week14 - STA 2023 c D.Wackerly Thought Why is...

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Unformatted text preview: STA 2023 c D.Wackerly - Lecture 21 281 Thought: Why is “abbreviation” such a long word? If you are taking BUL 4310 (Exam Conflict) or think you have some other exam conflict – contact me IMMEDIATELY!!! Assignments Chapters 7 & 8 All assigned problems from syllabus Today : P. 374 – 378 (Sec. 9.1, “Large Sample”) For Tuesday: Exer. 9.1, 9.13, 9.19, 9.22, 9.24, 9.25 Last regular 20 point quiz OPTIONAL Up to 20 BONUS POINTS- ¡ ¡ see explanation on intro page for Lecture 19. Wednesday : P. 378 – 383 (rest of Sec. 9.1) For Thursday : Exer. 9.7, 9.16, 9.17, 9.95, 9.105, 9.114 For Monday : P. 389 – 396 STA 2023 c D.Wackerly - Lecture 21 282 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 0 ) or (tail area) % OR # $¤ ¡ Small Sample (p. 292):  Confidence Interval for ©§ (¥ ¤ # ©§ (¥ ¤ ¤ &¤  ' ¡ STA 2023 c D.Wackerly - Lecture 21 283 Test statistic :   ¡ ¤  ©§ ¨¥ ¤ ) ¨ ¦¥ ©§¤£ £ £   ¥   %) ¨ ¦¥ ©§¤£  & ¤" # ¤ !   ¦¥ £ ¨  ¨¥ ¤£ From the table, £ "!¥ %   ¦¥ ¤©§¤£  % ¨ ¥ ¤¨ ¤£ ¤" # !  & ¤"   ¦¥ £ % ! # & ¤" ¡ ¡ ¡ ! From the table, ¨ ¦¥ ©§¤£ ) to ) ! ! and ¤" & ¤" # Closest values in table ( with d.f. %) # ¨ ¦¥ ©§¤£ Look at table, ¨ ¦¥ £ ¡ p-value are , %) ¥¤ In Everglades example, lower tail test, d.f.  ¢ ¡ (new) (like before) AND #d.f. £ ¤£ ¡ ) depends on ¢ # (and (looks just like !!) ¡ STA 2023 c D.Wackerly - Lecture 21 284 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 D.Wackerly - Lecture 21 285  £ ¥ ¤     "  £ ¥   £ ¥   ¥ ©¦ £ %  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, is ¡ ¡ STA 2023 c D.Wackerly - Lecture 21 286 ) # ¤  ¥ , £ "!¥ Ex. : New case, lower tail test, d.f. p-value = # & ¤"  . %  ¥ ¤¥ ¤£ % ! ¨ ¤¥  ¥ then .  ) ) # & ¤" ¡ ¡ ! So  £¥ If d.f. and  are that are in table for d.f. ¥ ¡ Closest values to and ¡ STA 2023 c D.Wackerly - Lecture 21 287 ) # ¤ £ , ¨ ¥ ¥  Ex. : New case, two-tailed test, d.f. p-value =  % .  % ) ¨ ¥ ¥ ) ¦ ¥ # ) & ¤" !  and & & tail area p-value £ ) & ¤" # % then % that are in table for d.f. ! and ¡ ¡ £ "!¥ ¨ & £ #  & ¤" ! ¡ & So And ¨ ¥ ¥ If d.f. ! are ¤" Closest values to ¨ ¤¥ ¥ ¡ tail area . . ¡ ¡ STA 2023 c D.Wackerly - Lecture 21 288 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 D.Wackerly - Lecture 21 289 Large Sample Inferences about Differences Between Two Population Means, Independent Samples (Section 9.1) Independent samples from two pop’s. © © ¢   ¢ ¢ ¢ © ¢ ¡ Sample variance  © © ¡ ©¢ Sample mean  . # ¢ ¡ ¢ # ¢ – Use the difference in sample means, ¡ ©¢ © Want: Estimate for © Sample size ¢ Pop. variance © Pop. mean Pop. 2 © ¡ Pop. 1 (p. 375) © [formula sheet, P. 375] © [formula sheet, P. 375] © © #  © © ¢ ¢ ¢    § ¥£ ¨¡ ¢ ¦¤¡ ¢ § ¡¢ ¥ £ ¡¢ ¡ STA 2023 c D.Wackerly - Lecture 21 290 ¢ ¡ ¡ ©¢ # ¢ ¡ ¢  © “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 D.Wackerly - Lecture 21 291 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 D.Wackerly - Lecture 21 © # # equals ? # © equals © by  ¢ © is ¢ ¡ ¢ What does it mean if than # ¡ is ¦ )¥ ¦ ¢ What does it mean if ? by © than © ¢ is equals ¦ ¥  What does it mean if 292 ? ¡ ¢ How can we interpret the 90% confidence interval, , for the difference in mean motor skills % #" ¦ ¥  ¦ ) ¥ ¦  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 left-handers) smaller than © ¢ ¡ could be as much as (the true mean dexterity score for right-handers) . or anything in between. ¡ STA 2023 c D.Wackerly - Lecture 21 293 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 D.Wackerly - Lecture 21 294 Consider testing (p. 376)  a fixed particular value of difference © # ¢    versus p-value score % ¡ " larger #  RR ¥ ! ¡  ©  ¡  # ¢ # " ¡ smaller score ! ¥ # ¡ & &©  ¡ # ¢ OR (tail area) 0 ©§ ¨¥ or ) ©§ (¥ # ¡ & ¡  ¡ ' © # ¡ hypothesized value ¤ ¦¤ NULL HYPOTHESIS #©¢ # © © © ©  © ¢ ¢  ¢ ¢ ¡ £  ¡ Formula Sheet ¡ standard error  # estimator ¡ ¢¡  © © © ¢ TEST STATISTIC % OR #©¢ #  © © ¢ ¢  ¢ ¢  ¡ ¤ ¡ STA 2023 c D.Wackerly - Lecture 21 295 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? ¢  # ©   # © ¢ ¢   )    ¢ Test Statistic ¦ ¥   Rejection Region :  ©  ¡ ¡ ¡ STA 2023 c D.Wackerly - Lecture 21 Conclusion : 296 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 D.Wackerly - Lecture 22 297 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 (Exam Conflict) or think you have some other exam conflict – contact me IMMEDIATELY!!! Tomorrow : Exer. 9.7, 9.16, 9.17, 9.95, 9.105, 9.114 Monday : P. 389 – 396 STA 2023 c D.Wackerly - Lecture 22 298 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 D.Wackerly - Lecture 22 299 Hypothesis testing (p. 376)  a fixed particular value of difference © # ¢    versus p-value score % ¡ " larger #  RR ¥ ! ¡  ©  ¡  # ¢ # " ¡ smaller score ¥ ! # ¡ & &©  ¡ # ¢ OR (tail area) 0 ©§ ¨¥ or ) ©§ (¥ # ¡ & ¡  ¡ ' © # ¡ hypothesized value ¤ ¦¤ NULL HYPOTHESIS #©¢ # © © © ©  © ¢ ¢  ¢ ¢ ¡ £  ¡ Formula Sheet ¡ standard error  # estimator ¡ ¢¡  © © © ¢ TEST STATISTIC % OR #©¢ #  © © ¢ ¢  ¢ ¢  ¡ ¤ ¡ STA 2023 c D.Wackerly - Lecture 22 300 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 D.Wackerly - Lecture 22 301 ¦ © (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 D.Wackerly - Lecture 22 302 Since (1) and (2) are valid ¡ © % is a -score : # used in Ch. 7-8 " ¢ £ © £ ©  %¡ ©¢ # # ©  ¡ ¢ ¢"  ¡ ¢ New Problem : how do I estimate this common © ? d.f. #© £ % # £ % © ¢   ¢ " © ©  ¡ Could use " Could use  variance, d.f. ¡ Will combine or “pool” these values, using BOTH The “pooled” estimator will have (p. 379)  ) #© © © #© £ % % © £ #©   ¢   © © " " ©  # d.f. d.f. % ¢ £ % £ %  #© £ # # ¢ © "   ¢ "  " % £   # © ¢  " ¡ ¡ STA 2023 c D.Wackerly - Lecture 22 303 © © is a “weighted average” of the individual ’s   MORE weight to the estimator based on the ¡ LARGER samp size. ’s ) ¥ © © ©   ¨ ¥ ¨   ¨  ©  "£  ¢    ¢ ¡ 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 D.Wackerly - Lecture 22 304 ¦ © © £ © ©  £ © ¡¡ £¢ ¨ § © ©  ©  © ¢ ©§ (¥ ¢ ¡ ¦ ¥¤ Large sample with C.I. for £©¢ # © £  £ © ¢  ©  ©§ ¨¥ ¤ £©¢ # ¢ ¡ ¢ ¢ Small sample (P. 379) ¢ ¡ STA 2023 c D.Wackerly - Lecture 22 305 Hypothesis Testing, P. 380  a fixed particular value of difference © #   ¢  versus p-value score % " larger # $¤  RR   &¤ ¥¤ #  ¤ ¥¤ !   © # ¢ # $¤ smaller score " ! &© # ¢ OR ) &¤ ©§ ¨¥ ¤ ' © # © £  #©¢ © £ ¢  # ¢ ¢ ¤ ©  ) #©  © ¢   d.f.  ¢ ¤ Test Statistic : 0 ©§ (¥ ¤ # or (tail area) % OR ¡ STA 2023 c D.Wackerly - Lecture 22 306 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 32.3 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?   £ ¥ "£  ©  ¥  © © ¢   £¥  )  ¥  ¨ ) )  ¢ ©¢  ¢   ©  ¢  ¡ © % © £ © © £   ¥ £  ¤£ ¥   ¢ © £   © %¡ ©¢ #  ¡ ¢ #  ¤ ¢" Test Statistic ¡ d.f.      ¦  ¥ ¥  ¦  %) #© #©  ¢ #  " © ¢ © ¢ " %  £ ©   ¢ #   "  © ©  # ¢    ¡ STA 2023 c D.Wackerly - Lecture 22 307 STA 2023 c D.Wackerly - Lecture 22 308 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 D.Wackerly - Lecture 22 309 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 ¡ STA 2023 c D.Wackerly - Lecture 22 310 Minitab? Punch in data values Basic Statistics 2-Sample ¤ ¡ Stat Select variable and click Radio button “Samples in ¡ ¡ different columns” Click in box labelled ”First”, double click on Variable ¡ 1. Click in box labelled ”Second”, double click on ¡ Variable 2. Choose alternative (Greater than in this case) Click (Check) in box ”Assume equal variances”. ¡ ¡ Two sample T for AtDisc vs Upstream AtDisc Upstream N 6 6 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) T-Test mu AtDisc = mu Upstream (vs >): T = 1.53 P=0.078 Both use Pooled StDev = 2.80 DF = 10 ...
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