Week14_001 - STA 2023 c B.Presnell & D.Wackerly -...

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Unformatted text preview: STA 2023 c B.Presnell & D.Wackerly - Lecture 22 Thought: Why is “abbreviation” such a long word? If you are taking BUL 4310 – contact me IMMEDIATELY!!! (Exam Conflict) Assignments Today : P. 374 – 378 (Sec. 9.1, “Large Sample”) For Tuesday: Exer. 9.1, 9.13, 9.19, 9.22, 9.24, 9.25 Wednesday : P. 378 – 383 (rest of Sec. 9.1) For Thursday : Exer. 9.7, 9.15, 9.16, 9.18, 9.20 For Monday : P. 389 – 396 264 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 dist. : ¨ ¦ ©§¥ ¤ ¢ £ Hypothesis Tests       versus p-value # larger score &  RR $ %¥ " ¦¥   !¥ ! smaller score # ¦¥ $ "  '¥ ' OR 1 0 or (tail area) & OR $ %¥ ¡ Small Sample (p. 292):  Confidence Interval for ¨ )¦ ¥ $ ¨ )¦ ¥ !¥ '¥  ( ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 22 266 ¡ Test statistic :   ¡ ¥  ¨ ©¦ ¥ 0 ¨ ¦¥ ©§¤£ £ £   ¥   &0 ¨ ¦¥ ©§¤£  ' ¥# $ ¥ "   ¦¥ £ ¨  ¨¥ ¤£ From the table, £ "!¥ &   ¦¥ ¤©§¤£  & ! ¥# ¨ ¥ ¤¨ ¤£ $ "  ' ¥#   ¦¥ £ & " $ ' ¥# ¡ ¡ ¡ " From the table, ¨ ¦¥ ©§¤£ ) to 0 " " and ! ¥# ' ¥# $ Closest values in table ( with d.f. &0 $ ¨ ¦¥ ©§¤£ Look at table, ¨ ¦¥ £ ¡ p-value are , &0 ¦¥ In Everglades example, lower tail test, d.f.  £ (new) (like before) AND #d.f. £ ¤£ ¢ ) depends on ¢ $ (and (looks just like !!) ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 22 267 0 -1.732  & ¢ £ ¥$ ¨  ¨¥ ¤£ ¥$ ¡ $# & $#   ¦¥ £ Thus, in this case, best we can say is that £ "!¥ ' Can’t be any more precise using these tables! ¢   £¥ Ho . ¢   £¥ Ho . ¢  ¥ ¢    ¥ ¢   ¥ Ho . ¢  ¨  ¥ Ho . ¥ ¦ ¡  ¥ ' p-value ¤ ¤ ¤ ¤ ¤ ¤ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 22  £ ¥ ¥     #  £ ¥ ¡   £ ¥   ¥ ©¦ £ &  if d.f.  ! ¥# " ¥ ©¦ ¤£ & ! ¥# " ¡ ¡ ¡ "  ¤£ ¥ & ! ¥# So p-value that is in table for d.f.  ¡ Closest value to  " p-value = & ¡ d.f. , Ex. : New case, upper tail test, 268 is STA 2023 c B.Presnell & D.Wackerly - Lecture 22 0 $ ¥  ¥ , £ "!¥ Ex. : New case, lower tail test, d.f. 269 p-value = $ ' ¥#  . &  ¥ ¤¥ ¤£ & " ¨ ¤¥  ¥ then .  0 ¡ 0 $ ¡ ¡ So  £¥ If d.f. and  are that are in table for d.f. ¥ ¡ Closest values to and ' ¥# " STA 2023 c B.Presnell & D.Wackerly - Lecture 22 0 $ ¥ £ , ¨ ¥ ¥  Ex. : New case, two-tailed test, d.f. 270 ¡ p-value =  & .  & ¨ ¥ ¥ " ¡ ¡ 0 0 ¦ ¥ $ 0 ' ¥# "  and ' ' tail area p-value £ 0 ' ¥# $ & then & that are in table for d.f. and £ £ "!¥ ¨ ' ' $  ¡ ¡ So ¡ And ¨ ¥ ¥ If d.f. " are ! ¥# Closest values to ¨ ¤¥ ¥ tail area . . ' ¥# " STA 2023 c B.Presnell & D.Wackerly - Lecture 22 271 Ex. Investigation between “handedness” and motor competence in preschool children. Scores on motor skills tests. £   41 97.5 17.5 Right-handed (2) 41 98.1 19.2   Find a ¢ Left-handed (1) confidence interval for the true difference in mean motor skills scores for left and right handed preschoolers. ¡ Samples from two populations ¡ Objective is to compare two means ¡ HOW???? STA 2023 c B.Presnell & D.Wackerly - Lecture 22 272 Large Sample Inferences about Differences Between Two Population Means, Independent Samples (Section 9.1) ¡ Independent samples from two pop’s. Pop. 1   ¢   ¢    ¢ £ ¢ ¢  £ ¢ Sample variance  . $ ¢ ¢ £ $ ¢ – Use the difference in sample means, ¢ £  Want: Estimate for  Sample mean  Sample size ¢ Pop. variance  Pop. mean Pop. 2 (p. 375)  [formula sheet, P. 375]  [formula sheet, P. 375]   $   © ¢ ¢ ¢    § ¥£ ¨¡ ¢ ¦¤¡ ¢ § ¡¢ ¥ £ ¡¢ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 22 273 £ ¢ ¡ ¡ ¢ £ $ ¢ ¢ £   “large”, then ¢  and $ If ¢ £ If both populations normal, then normal. approx normal, ¢ regardless of shapes of sampled populations. (P. 375)  large.   formula sheet   ©  ¢ ¢ ’s          ¨ )¦ ¡    ¢  ¤ &¢ £ $ ¢ ¢ £# ’s, use them, otherwise use  ¤ ¡ table standard errors &  formula sheet  Know ¨ # ©¦ estimator ¥  and © ¦ ¥¤ £¢ ¡¡ Both © ¨§ (p. 376) ¦ Large Sample CI for STA 2023 c B.Presnell & D.Wackerly - Lecture 22 274 Ex. Investigation between “handedness” and motor competence in preschool children. Scores on motor skills tests.  £  41 97.5 17.5 Right-handed (2) 41 98.1 19.2   Find a ¢ Left-handed (1) confidence interval for the true difference in mean motor skills scores for left and right handed preschoolers.  ¡   ¡ ¨ ©¦  ¡ ¢ 90% CI is      © ¢ ¨ )¦  ¢ ¡ &¢ £ ¤  ¤ ¦ ¥  ¤ $ ¢ ¢ £# STA 2023 c B.Presnell & D.Wackerly - Lecture 22  $  equals ?  $ ¢ equals  than  ¢ ¡ is $ ¢ What does it mean if $ ¡ is ¦ 0¥ ¦ ¡ ¢ What does it mean if by  than ?  ¢ is equals ¦ ¥  What does it mean if 275 ? by ¢ How can we interpret the 90% confidence interval, , for the difference in mean motor skills & $# ¦ ¥  ¦ 0 ¥ ¦  scores for left and right handed preschoolers? (the true mean dexterity score for left-handers) larger than  ¡ ¢ could be as much as (the true mean dexterity score for right-handers) . (the true mean dexterity score for right-handers) ¡ true mean dexterity score for left-handers). or anything in between. ¢ larger than  could be as much as (the ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 22 276 ¡ Claim : no difference in mean dexterity scores for left and right handers. This claim   rejected at the be confidence level. ¡ Is it plausible that the mean dexterity score for left handed preschoolers is larger that the mean for RH preschoolers by 5 points? ¡ Is it plausible that the mean dexterity score for left handed preschoolers is smaller that the mean for RH preschoolers by 5 points? Is it plausible that the mean dexterity score for left handed preschoolers is larger that the mean for RH preschoolers by 7 points? ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 22 277 Consider testing (p. 376)  a fixed particular value of difference  $ ¢    versus p-value score & ¡ # larger $  RR ¦ " ! ¡ !  ¡  $ ¢ $ # ¡ smaller score " ¦ $ ¡ ' '  ¡ $ ¢ OR (tail area) 1 ¨ ©¦ or 0 ¨ )¦ $ ¡ ' ! ¡  ¡ (  $ ¡ hypothesized value ¤ ¦¤ NULL HYPOTHESIS $£ $      © ¢ ¢  ¢ £ ¡ £  ¡ Formula Sheet ¡ standard error  $ estimator ¡ ¢¡     $£ $   ¢ TEST STATISTIC & OR © ¢ ¢  ¢ £  ¡ ¤ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 22 278 Ex. Investigation between “handedness” and motor competence in preschool children. Scores on motor skills tests. £  ¢  Left-handed (1) 41 97.5 17.5 Right-handed (2) 41 98.1 19.2 Significant difference between mean motor skills scores £ "!¥  ¢ for left and right handed preschoolers level? ¢  $    $  ¢ ¢   0   ¢ Test Statistic ¦ ¥   ¡ Rejection Region :  ©  ¡ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 22 ¡ Conclusion : 279 claim a difference in mean motor skills scores for left and right handed £ "!¥  ¢ pre-schoolers at the level! p-value = ?  ¥  scores for left and right handed pre-schoolers for any value of ¢ ¤ ¢ Cannot claim a difference in mean motor skills that is less than !! ¤ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 23 280 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) If you are taking BUL 4310 – contact me IMMEDIATELY!!! (Exam Conflict) For Thursday : Exer. 9.7, 9.15, 9.16, 9.18, 9.20 For Monday : P. 389 – 396 STA 2023 c B.Presnell & D.Wackerly - Lecture 23 281 Last Time: Large Sample Inferences about Differences Between Two Population Means : Independent Samples (Section 9.1) ¢ standard errors  formula sheet    ©  ¢ ¢ ’s     ¨ # )¦ ¤ ¡   ¨ ©¦ ¡ $ ¢ &    ¤ &¢ £ $  ¢ ¢ £# ’s, use them, otherwise use & £ # ¤ £   Know table formula sheet $ estimator ¥  (p. 376) Large Sample CI for ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 23 282 Hypothesis testing (p. 376)  a fixed particular value of difference  $ ¢    versus p-value score & ¡ # larger $  RR ¦ " ! ¡ !  ¡  $ ¢ $ # ¡ smaller score ¦ " $ ¡ ' '  ¡ $ ¢ OR (tail area) 1 ¨ ©¦ or 0 ¨ )¦ $ ¡ ' ! ¡  ¡ (  $ ¡ hypothesized value ¤ ¦¤ NULL HYPOTHESIS $£ $      © ¢ ¢  ¢ £ ¡ £  ¡ Formula Sheet ¡ standard error  $ estimator ¡ ¢¡     $£ $   ¢ TEST STATISTIC & OR © ¢ ¢  ¢ £  ¡ ¤ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 23 283 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 also upstream from the plant. Each specimen analysed 5 times, average bacteria count for each specimen reported. Six specimens at each location At Discharge(1) Upstream (2) 30.1 36.2 33.4 29.7 30.3 26.4 28.2 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?     HOW??? ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 23 284 ¦ © (Section 9.1, last part) © Small sample inferences about Useful when one or both sample size(s) less than 30. ¢ £¡ Assumptions: (p. 346) 1. Both pop.’s have the Same variance,   &  Both populations Normally distributed with   , respectively. , standard error of   ¢ is     ©       ¢ £   ©  ¢ £ ¢    ¢ £ § ¡ ¢ ¥ ¤¡ ¢ £  ¢ © & ¢ # # Since assuming and  & ¡ (unknown) means ¢ ¢ Samples are Independent.  # 3.  2. (unknown).  $ ¢ ¢ £ STA 2023 c B.Presnell & D.Wackerly - Lecture 23 285 Since (1) and (2) are valid ¡  & is a -score : $ used in Ch. 7-8 # ¢ £  £ ©  &¢ £ $ $   ¢ ¢ £#  ¡ ¢ New Problem : how do I estimate this common  ?   $ £ $ £ # variance, ¡ Could use ¡ Could use ¡ Will combine or “pool” these values, using BOTH ¡ The “pooled” estimator will have (p. 379) 0   $ £ & & © £ $   ¢   ¢  © # # ©  # d.f.  & d.f. $ & d.f. d.f. & ¢  # £ &    $ £ &  ¢ £ $ $ ¢ © #   ¢ #  #  & £   $  ¢  # STA 2023 c B.Presnell & D.Wackerly - Lecture 23 286   is a “weighted average” of the individual ’s   ¡ MORE weight to the estimator based on the LARGER samp size. ’s 0 ¥      ¨ ¥ ¨   ¨    "£  ¢    ¢ ¡ Ex. :  Always between the two        # d.f. 3 and 4.2, closer to $ # & ©£$ ¢ # ¥ &   $ £ ¢  # $ £ © gives the d.f. for this statistic. £  deg of freedom & has a dist with  ¢     &¢ £ $ ¢ ¢ £# ¥  Note : divisor in ¥ Note: 3.3 is STA 2023 c B.Presnell & D.Wackerly - Lecture 23 ©  £ ©   £ © ¡¡ £¢ ¨ § C.I. for ¦       ¢ ¨ )¦ ¢ ¡ ¦ ¥¤ ¤£ $ £ ¢  Small sample (P. 379) £  £ © ¢    ¨ ©¦ ¥ ¡ Large sample with 287 ¤£ $ ¢ £ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 23 288 Hypothesis Testing, P. 380  a fixed particular value of difference  $   ¢  versus p-value score & # larger $ %¥  RR ¦¥ "    !¥  ! $ ¢ $ %¥ smaller score # ¦¥ $ " '¥ ' $ ¢ OR 0 ¨ ©¦ ¥ '¥ (  $  £  $£ © £ ¢  $ £ ¢ ¥   0 $  © ¢   d.f.  ¢ !¥ Test Statistic : 1 ¨ )¦ ¥ $ or (tail area) & OR ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 23 289 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 also upstream from the plant. Each specimen analysed 5 times, average bacteria count for each specimen 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 32.2 ¡ 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 discharge location exceeds that for the upstream ¥ location?   £ ¥ "£    ¥    ¢   £¥  0  ¥  ¨ 0 0  ¢ £  £     ¢  ¡  &  £   £   ¥ £  ¤£ ¥   ¢ © £    &¢ £ $  ¢ ¢ $  ¥ £# Test Statistic ¡ d.f.      ¦  ¥ ¥  ¦  &0 $ $  ¢ $  # © ¢ © ¢ # &  £    ¢ $   #     $ ¢  ¡  STA 2023 c B.Presnell & D.Wackerly - Lecture 23 290 STA 2023 c B.Presnell & D.Wackerly - Lecture 23 291 ¡ Rejection Region :  ¡ ¥   d.f. ¢ Rejection Region :  ¥  ¢ ¡ Conclusion : At the level, there is evidence to conclude that the mean bacteria count is greater at the discharge point than it is upstream. P-value ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 23 292 95% CI :  £  © £  ¢  ¤£ ¨ )¦ ¥  $ £ ¢ ¤  ¤¥ ¨  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.

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