Unformatted text preview: STA 2023 c B.Presnell & D.Wackerly  Lecture 24 293 Assignments Today : P. 389 – 396
For Tuesday : Exer. 9.29, 9.33, 9.35, 9.38, 9.39, 3.42
Wednesday : p. 402 – 406
For Thursday : Exer. 9.46, 9.52–54, 9.56, 9.59,
9.60, and rest of Chpt 9 problems on syllabus! Assumptions: (p. 379)
1.
: Same variances, ©
¥
¦
© : Independent samples. ©
§ 3. §
¨¦ ¥ 2. : Normal distributions. ¤¡ £¢¡ Last Time : Small sample inferences about STA 2023 c B.Presnell & D.Wackerly  Lecture 24 294 Pooled estimator for common variance
¤
¤ ¥
¥ £ ¤ ¤©
£
£ ¤ ¤©
£ ¦ ©
¦ ¨ ¥ ¢
¤
¥ £ ¢ ¤© §
£
£ ¢ ¤©
£
¢ ¤¡ # d.f.
$ " ¥ ¤ # ¥
£ ¦ ¢ ¤¡ " £
!
¤ 0(
1)' % ¤
& (table value)(standard error)
& £¢
% estimator C.I. for Test Statistic
3 ¢ 7
4 % £¤
2 ¢ 5
64
¦ £ % £¢
¤¡ ¢
' ¢ test stat estimator hyp. value (standard error) STA 2023 c B.Presnell & D.Wackerly  Lecture 24 295 Ex. : #9.26, p. 389 Sea urchins were starved for 48
hours, then fed a 5 cm blade of turtle grass. 10 urchins
were randomly selected and fed fresh grass, another
independently selected 10 were fed decaying grass.
The measurements were the time (in hours) necessary
to ingest the grass blades. Green Blades Decayed Blades (1) (2) Number of urchins 10 10 Mean Ingestion Time 3.35 2.36 Standard Deviation 0.79 0.47 ¤¢
¥£¡ Construct a conﬁdence interval for the difference in mean ingestion times for urchins fed
green and decaying grass. STA 2023 c B.Presnell & D.Wackerly  Lecture 24 296 90% CI :
¥ ¤
¤ £¤ £ ¤ ¤©
£ ¦ ¢ ¤©
£
¢
£
¤ ¥ £ ¢ ¤©
¦ £ ¤¡
¢ ¢ ¦ ¤¢
§¢ ¥£¡
¢ ¢
©
¨ § d.f.
' ¥ ¤ ¥
£ ¦ ¢
£ ¤¡ ¤ 0(
1)' ¤
& % £¢ ¨
©¥ , ¢ ¢ d.f. % & or ¦
§¢ ¦ ¢
& (A bunch of “scholarly” words about sea urchins, green
grass and decayed grass) Is the average time it takes
sea urchins to consume green grass larger than the
average time to consume decayed grass? STA 2023 c B.Presnell & D.Wackerly  Lecture 24 ¤¢¡ , at the 297 conﬁdence level. All “plausible values” for the difference in means (green vs.
decayed) are . In terms of this example, what are the assumptions
necessary for the above test and CI to be valid? §
¨¦ ¥ – ¥
¥ ame for are approximately the
grass. : the sea urchins in the study were ¦ – : the POPULATION variances of assigned to green and decayed grass. § – : the a 5 cm blade of sea grass is approximately
for both green and decayed grass. distributed STA 2023 c B.Presnell & D.Wackerly  Lecture 24 298 Ex. : #9.101, P. 430 Study to assess the effect of
biofeedback exercises on blood pressure. Five subjects
were taught biofeedback. Blood pressure measured
(millimeters of mercury) were taken before and after the
training. Mean blood pressure after learing biofeedback ¢¢
¥ . ¢ less than before? Use ¥ © © Before After ¡ ¢ ¤ ¢¥
¡ ¢ £¢¥
¢¡ £
¤¢ ¢¨
¡ ¢ £©¥ ¨¢ ¤ ¤
¥ ¤¢ ¡ £
¥ ¢ ¥¦
¡
¥ ¤¢ ¤
¥ ¢¢¤
¥ Subject
¥ ¢ ¥ ¡
¥ ¢¢ ¢ ¦ £
¥ ¢ ¡ ¡
¡¢ ¡ ¤ ¦ ¥ ¥ ¤ STA 2023 c B.Presnell & D.Wackerly  Lecture 24 299 Sample sizes are small. Each individual subject had BP taken before AND
after learning biofeedback. What happens if someone is careful about diet and
not overweight? Is likely that both before and after BP measurement
will be low. Samples are Why take before and after BP’s on same subject? What “factors” can impact BP? independent!!! –
–
–
–
HOW do we analyse the data??? STA 2023 c B.Presnell & D.Wackerly  Lecture 24 Objective : Compare durability of two brands of steel
belted radial tires. Randomly select some tires of each brand – install on
cars – drive around – record mileages. Things that inﬂuence mileage: ¡¡ 1. Quality of the tires
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?? 300 STA 2023 c B.Presnell & D.Wackerly  Lecture 24 301 Method to consider : Randomly select one tire of each brand to be
installed on the front of each car. How about left and right sides? This strategy will control for (2) – (6). Problem : Samples are NOT independent Can’t analyse data using method of Section 9.1 PAIRED DIFFERENCE EXPERIMENT Treated wood last longer than untreated? Two brands of solar collectors : one better? Before/After Experiments, Biofeedback #9.101 Why Pair?
– Someone else collected the data in a paired
manner.
– By design to control for other factors STA 2023 c B.Presnell & D.Wackerly  Lecture 24 Popn. 2 £ ¤ ¢ ¤ £ ¡ £ £ depen. .
.
.
4 ¡ .
.
. ¤ £ £ ¢ .
.
. ¢ £ Indep. £ depen. £ depen. Difference
¤ Popn. 1 302 ¢ ¡ ¤¡ ¢¡ differences
¢ Pairs
¢ mean of the population of DIFFERENCES ¢
¥ ¡
§ 3 ¦ 3 ¤¡ £¢¡
¢ 2 ¡ ¢
¥ ¢ ¡ § ¥ 3 ¦ NORMALLY distributed
¨¨¨
©¨ is NOT required 2 ¤¡ £¢¡ £ ¥ ASSUMPTION : the DIFFERENCES are approx. 3 same as “2”. £ if differences are taken “1” . ¤
¤ ¢ ¤ ¢ ©
¨¨¨
©¨ STA 2023 c B.Presnell & D.Wackerly  Lecture 24 303 Method of Analysis : do a onesample “t” ON THE
DIFFERENCES Ex. : #9.101, P. 430 Study to assess the effect of
biofeedback exercises on blood pressure. Five subjects
were taught biofeedback. Blood pressure measured
(millimeters of mercury) were taken before and after the
training. Mean blood pressure after learing biofeedback
¥ ¢¢ ¢ less than before? Use . ¥ © © Subject Before After ¥ ¡¢ ¤ ¡ ¢ ¢ ¢¥
¡ ¡
£¨ ¢ £ £
©
¤ ¨
¡ ¢ ¢ ©¥ ¡¢ ¢ ¡ £¢ ¨ £ ¢ ¡ £ ¨¢ ¦ ¡ Diff. ¤ ¡¢ ¤
¥¢ ¡ £ £
¤¢ ¥ ¥
¥ ¤¢ ¢ ¢ ¡ £¤
¥¥¢ £ ¨ ¢ ¥ ¤ ¤ ¤¢ ¤ ¡ ¡
£ ¥ ¢¢¤ ¥
¥ ¥
¥
¥ ¡¢ ¡ ¤ ¥ ¢ ¡ ¡ ¥ ¤
¥¢ ¤
¥ ¢ ¥ ¢¢ ¢ ¦ ¢ ¡ ¦
¥ ¥ ¨¢ ¤ ¤
¥ Totals £ ¢ ¢ Diff. ¡
£ ¦
¤ STA 2023 c B.Presnell & D.Wackerly  Lecture 24 304 % ¢ ¤
¤ © £ £ £¢ ¢ ¡ £ £
¥ ¤ ¥ ¥
¤ ¥ £ ¢
£ ¥ ¢ ¥ ¤ ¢
¢
¥ d.f.
¢ Mean blood pressure larger before than after? ¢
¢ § ¤¡ £¢¡
§ ¡
¥ ¢
¦ 3
3 ¦ ¡ § ¤¡ £¢¡
§ ¡ ¦
¥ 3 ¦ £
2 ¥ %
¥ '
' £
¥ ¢ ¦
¦ ¢¢ ¢
¢ ¦
§¢ ¢
¢ Conclusion : At the ¢ ¢ ¢ ¢ d.f. level there that the mean blood pressure reading is higher before
learning biofeedback. STA 2023 c B.Presnell & D.Wackerly  Lecture 24 305 Pvalue? 0 ¤ '
(2.571) £
$ " # (table value)(standard error) ¥ ¢ £ 0
6( ' & ¤ ¤ '
¢
& or ¥ ¤ & " d.f. " ¡
¢ estimator (3.365) ¤ C.I. for ¢ '
2.977 ¨ ¡¢ ¨
& ¢ ¢
¥ STA 2023 c B.Presnell & D.Wackerly  Lecture 24 306 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 ¦
§¢ ¢
¢ difference at the (and others) level. CAN tell how big the difference is from the CI
– Between ¤ §¡
¦ mercury with and milliliters of conﬁdence. – MORE information in CI, no more work!!! Why did we analyse the data using the
paired difference test? The manner in which the data was collected
dictated the method of analysis STA 2023 c B.Presnell & D.Wackerly  Lecture 24 307 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!!! STA 2023 c B.Presnell & D.Wackerly  Lecture 25 Assignments Today : p. 402 – 406
Tomorrow : Exer. 9.46, 9.52–54, 9.56, 9.59,
9.60, and rest of Chpt 9 problems on syllabus!
Last Time : PairedDifference Experiments
Assumptions: Differences appr. Normally dist.
Method of Analysis : do a onesample “t” ON THE
DIFFERENCES
Test Statistic:
3
2
¢ ¤ £ £
¥ ¥ ¢ ' Conﬁdence Interval:
(table value)(standard error) ¥ ¥ £ ¤ 0
6( ' & ¤ & estimator 308 STA 2023 c B.Presnell & D.Wackerly  Lecture 25 309 Ex. # 9.54, p. 407 Managerial careers of men and ¦¡¡ from Fortune 500 corporations. ¤¦£ managers married, female managers ¤¤¨ male managers, ¡
¥ women. of the male of females managers married. Find a 95% CI for difference in proportions of male and
female managers who are married.
(Male = “1”, Female = “2”). ¢
¢ ¢ ¢
£ ¤
£ % ¢ ¡ ¢
¢ ¤ ¤ ¢
¢ % £ £ ¡ ¢ ¤ STA 2023 c B.Presnell & D.Wackerly  Lecture 25 310 Comparing Two Population Proportions
Independent Samples (p.402)
Have: Two populations ¢ ¡ ¤
¢ © attribute ¢ © Pop 2 attribute ¡ Pop 1 Independent samples:
# with attribute
¢ # with attribute
¡ ¢
¡ ¤
¡ £¢ ¤ (p. 402) ¤
£ ¤ ¢
¢ £ ¤ ¡ £ ¤
¢ ¡ ¡ £¢
¡ ¡ £¢ ¡ ¤£
¤ ¥¤
£ £ % ¡ ¤ ¢ ¡ ¢
estimates (p. 402) ¢ ¡
¢5 ¡ 7¡ ¦ ¡ ¢£
¢ ¤¢
£ is approx. normally dist’d when both ¢ 7¡ ¡ ¤ ¡5¡ ¡ £¢ are large. (p. 402) £ % ¢ estimates ¤ % ¤ estimates ¡ % ¢ from pop 2, ¢ from pop 1, ’s STA 2023 c B.Presnell & D.Wackerly  Lecture 25 CI for (P. 403)
standard errors ¢ ©
¨ formula sheet ¡
©§ ¤ $ #
& ©
¨ §
0
6(
¢ ©
¨ § ¡ ¤£
¤£¤
¡ table
formula sheet estimator
! Large Sample 311 ¡ ¦ ¢£
¢£¢ ¡ ¤ 0
6( ¡ ¤
& ¢ £¢© Ex. # 9.54, p. 407 Managerial careers of men and
from Fortune 500 corporations. ¤¦£ managers married, female managers ¤¤¨ ¦¡¡ male managers, ¡
¥ women. of the male of females managers married. Find a 95% CI for difference in proportions of male and
female managers who are married.
(Male = “1”, Female = “2”).
¢ ¦
§¡ ¡
¢ ¡ ¢
¤ £
£ ¢ ¤ ¨¢
¢ ¦ £¢ % ¢ ¢ ¡ ¤ ¤ ¢
¢ % £ £ ¡ ¢ ¤ STA 2023 c B.Presnell & D.Wackerly  Lecture 25 312 ¢ ¤ ¢
¢ ¤ 0
1( ¤ §¡
¦ Conﬁdence interval: ¦
§¢ ¢ ¦ §¢ ¢ ¢ ¡ ¤£
¤£¤ ¢ ¢ ¦ ¡ ¢£
¢£¢ ¡ ¤ 0
6( &
¢ ¤ ¡ £¢©
¢ &
¦
¢ ¢ ¡ ¢¢ ¢
& ¤ §¡
¦ At the conﬁdence level , the proportion of male managers who are married exceeds the proportion of
female managers who are married by between
and STA 2023 c B.Presnell & D.Wackerly  Lecture 25 313 Consider testing (p. 404)
a ﬁxed particular value of difference ¡ ¦ ¡ £¢
¤ ¢
2 § 3 versus
pvalue
score £
¢ larger RR © § ¦ ¡ ¡
¡ (
¢ ¡ ¤ £¢ 2 ¢ £
¢ smaller score © ( £
¢
2 ¢ ¡ ¡ ¤ £¢ OR ¢ or (tail area) ¤ 0
1( £
¢ ¡
¢ ¤ ¡¢
2 ¡ £¢ ¡ ¤ 0
1(
¢ ¢ TEST STATISTIC
hypothesized value £ £ standard error £
¤£ £ estimator OR ¢
¢ STA 2023 c B.Presnell & D.Wackerly  Lecture 25 ¡ ¤
£ ¢ ¤ ¢ ¡ ¤
¦ £ ¢ ¢ ¡ £ ¢ ¢
¡ £ ¢
¢ ¡ ¡ ¡ ¡ ¤ ¢ ¢ , estimate this with
¡ ¢ total # “S” in experiment
¢ total sample size
% ¡ ¤
¤ ¥
£ ¥
¦ ¢ ¢
£ ¦
£ £ ¢
¦ ¢ % ¢
¢ £ ¢ ¢ ¢ ¡ ¢ ¢ 2 2 ¡ ¡ on formula sheet. ¤ ¡¢ standard error ¡ ¢ ¢
, then ¥¦ ©
¤§ £¢ , use the individual ’s COMMON value of £ £¤
2 If ¢ ¡ If NULL HYPOTHESIS ¢ Formula Sheet 314 ¢ STA 2023 c B.Presnell & D.Wackerly  Lecture 25 315 Ex. : # 9.107, p. 431 Does inositol (found in breast milk)
reduce the risk of eye damage in premature infants? ¢¢ . ¢ ¥
¥ ¢ £ of ¥ Use premature infants given inositol had an ¥ eye injury to to high oxygen levels used to
compensate for poorly developed lungs ¢ ¥
¥ ¡ of
¢ true prop. of premi’s given inositol with eye on standard diet had eye injuries. ¡ ¢ injuries
true prop. of premi’s not given inositol with ¡ ¤ ¢ eye injuries ¤ (2) ¡ ¦ ¡ £¢
¡ § ¦ ¡ £¢
§ 3 ¢ RR : ¢ ? ¦
§¡ ¥ ¢
¢ ¢
¢ ¢ ¤ ¢ % ¤
¤ £ % ¢
¦
¦
¢ ¢
£ 2 ¢ ¢
¡ ¡
¡ ¢ ¢ Is ¤ tail test, (1) ¢ ¢
¡ STA 2023 c B.Presnell & D.Wackerly  Lecture 25 ¥ ¤
£ £¤
2 ¥ ¦ ¢ ¢ £¢
¢ £ ¢ ¡ ¡ ¢ ¢
£ Test Statistic Decision : at the 316 ¢ ¢ ¡ ¢ £ ¡ ¦ ¢¢
¢ level, there
¢ ¢ ¢ evidence to claim a difference in the
proportion of premature infants with eye injury due
to high oxygen levels for infants given inositol and
those not given inositol.
Pvalue : Lower tail test –
¢ pvalue STA 2023 c B.Presnell & D.Wackerly  Lecture 25 ¦¢
§£¢ ¢ value larger than ¥ For any 317 , claim there is a difference in the proportions with breathing irregs. ¦
§¢ ¢ ¢ ¥ ¢¢ ¢ – Claim ¦ ¢ ¢¢ ¢ – ? ? – Claim ¦ ¢ ¢¢ ? ¥ No signiﬁcant difference for any ...
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 Spring '08
 Ripol
 Statistics, Normal Distribution, Statistical hypothesis testing

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