Week13 - STA 2023 c D.Wackerly - Lecture 21 289 Thought: If...

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Unformatted text preview: STA 2023 c D.Wackerly - Lecture 21 289 Thought: If you pick up a starving dog and make him prosperous, he will not bite you; that is the principal difference between a dog and a man. -Mark Twain Assignments Monday : P. 378 – 383 (rest of Sec. 9.1), For Tuesday: Exer. 9.7(a)(c), 9.26, 9.114(a)(b)(d) Wednesday : P. 378 – 383 (rest of Sec. 9.1), P. 389 – 396 For Thursday : 9.7, 9.16, 9.17, 9.95, 9.105, 9.114, 9.29, 9.33, 9.35, 9.38, 9.39, 9.42, 9.94, 9.96, 9.101, 9.110 For Monday : p. 402 – 406 STA 2023 c D.Wackerly - Lecture 21 290 Last Time: Large Sample Inferences about Differences Between Two Population Means : Independent Samples (Section 9.1) ¥  ' 6  % 2# "    !  )1  ¨  0¥ )1 ’s, use them, otherwise use  4 5 3    '  %# &$" ! 3 Know formula sheet table 4  3 ¨ §¦¤£¡ ¡¥ ¢ ¢ formula sheet standard errors ’s (   © estimator ¨ (p. 376)  Large Sample CI for STA 2023 c D.Wackerly - Lecture 21 291 Consider testing (p. 376) a fixed particular value of difference £ ¨ ¢ ¡ ¥¤ versus p-value ¨" larger score  ¢¦ RR #" ¥¨ #" ¨ ¥¨ " § ¨ § ¥¤ ¨" smaller score " © © ¥¤ ¨  OR or (tail area)   " %&# "  © " % 2# " ¨ £   ¥¤ ¨ § TEST STATISTIC   NULL HYPOTHESIS ¤  6 4 ¨1 4 5 6  ¨1  ¡ Formula Sheet  " £ standard error ¡ ¨ hypothesized value   estimator  OR ¤  3 4 ¨1 4 5 3 ¨1 £ "  STA 2023 c D.Wackerly - Lecture 21 292 Ex. Investigation between “handedness” and motor competence in preschool children. Scores on motor skills tests. )1 6 4 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 : 5 £ " STA 2023 c D.Wackerly - Lecture 21 Conclusion : 293 claim a difference in mean motor skills scores for left and right handed ¢ ¡( £ © pre-schoolers at the level! p-value = ? ¢ ¥¢ ( £ ©  Cannot claim a difference in mean motor skills scores for left and right handed pre-schoolers for © any value of that is less than !! Can only claim a difference in mean motor skills scores for left and right handed pre-schoolers for © values of that are .8808!!  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 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 21 295 £ ¢¡ Small sample inferences about (Section 9.1, last part) Useful when one or both sample size(s) less than 30. Assumptions: (p. 346) 1. Both pop.’s have the Same variance, 3 3 £ £ 3  ¤¥  ¥¥ Samples are Independent. Both populations Normally distributed with , respectively. 3 4 5 3 4 £  3 , standard error of 4 5 3 4 ¡ 4 5 £ ¡ 4 3 3 £ is 3 Since assuming 3 and  ¦¥ (unknown) means   3. ¦ §¥ ¤ 2. (unknown).  )1   ¨ © ¨ © 3 £ £ ¨  )1 STA 2023 c D.Wackerly - Lecture 21 296 Since (1) and (2) are valid "   is a -score : used for L.S. inf. 4 4 3 5 ¡ ¡ ¨   ¥ ¨   )1 ¨  )1 ¥ £ " New Problem : how do I estimate this common 3 ? d.f. 6 Could use d.f.  Will combine or “pool” these values, using BOTH The “pooled estimator will have (p. 379)  $¡ 6   $¡ 5 4 ¨  4¥ ¨  4¥ 4 ¥ £  $¡ 6 ¨  4¥ '   5 $¡ ¨ 4 ¥ 6   5  $¡ ¨ 4 ¥ # d.f.  ¨  4¥ ¡  Could use ¨4 ¨ ¡ variance, d.f.  5 $¡ £  ¨  4¥ 6 STA 2023 c D.Wackerly - Lecture 21 297   is a “weighted average” of the individual ’s 6 6 MORE weight to the estimator based on the LARGER samp size. ( ¡  £ 6 ¡ ( £ 4 6  ¡ £ 6 ¡ £ ¡    £ £   4 6 Note: 3.3 is 6 Ex. : and  Always between # d.f. 3 and 4.2, closer to ¢  $¡ ¨  4 ¥ 5 $¡ ¨  4 ¥    ¡ ¨  ¥ 4 4 6 5 ¡ ¨   ) 1 ¨  0¥ )1 deg of freedom ¢ ¢ £  6 Note : divisor in has a dist with gives the d.f. for this statistic. STA 2023 c D.Wackerly - Lecture 21 298 £ ¢¡ 3 ¡ 4 5 ¡ 4 ¥¥ §¦¡ ¤ £ £ 3 3 £ 3  ¢ ¢ ¡ % &# " ! 1 ¨1 Small sample (P. 379) ¡ 4 5 ¡ 4 6  % 2# ¢ Large sample with C.I. for ! 1 ¨1 STA 2023 c D.Wackerly - Lecture 21 299 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. Want : 95% a CI for the difference in mean bacteria counts at the discharge location and the upstream location? ( ¡¢ ¢ ( ¡¡ ¢¡  £ 6  £ 6 ¢ £¡ (  ¡ ( ¢ £ £ 1 £ 1 4 £ £ 4 6   $¡ 4 ¡ 5 or ¢ ¥¢ ( ¡   £  £ ¤ ¥¢ ( ¤ £   ¨  4 5  4¥  ¨  4 ¥ 5  6 $¡ ¨  4 ¥  4 ¡ 6 ! ! d.f.  % &# ¢ ! £ 1  6 ¨1 95% CI : STA 2023 c D.Wackerly - Lecture 21 300 STA 2023 c D.Wackerly - Lecture 21 301 Ex. : #9.23, p. 388 : Can it be concluded that the mean count at the discharge location exceeds that for the upstream 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 32.3  ( ¡¡ ¢¡  ( ¡¢ ¢ £ 6 ¢ £¡ (  ¡ 4 £  ¨ 5 £ £6   ¡ ¢ ( ¢ ' ¢ ¥ 5  ¡ ¡ ( ¢ §' ¢ ¥ ¥ ¡¥ £ 6  ( £ £  ¢ £ 1 4 ¤ £¢ ( ¤ £ 1 d.f. =  ¨ ¢ ¡  ¨ ¢ ¦ STA 2023 c D.Wackerly - Lecture 21 302 Rejection Region : ¡ Test Statistic 4 4 6 5 ¡ ¡ ¨   ) 1 ¨  0¥ )1 ¤ ¢ £ £ £ ( ¡ ¡ £ ¢ £ ¥ £ ¢ £¢ ( £ © £ © Conclusion : At the ¡ d.f. Rej.Reg : level, there is evidence to conclude that the mean bacteria count is greater at the discharge point than it is upstream. STA 2023 c D.Wackerly - Lecture 21 303 P-value In terms of this example, what are the assumptions necessary for the previous CI hypothesis test to be valid? ¦ §¥ ¤ – : the POPULATION variances of are approximately ¤ the : the samples were ¥ – for the two locations. 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 22 304 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 Tomorrow : Exer. 9.7, 9.16, 9.17, 9.95, 9.105, 9.114, 9.29, 9.33, 9.35, 9.38, 9.39, 9.42, 9.94, 9.96, 9.101, 9.110 Monday : p. 402 – 406 Tuesday : Exer. 9.46, 9.52–54, 9.56, 9.59, 9.60, 9.97, 9.100, 9.107 and rest of Chpt 9 problems on syllabus! For Wednesday : Print out a copy of the ANNOTATED FORMULA SHEET from the course web page. The link is right under the “Formula Sheet” link near the top of the page. STA 2023 c D.Wackerly - Lecture 22 305 Last Time: Small sample inferences about £ (Section 9.1, last part) ¢¡ Useful when one or both sample size(s) less than 30. Assumptions: (p. 346) 1. Both pop.’s have the Same variance, 3 3  ¥¥ £ 3  ¤¥ £ Samples are Independent.  Both populations Normally distributed with , respectively. ¡ 6  ¡  5  ¡ ¨ 4¥ 6  5   ¡ ¨ 4¥ ¨  4¥ ¨  4¥ degrees of freedom. (P. 379) ¡ 4 5 ¡ 4 6 ¨  % &# ¢   6 ¨4 5 4 Small sample CI for  £ has and  (unknown) means  ¦¥ 3. ¦ §¥ ¤ 2. (unknown). ! 1 ¨1 STA 2023 c D.Wackerly - Lecture 22 306 Hypothesis Testing, P. 380 ¡ a fixed particular value of difference £  ¨ ¢ ¡ ¤ versus p-value ¨ ¢ larger score  ¢¦ RR # ¥¨ ¢ ¡ ¢ § ¨ § ¤ ¨ ¢ smaller score # ¥¨ ¨ ¢ ¡ ¢ ¨ © ¤ © OR % &# or   %2#  ¨ ¢ ¢ ¡ ¢ © ¨ £   ¤ ¢ § Test Statistic : ¡ ¤ ¡ 4 5 ¡ ¨1 4 ¨1 ¢ £ 6  ¨4 5 4 £ d.f. (tail area)  OR STA 2023 c D.Wackerly - Lecture 22 307 Minitab? Punch in data values Stat ¢ Basic Statistics 2-Sample 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 STA 2023 c D.Wackerly - Lecture 22 308 Ex. : #9.101, P. 430 Study to assess the effect of biofeedback exercises on blood pressure. Six subjects were taught biofeedback. Blood pressure measurements (millimeters of mercury) were taken before and after the training. Mean blood pressure after ¡ $§¥  ¥ After © Before £ ( ¢ ¡ ¡ ¡ ¡ ¢( Subject ¢ ¥¢ ( learning biofeedback less than before? Use . ¡  ( ¡ ¡¢ ¢( ¢ ¢ ¡ ( ¢¡ (¢ ¡ ¡ ¤( ¢ ¡ ¤¡ ¡ ( ¡( ¢ ( ¡ ¡ ( ¡ ¡ ¡ (¢  ¡ ¡ ¡ ¢ STA 2023 c D.Wackerly - Lecture 22 309 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 D.Wackerly - Lecture 22 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?? 310 STA 2023 c D.Wackerly - Lecture 22 311 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 D.Wackerly - Lecture 22 312 Popn. 1 Popn. 2 Difference £££ ¤ ¢¡ ¢¡ . . . £££ depen. . . . ¥ ¦¤ . . .  depen.  £££ ¤ Indep. depen. ¢¡   differences £ Pairs £ £ ¢¡ ¡ ¤  £ ¨ £ £ § ¨ ¢ ¡ ASSUMPTION : the DIFFERENCES are approx. NORMALLY distributed ¨¨¨ ©¨  is NOT required ¤ § same as “2”. ¡ if differences are taken “1” ¨ §4 § mean of the population of DIFFERENCES .  3 £  3 ¥ ©¨ ¨¨¨ STA 2023 c D.Wackerly - Lecture 22 313 Method of Analysis : do a one-sample “t” ON THE DIFFERENCES Ex. : #9.101, P. 430 Study to assess the effect of biofeedback exercises on blood pressure. Six 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 After Diff. Diff. ¡¡ ¢( ¡ ¡ ¢ ¡ £¡ ( ¢ ¡ ¢ ¢(  ( ¢ (¡¡ ¡ ¡ ¡ ¡ ( ( ¤¢ ¡ (¢¡ ( ¡ ( ¢ ( ¡( ¢  ¡ ( ¡ ¡ ¢( ¢ ¢ ¡ ¡ ¡ ¡  ( ¡ ¡¢ ¡ ¡ ¢( ( ¢¡ (¢ ¡ ¡ ¤( ¢ ¡ ( ¤¡ ¤( ¢  ( ¡ ( ¡ Totals ¥ Before  ¡ $§¥  ¥ Subject . ¡ ¤¡ ¡ (¢ ¡ ¢ STA 2023 c D.Wackerly - Lecture 22 314 §1 £  ¤ ¥ §4 ( ¡¡¡ ¢¤ ¡ ¨ ¤ ¡ §4 ¨ § 6 £ £ £ §6 £ £ d.f. Mean blood pressure larger before than after? ¢ £ ¨ ¢ ¡ ¢¡ § ¢ § ¨ ¢ ¦ ¢¦ § ¡ ¤ §4 §6 §1 ¨ £ ¢ ¡ ¥ £ ¢ ¢ £¢ ( £ £ © ¢ ¥¢ ( £ © Conclusion : At the £ ¢ £ d.f. level there that the mean blood pressure reading is higher before learning biofeedback. STA 2023 c D.Wackerly - Lecture 22 315 P-value? 0 ¥ (3.365) ¤ ££ ¢¡ (table value)(standard error) §6  % 2# §4 ¢ ! ! ¡ ¢ estimator §1 ¢ £ ¥ £ ! or ¡ d.f.  ¥ ¢ ¡ (2.571) C.I. for ¢ 2.977 ¡ ( ! ( ¢ £¡ STA 2023 c D.Wackerly - Lecture 22 316 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 ¢ ¥¢ ( £ © 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 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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