This preview shows page 1. Sign up to view the full content.
Unformatted text preview: For Monday : P. 389 – 396 For Thursday : Exer. 9.7, 9.15, 9.16, 9.18, 9.20 Wednesday : P. 378 – 383 (rest of Sec. 9.1) For Tuesday: Exer. 9.1, 9.13, 9.19, 9.22, 9.24, 9.25 Today : P. 374 – 378 (Sec. 9.1, “Large Sample”) Assignments IMMEDIATELY!!! (Exam Conﬂict) If you are taking BUL 4310 – contact me Thought: Why is “abbreviation” such a long word? ¢ Population Mean, ¢ based on SMALL samples. Last Time : Tests and Conﬁdence Intervals for a STA 2023 c B.Presnell & D.Wackerly  Lecture 22 versus Hypothesis Tests Small Sample (p. 292): Conﬁdence Interval for dist. OR OR : Assumption : POPULATION approx. NORMALLY ¡ 264 ¡
¡ %
&
¡
¡
¡ ¢ ¡ ¦
¦
¦ RR ¦
%
#¦
¦ STA 2023 c B.Presnell & D.Wackerly  Lecture 22 ¡ §
%
#¦
§ ©¨ or ¡ 265 score (tail area) smaller score pvalue
larger !"
!"
)('
¡
§
§ ©¨ ¦ £¤
¦¥
§ ©¨
¦
# # ¦ $
$ (and ¦ pvalue From the table, and Closest values in table ( with d.f. are From the table, 266 (like before) AND #d.f. (looks just like !!) ) depends on Look at table, ¤
¥§£ ¢ ' ¨§ ¢ ¤#
¥£ ¢ ¤£ §¦ ¦!" ¢ ¦!" ©¦ %
#
¤¥£
©¦
¨$ ¨ ¦!" %
#
¤¥§£
¨§
$ ¤ %
#
¤¥£
§¦
'$ ¦!"
%
¤# £ In Everglades example, lower tail test, d.f. (new) § ¢¦ ¦ ¦ § ©¨ #
¡ £¤ Test statistic : ¢ ¢ STA 2023 c B.Presnell & D.Wackerly  Lecture 22 §¦
'$
¦!"
) to ££
¤£
©¦
¨$
¤
£ ¦!" ¡ , ¤£ 1.732 0 STA 2023 c B.Presnell & D.Wackerly  Lecture 22 pvalue Thus, in this case, best we can say is that Ho . Ho . Ho . Ho . Can’t be any more precise using these tables! ¤ ££
§¦
'$
§¦
¤
¥£ "
%
¢
¢
¢
¢
¢
¢ £
¤
¤
¤
¤
¤
§
¤ ' £
¤% ¢ #¦
#
¤¥£
©¦
¨$ £
¨ ¢ "
#
¤¥§£ # ¦
¨§
$ 267 ¢ ¢ Closest value to pvalue = ¦!" ¦!" ¢ ¢ ¢ ¤
¥£ So pvalue ¢ !" ¤¥£
¦
¨$ $ if d.f. that is in table for d.f. , d.f. ¤
¥£ Ex. : New case, upper tail test, ¦!"
$ ¦¨
¤¥£
¦
¨$ is 268 STA 2023 c B.Presnell & D.Wackerly  Lecture 22 If d.f. are So and Closest values to pvalue = Ex. : New case, lower tail test, d.f. then , . that are in table for d.f. STA 2023 c B.Presnell & D.Wackerly  Lecture 22 ¢ ¤
¥£
¤ ¢
¢
¢
¦!"
£
%
¦
. %
¤ # £
¨
$ ¤' ¦!" ¤# '
and £
¤ # §'
¨$ £
269 pvalue = Ex. : New case, twotailed test, d.f. STA 2023 c B.Presnell & D.Wackerly  Lecture 22 ¢ ¢ ¢ If d.f. are So And and Closest values to tail area ¢ ¢ then £ ¦!" £ , . pvalue . . that are in table for d.f. tail area %
¤# §
¨©£
$ % ¢ ¦!"
%
¤# ' ¦
¤ '
§ $
% ¤ '
§ ¤ # '
§
$
§ ¦!" % 270 and 98.1 19.2 17.5 271 conﬁdence interval for the true difference in 41 Righthanded (2) 97.5 Objective is to compare two means
HOW???? Samples from two populations preschoolers. mean motor skills scores for left and right handed Find a 41 Lefthanded (1) skills tests. competence in preschool children. Scores on motor Ex. Investigation between “handedness” and motor STA 2023 c B.Presnell & D.Wackerly  Lecture 22 ¢ %
¢ ¦!" £¤
¢ ¤# '
¨¦
$ % £ © Want: Estimate for Sample variance Sample mean Sample size Pop. variance Pop. mean Pop. 1 Independent samples from two pop’s. Pop. 2 . ¡ £¡ £¡ £¡ £¢
¡ (p. 375) 272 [formula sheet, P. 375] [formula sheet, P. 375] – Use the difference in sample means, ¤¥ ¤¥ ¢ Independent Samples (Section 9.1) ¦ ¦ ¡ # © If both populations normal, then STA 2023 c B.Presnell & D.Wackerly  Lecture 22 and (P. 375) Know Both large. estimator and Large Sample CI for
(p. 376) formula sheet table standard errors formula sheet ’s ’s, use them, otherwise use 273 approx normal, normal. regardless of shapes of sampled populations. “large”, then If Between Two Population Means, § ¢ ¡
©
£¤
© Large Sample Inferences about Differences ¡
# ¡© ©©
¨ ¡©
©©
©
£¤©
© © £¤ " ¨© ©
£¤
" £¤©
# " ©
! ¥
$ § ©¨ ¡
! ¥ £¤© ¢ ¢ © # £¤ "
§ ©¨ ! ©
# £¤
§ © © ¡
¤$
" STA 2023 c B.Presnell & D.Wackerly  Lecture 22 £¤© £¤©
#
©©
© © ¡©
90% CI is ¢ preschoolers. # £¤ " 41 41
98.1 97.5
19.2 17.5 274 conﬁdence interval for the true difference in Righthanded (2) Lefthanded (1) ¥ ¥
¤¨
¨¦ £¤© $ ¥ § ©¨ ¡ § ©¨ ¡ mean motor skills scores for left and right handed Find a © skills tests. competence in preschool children. Scores on motor Ex. Investigation between “handedness” and motor STA 2023 c B.Presnell & D.Wackerly  Lecture 22 ¡
©
is is What does it mean if than What does it mean if is by equals ? ? How can we interpret the 90% conﬁdence interval, than What does it mean if equals ? by equals STA 2023 c B.Presnell & D.Wackerly  Lecture 22 " £¤
© © , for the difference in mean motor skills (the 275 larger than or anything in between. true mean dexterity score for lefthanders). could be as much as (the (the true mean dexterity score for righthanders) true mean dexterity score for righthanders) . could be as much as larger than (the true mean dexterity score for lefthanders) scores for left and right handed preschoolers? ¤# ¦
¢
¢
¢ ¦ $ ¢ ¡ ¡ ¡ ¢
¢ ¡©
#
¡©
¡©
# ¡
¡
¡
'¦ ¤¨ ¡
¡© § © ¤# ¦ ' ¡©
¡©
#
¡© ¤¨ ¦
¦
¡© ¡ ¢ ¢ be preschoolers by 7 points? handed preschoolers is larger that the mean for RH Is it plausible that the mean dexterity score for left RH preschoolers by 5 points? handed preschoolers is smaller that the mean for Is it plausible that the mean dexterity score for left preschoolers by 5 points? handed preschoolers is larger that the mean for RH Is it plausible that the mean dexterity score for left rejected at the
conﬁdence level. left and right handers. This claim Claim : no difference in mean dexterity scores for © versus OR OR Formula Sheet estimator TEST STATISTIC 277 RR NULL HYPOTHESIS standard error score (tail area) smaller hypothesized value or larger score pvalue a ﬁxed particular value of difference Consider testing (p. 376) STA 2023 c B.Presnell & D.Wackerly  Lecture 22 ¡
¢ ¢ ¢ ¡
#
# ¡
# ©¡
©¡
¡
¡
¡
# ¡©
%
&
©¡ 276 ¡ ¤
# ©
§ ¡ %
#
#
©©
© STA 2023 c B.Presnell & D.Wackerly  Lecture 22 ¢ ¡
¡ % ¡
¤
# ¤© §¡ ¡ #
§ ©¨ ¡ ¡ #
§¡
§ ©¨ ¡ !"
)('
¤© ©
#
© © ©
§ # !"
¡ ¡ # ¡ $
$
41 41
98.1 97.5 £¤ ¢ ¡ ¢ ¢ Test Statistic §
¡
#
¡©
#
¡ Rejection Region : ' ¢ 19.2 17.5 ¡© for left and right handed preschoolers ¢ ¤ level? £ Signiﬁcant difference between mean motor skills scores Righthanded (2) Lefthanded (1) skills tests. competence in preschool children. Scores on motor claim a difference in mean pvalue = level! 279 any value of that is less than !! scores for left and right handed preschoolers for Cannot claim a difference in mean motor skills ? preschoolers at the motor skills scores for left and right handed Conclusion : STA 2023 c B.Presnell & D.Wackerly  Lecture 22 Ex. Investigation between “handedness” and motor 278 ¦
¤ ¢
¢ STA 2023 c B.Presnell & D.Wackerly  Lecture 22 ¢
¢
¤ ¢ ¤ £ For Monday : P. 389 – 396 For Thursday : Exer. 9.7, 9.15, 9.16, 9.18, 9.20 IMMEDIATELY!!! (Exam Conﬂict) If you are taking BUL 4310 – contact me Today : P. 378 – 383 (rest of Sec. 9.1) Assignments : Large Sample CI for estimator (p. 376) £
¢ 9.1) formula sheet until you can ﬁnd a rock. # £" 281 Two Population Means : Independent Samples (Section Thought: Diplomacy is the art of saying “Nice doggie” – Large Sample Inferences about Differences Between " Last Time: STA 2023 c B.Presnell & D.Wackerly  Lecture 23 ! ¥
Know table ¢$ " ! ¥ 280 standard errors formula sheet STA 2023 c B.Presnell & D.Wackerly  Lecture 23 ’s, use them, otherwise use § ©¨ ¡ "
§ ©¨ ¡ ! ©
$
£¤©
# £¤
© § © ©©
’s © " ¡
# ¡© ¤$
" ¡ OR OR Formula Sheet 282 RR NULL HYPOTHESIS standard error score (tail area) smaller hypothesized value or larger score pvalue a ﬁxed particular value of difference estimator TEST STATISTIC ¡ ¢ ¡ # ¡ # ©¡ versus ¤ 283 29.8 28.2 29.7
27.3 33.4
34.9 31.7 30.3 Upstream 32.2 26.4 (2) HOW??? location? discharge location exceeds that for the upstream Can it be concluded that the mean count at the 36.2 30.1 At Discharge(1) reported. Six specimens at each location 5 times, average bacteria count for each specimen also upstream from the plant. Each specimen analysed collected water specimens at the point of discharge and “puriﬁed” liquid waste into a river. EPA inspector Ex. : #9.23, p. 388 Manufacturing plant discharges STA 2023 c B.Presnell & D.Wackerly  Lecture 23
¡ # ¡© ©¡ #
% ©¡ & ¡ ¤© # ¡ ¡ % ¡ ¡ %
#
©© Hypothesis testing (p. 376) ¢ © § © §¡ ¡ #
§¡
¤
#
# ¡ #
§ ©¨ ¡
# )('
¤© ©
§ ©¨ ¡ !"
!"
¡ #
© ©
§ $
$ ¡
# ¡ © ¢
¢ STA 2023 c B.Presnell & D.Wackerly  Lecture 23
¡
Useful when one or both sample size(s) less than 30. (Section 9.1, last part) Small sample inferences about ¡ $" © ¡" (unknown) means is £¡ ¤¥ £¡ # £¤© and 284 Since (1) and (2) are valid STA 2023 c B.Presnell & D.Wackerly  Lecture 23 score : Could use variance,
? New Problem : how do I estimate this common used in Ch. 78 is a d.f. Will combine or “pool” these values, using BOTH
The “pooled” estimator will have (p. 379) d.f. , standard error of Could use , respectively. Both populations Normally distributed with ¦ Since assuming 3. Samples are Independent. © © ¢$ $ © 2. © £¤ (unknown). © © ¡ © ©
§ £ £
© ¡©
§ ¤ § " Assumptions: (p. 346)
1.
Both pop.’s have the Same variance, ¢
¡
¢ © © ©
"
# ¥
¦© © © STA 2023 c B.Presnell & D.Wackerly  Lecture 23 £ © © £$ # d.f. "
"
#
£
#
£$ $© © © " £¤
# £¤©
§ © ©
# " ¢ ! " ¢ § £$ #
§ ""
© ©
##
£
© $© ©© " ¢ # £$
£$
§ © £$ " © $
# £ £ " ¡
# £
§ $
¡© ¤
© ¡ d.f. '
# 285 ¥© ’s Always between the two LARGER samp size. Note: 3.3 is Note : divisor in ’s MORE weight to the estimator based on the Ex. : ¢ ¢ is a “weighted average” of the individual 3 and 4.2, closer to # d.f. 286 has a dist with deg of freedom gives the d.f. for this statistic. Large sample with C.I. for STA 2023 c B.Presnell & D.Wackerly  Lecture 23 ¨ STA 2023 c B.Presnell & D.Wackerly  Lecture 23 " £¤ ¥© ¥ © ¨£ $ ¥© # © § § $ £
#
"
¡
£ # © £¤© ! ¡© ¡ "
¦ ¦ ©
©
" ¨
¦ £$
# © © ©
¤ §§
"
§ ¤ Small sample (P. 379) ¤ '
#
# ¤© ¢
¢ © ¥ ¨© ¤© ¦¥ © ©
© ©
§ ©¨ ¡ © £
£ £
¥©
§ ©¨ ¤
©
#
£$ §
§ © £
©
£
£ ¤
¤ 287 versus ¡ ¡ # ¡ OR OR d.f. Test Statistic : # ¢ ©¡ # % ¦ §
¡ # ¡©
©¡ & ©¡ ¦ © ¦ ¦ ¦ RR ¦ ¤
#
or larger score (tail area) smaller score pvalue ¦ ¦
§
%
#¦
§
§ ©¨ £ ¥© # § ¤© £ # £ ¦
# !"
© 289 33.4
34.9 36.2
29.8 27.3 29.7 26.4 (2) 32.2 30.3 Upstream 31.7 location? discharge location exceeds that for the upstream Can it be concluded that the mean count at the Each is the average of ﬁve actual measurements. approximately normally distributed? Why might the recorded values tend to be 28.2 30.1 At Discharge(1) reported. Six specimens at each location 5 times, average bacteria count for each specimen also upstream from the plant. Each specimen analysed collected water specimens at the point of discharge and ¢
¢ !"
'( ¤ %
#¦
§ ©¨ '
© a ﬁxed particular value of difference ¤
¤© # “puriﬁed” liquid waste into a river. EPA inspector ¨
¨ § $ Ex. : #9.23, p. 388 Manufacturing plant discharges ¤'
¤ '© Hypothesis Testing, P. 380 £
'¨ $ STA 2023 c B.Presnell & D.Wackerly  Lecture 23 © 288 ¤ £
© ©
¤ STA 2023 c B.Presnell & D.Wackerly  Lecture 23 ¨£ ¨
¡ Test Statistic ¦ ¢ d.f. § ¢
STA 2023 c B.Presnell & D.Wackerly  Lecture 23 ¤£ £¨ ! " £¤ " ¤ ¦
¦
£¤©
# £ ¥© § $
#
§ ¥© ©¡ # ¡© 290 Rejection Region : level, there is 291 Pvalue it is upstream. bacteria count is greater at the discharge point than evidence to conclude that the mean Conclusion : At the Rejection Region : d.f. STA 2023 c B.Presnell & D.Wackerly  Lecture 23 ¢
¡ £
© ¢ ¦ # "
" #
£$©
©
"
#©
$' #
£
© $©
¡ ¢
¢
¢
¤ ¤ 95% CI : STA 2023 c B.Presnell & D.Wackerly  Lecture 23 ¤© # ¥
¤§ § ©¨ ¦¥
or ¢ – – for the two locations. the two locations. approximately distributed for both they are averages of ﬁve readings) are : the 292 taken at measurements (remember, : the samples were ¢ are approximately : the POPULATION variances of the ¡ – ¢ ¡ necessary for the above test and CI to be valid? In terms of this example, what are the assumptions ¨ ¥© ¥ § © £
£
¤
£ ...
View
Full
Document
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

Click to edit the document details