢ Click in box labelled ”Second”, double click on
2©¢§ 1. d§ ¡ Click in box labelled ”First”, double click on Variable t
u¤ 5 Q ¡ Punch in data values . and 1£ X £ OR Both populations Normally distributed with pvalue §Y §
¨§
! A Q ! ! A Q ! A
§§ Samples are Independent. RR £ ¨
©§ (unknown). (unknown) means has !
F S X " P
HVE 3. a ﬁxed particular value of difference versus Assumptions: (p. 346)
1.
Both pop.’s have the Same variance, 2. Hypothesis Testing, P. 380 ¥ £
¦¤¢¡ (Section 9.1, last part) 306 309 STA 2023 c D.Wackerly  Lecture 22 STA 2023 c D.Wackerly  Lecture 22 310 ¡ Sample sizes are small. Objective : Compare durability of two brands of steel Each individual subject had BP taken before AND belted radial tires. ¡ after learning biofeedback. ¡ What happens if someone is careful about diet and ¡ Is likely that both before and after BP measurement cars – drive around – record mileages. not overweight? Things that inﬂuence mileage:
1. Quality of the tires independent!!! ¡¡ will be low. 2. Weight of cars
3. Speed driven – 4. Road conditions and Surface – 5. Driving habits of drivers – 6. Etc. – Want: information about (1), compensating (controlling
for) (2) – (6). HOW?? STA 2023 c D.Wackerly  Lecture 22 Method to consider : Popn. 1 ¢¢
££¢
¢¢
££¢ depen. Indep. .
.
.
P F S9Q ! H F E
! Q
Q
Q B
Q ¡
¡
¡ if differences are taken “1”
same as ¡ ASSUMPTION : the DIFFERENCES are approx.
NORMALLY distributed is NOT required
!! A Q ! §A
¡
¡ ¡
¡ – By design to control for other factors differences mean of the population of DIFFERENCES Two brands of solar collectors : one better? manner. Pairs
¡ Treated wood last longer than untreated? PAIRED DIFFERENCE EXPERIMENT .
.
. !
" Can’t analyse data using method of Section 9.1 – Someone else collected the data in a paired depen. ¢¢
££¢ Problem : Samples are NOT independent Why Pair? .
.
. ¥
¦¤ ¡ This strategy will control for (2) – (6). Difference §§
¨¨§ ¡ How about left and right sides? depen. ¥
¦¤ installed on the front of each car. Before/After Experiments, Biofeedback #9.101 Popn. 2 §§
¨¨§ ¡ Randomly select one tire of each brand to be 312 ¥
¦¤ 311 §§
¨¨§ STA 2023 c D.Wackerly  Lecture 22 ¡ HOW do we analyse the data??? F S9Q H F E
¡ What “factors” can impact BP?
© ¡ Why take before and after BP’s on same subject? !©
© ¡ Samples are Randomly select some tires of each brand – install on “2”.
. STA 2023 c D.Wackerly  Lecture 22 (millimeters of mercury) were taken before and after the
training. Mean blood pressure after learing biofeedback Conclusion : At the level there that the mean blood pressure reading is higher before
learning biofeedback. d5 "
¤ 5 ¥¢ d5 ¢ ¡5 " ¢
d5 ¡ v ¢
¤5 ¤ t ¢ t d ¡s 315 d5 v ¢
v5 ¤ d STA 2023 c D.Wackerly  Lecture 22 d.f. ¢ 5 ! ¢ 5¢¤d
"s 5 ¡ ¢ ¢5 ¤ ¢ 5 d ¢
d5 ¡ t ¢ 5 " s ¢
v5 ¤ ¢
d5 ¤ ¡ ¢ Totals Diff. ¡ Diff. Mean blood pressure larger before than after? ¥£
¦¤ After s5 v "
s 5 s
¢ ¡ 5 d ¦¢
s d5 ¤ ¢
"
5 t #d
"s 5 d
" s 5 us s
s
! § ©¢§ Before d§
t
u¤ 5 Q Subject . d.f. t
u¤ 5 Q
Q T£ t Q
t
u¤ 5 Q
Q
Q
DQ
B
F S 8 £
HFE
HVE
X
¤ Q ! H F E ¤ p! H V E less than before? Use Q D were taught biofeedback. Blood pressure measured ss
u¦s 5 ¤ v Q biofeedback exercises on blood pressure. Six subjects Ex. : #9.101, P. 430 Study to assess the effect of Q
Q
¢ B Q ! D
!¢
Q 8 DIFFERENCES ¡
¢ Method of Analysis : do a onesample “t” ON THE 314 " B
! © § ! © 313 STA 2023 c D.Wackerly  Lecture 22 STA 2023 c D.Wackerly  Lecture 22 Pvalue? 316 Is there a “big” difference in the mean BP readings
before and after learning biofeedback? ¡ Can’t tell from hypothesis test. CAN say there is A ¡ CAN tell how big the difference is from the CI T £ ¡ (3.365) – Between
mercury with paired difference ¡ (
Q !T £
BD ! £ ( 8
3
c1
¥ ¡ or conﬁdence. Why did we analyse the data using the (table value)(standard error) ( Q d.f. milliliters of ¥ ¤£ ¡
estimator and – MORE information in CI, no more work!!!
C.I. for (and others) level. £ !
7T £ (2.571)
ut " 2.977 t
u¤ 5 Q difference at the 0 test? The manner in which the data was collected
dictated the method of analysis ¢ 5 ( d 5 ¥¢
¤ ¨
©§ STA 2023 c D.Wackerly  Lecture 22 317 Why were the data collected this way? What “factors” could have an impact on the BP
measurements? ¢¡ 1. Learing biofeedback . 2. The age of the patients
3. General physical condition
4. Weight of patients
5. Lifestyle
6. Stress level
7. Gender
8. Ethnic background
9. Etc. §
¢ Can assess the impact of , controlling for the others by collecting the data in this manner!!! ...
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 Statistics

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