Week13-2up - Last Time: Large Sample Inferences about STA...

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Unformatted text preview: Last Time: Large Sample Inferences about STA 2023 c D.Wackerly - Lecture 21 For Monday : p. 402 – 406 9.110 9.29, 9.33, 9.35, 9.38, 9.39, 9.42, 9.94, 9.96, 9.101, For Thursday : 9.7, 9.16, 9.17, 9.95, 9.105, 9.114, P. 389 – 396 Wednesday : P. 378 – 383 (rest of Sec. 9.1), estimator ¥ ¤ £ ¢£ Know formula sheet $ #¤ "  ¢  % table standard errors ¤ § ¦  formula sheet ’s, use them, otherwise use  For Tuesday: Exer. 9.7(a)(c), 9.26, 9.114(a)(b)(d)  Monday : P. 378 – 383 (rest of Sec. 9.1),  $" ¥    (p. 376) Large Sample CI for %  & Assignments §  ¨ ¡ Independent Samples (Section 9.1) Differences Between Two Population Means : difference between a dog and a man. -Mark Twain ' (  © prosperous, he will not bite you; that is the principal Thought: If you pick up a starving dog and make him 289 STA 2023 c D.Wackerly - Lecture 21 %    ¥ ’s &    © § !  290 ¡ ¢ £ © © OR © OR  Formula Sheet estimator TEST STATISTIC  291 RR NULL HYPOTHESIS standard error score (tail area) smaller hypothesized value or larger score p-value § © ¥ ¥ ¥ © ©  © © ¤  $ ¡ ¥ ¨¢ £ © ¦ §¥  ¦ §¥    versus %  ¤ ¦ §¥ a fixed particular value of difference ¡¥  ¤ ¦ §¥ ¤   ¥   &  $  (  $ %  &   ©   ¥ ©     ¥ ' $ ¥ ¥ ¥    ¤  41 41 98.1 97.5 19.2 17.5 292 for left and right handed preschoolers Test Statistic Rejection Region : level? Significant difference between mean motor skills scores Right-handed (2) Left-handed (1) skills tests. competence in preschool children. Scores on motor Ex. Investigation between “handedness” and motor STA 2023 c D.Wackerly - Lecture 21 ¨ ¦  & Consider testing (p. 376) ' ¡¢ £ £ ¤  & ¥ ¡ ¡  © © ¤ ¥ ¥ ' ¥ $"  © ©  ¦ ¦ ¤ ¤ ¥ ( ¢£ ! ¤ § ¦§¥ ¤ ¤  ¡¥ (  & £! ! STA 2023 c D.Wackerly - Lecture 21 " ¡ p-value = claim a difference in mean level! that is less than !! values of that are .8808!! scores for left and right handed pre-schoolers for Can only claim a difference in mean motor skills any value of scores for left and right handed pre-schoolers for Cannot claim a difference in mean motor skills ? pre-schoolers at the   ¦ ¡ motor skills scores for left and right handed ¦ 294 36.2 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 29.8 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 Conclusion : ¦  ¤ £! ! ¦ ¤ ¢£ ! STA 2023 c D.Wackerly - Lecture 21 ¡ ¡ 293 ¨ £ STA 2023 c D.Wackerly - Lecture 21 ¡ £ £ ¤ %  (unknown) means ¥ $"  $" is and , standard error of , 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, § Useful when one or both sample size(s) less than 30. ¦¥ '  Since (1) and (2) are valid STA 2023 c D.Wackerly - Lecture 21 -score : used for L.S. inf. is a ? d.f. d.f. The “pooled” estimator will have (p. 379) Will combine or “pool” these values, using BOTH Could use Could use variance, New Problem : how do I estimate this common  ¡ ¡ ¤ ¢ ¤ " $ #¤  ¤ ¡ ¡ % & ¤ ¢ % ¥ (  % §  § ¢ ¤ %  ¤  & & ' ¤ & ¤ & ¤ (Section 9.1, last part) ' ¥ $"  (  (  ¤ ¤ ¥ Small sample inferences about 295 ¢ © ¤ ¢ &  & & § ¢ ¢ § (  ¥ % © ' ¥ & &  © # d.f. § STA 2023 c D.Wackerly - Lecture 21  & ¤ ' ¡ ¢ ' § ¥ ¥ § ¢ § ¢ ¤ ' ' ¤ & & ¢ § (  § ¢  % & ¥ ¢ & ¥ ¢ &  & ¥ ¥ d.f.  ¥ 296  ( ¡ Note: 3.3 is Ex. : ¢ Always between Note : divisor in and LARGER samp size. ¡ ¤ ’s has a dist with 3 and 4.2, closer to # d.f. MORE weight to the estimator based on the (  & $ #¤ "  (  ¤ ¤ $" ¥ ( (  deg of freedom gives the d.f. for this statistic. © 297 STA 2023 c D.Wackerly - Lecture 21 ¨ £ Large sample with Small sample (P. 379)  is a “weighted average” of the individual ¤ ¦© STA 2023 c D.Wackerly - Lecture 21 § £ &  § ¤   ¢  © ¤ ¤ & (  ¤ ' ¡  ¥ ¢ & ¥ & (  ¢ ¢ ( (  ¤ ¢ ¡ ! ¦ $ ¤ ' ¥ $   ¡ ¤ & $ ¥ $ ¥ ! §   ¡ ¡ ©    ¥ § ¢ ¥ § ¢  %  ¢ C.I. for ¤  %  % (    ¤   ¡ % ¢ & ¢ & £ ' ' ¤ ¢ ¢ & ¢ &   298 299 29.8 28.2 34.9 33.4 27.3 29.7 31.7 30.3 Upstream 32.3 ¤ ¤ & & location? ¡ counts at the discharge location and the upstream Want : 95% a CI for the difference in mean bacteria Each is the average of five actual measurements. approximately normally distributed? (2) 26.4 Why might the recorded values tend to be 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 ¢£  !  ¢£ ¤ ¤! ! 95% CI : STA 2023 c D.Wackerly - Lecture 21 d.f. or ' $  “purified” liquid waste into a river. EPA inspector (  Ex. : #9.23, p. 388 Manufacturing plant discharges (  (  ! ¢ ! ! ¤ ¤ ¤  ¡ (  $ ¥   ¤ & !" ¢  ! £! !  ¥ ¤ & ¤  "    ¢ § (  '  ¥ ' & ¤ & ¢ & ¥ § ¢ & ¢ § (  STA 2023 c D.Wackerly - Lecture 21  300 28.2 30.1 29.8 36.2 d.f. = $ 34.9 33.4 At Discharge(1) § 27.3 29.7 31.7 30.3 Upstream 32.3 26.4 (2) ¢ ¡ ¡ &  ¤ © & ¤! © upstream location? count at the discharge location exceeds that for the ¤ ( ¨   ¢ ! § ¤ £ ¡ $ ¢£ ¤ ¤ ¡  ¡  ' ¢£ ! ! ! ¥ ' ¤  !§  £ Test Statistic STA 2023 c D.Wackerly - Lecture 21 level, there is 302 Rej.Reg : it is upstream. bacteria count is greater at the discharge point than evidence to conclude that the mean Conclusion : At the d.f. Rejection Region : ¢ 301 § Ex. : #9.23, p. 388 : Can it be concluded that the mean ¤ ¤ ¤ STA 2023 c D.Wackerly - Lecture 21 © ¤ ¥ !¤ ! § © (  (  ¥ ¡ ¡ ¡ ¢£ ¤ ¤! ! ¤ " ! ! !" ! ¦ ¤ ¦ " $ #¤  (  ¢ ! $" ¥ £ ¢ ¤  ¤! £ ! ¡ ¢ & ¥ ' ¤ ¢ ¡¥ ¢ & ¦ ¤ ¤ 304 ¡ ¦¥ – – locations. approximately taken at measurements (remember, distributed for both they are averages of five readings) are : the the two : the samples were for the two locations. are approximately : the POPULATION variances of the ¦ – § valid? the page. right under the “Formula Sheet” link near the top of ANNOTATED FORMULA SHEET from the course web page. The link is For Wednesday : Print out a copy of the problems on syllabus! 9.60, 9.97, 9.100, 9.107 and rest of Chpt 9 Tuesday : Exer. 9.46, 9.52–54, 9.56, 9.59, Monday : p. 402 – 406 9.110 9.29, 9.33, 9.35, 9.38, 9.39, 9.42, 9.94, 9.96, 9.101, Today : P. 378 – 383 (rest of Sec. 9.1), P. 389 – 396 Assignments : until you can find a rock. Thought: Diplomacy is the art of saying “Nice doggie” – STA 2023 c D.Wackerly - Lecture 22 necessary for the previous CI hypothesis test to be 303 Tomorrow : Exer. 9.7, 9.16, 9.17, 9.95, 9.105, 9.114, P-value ¥ § In terms of this example, what are the assumptions ¡ STA 2023 c D.Wackerly - Lecture 21 £ ¡ Useful when one or both sample size(s) less than 30. ¤ (unknown) means $  $ and , respectively. Both populations Normally distributed with ¥ (P. 379) degrees of freedom. Small sample CI for has ¤ & 3. Samples are Independent. ' (  & ¤ 2. (unknown). Assumptions: (p. 346) 1. Both pop.’s have the Same variance, %  ¥ %  ¦¤ ¤ % § ¤ § § § ¤ ¥ ¤  © ¤ &  ¥ & ¢ versus OR OR d.f. Test Statistic : 306 RR or score score (tail area) smaller larger p-value a fixed particular value of difference Hypothesis Testing, P. 380 STA 2023 c D.Wackerly - Lecture 22 ¢ (Section 9.1, last part) ¦¥ ¢ © ¡¢ £ © § © ' ' ¤ & ¤  © © ¡¥  ¡¥ ¢ Last Time: Small sample inferences about  © ¢ § (  ¥ ¥    ¥ ¥ ' § (  ¢ & & ¢ & ¢ § (  § ¢ ¢ © ¥ ¥ ¥ © © © © © ¡ ¥ ¨¢ £ ¤ ¢ 305 ¥ ¢ STA 2023 c D.Wackerly - Lecture 22  ¤ ¡¥ & ¢ $  ( ¤ ¡¥ ' ¢ ¤ & ¢   ¥ © ¢  ¥  ¢ $ & ¥ ¢  ¢ ' ©  ¥      ¡¥ ¢ & ¥ ¥ ¤ ¤   § § 2-Sample ¡ ¡ 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 AtDisc Upstream Two sample T for AtDisc vs Upstream DF = 10 Click (Check) in box ”Assume equal variances”. Choose alternative (Greater than in this case) Variable 2. Click in box labelled ”Second”, double click on 1. Click in box labelled ”First”, double click on Variable different columns” Select variable and click Radio button “Samples in Basic Statistics ¢ 308 Subject learning biofeedback less than before? Use Before After . before and after the training. Mean blood pressure after measurements (millimeters of mercury) were taken were taught biofeedback. Blood pressure biofeedback exercises on blood pressure. Six subjects Ex. : #9.101, P. 430 Study to assess the effect of STA 2023 c D.Wackerly - Lecture 22 Stat Punch in data values Minitab? 307 ¦ ¡ ¡ ¡ ¡ § £ STA 2023 c D.Wackerly - Lecture 22 ! ¢  ¢ ¡  ! ¢ ¡ ¤ ¢ £ ¢ ¢ ¡ ¡ ¢ ! ! ¡ ¢ £ " ¡  ! ! £ ¢ ¢ ¢ ¢ ! ¢ ¢ £ ¡ ¡ £ ¡ ! !  §¤  £ ¡ ! " ! !  !  ! ¢ ! ¤! ! ¡ ¡ ¡ independent!!! HOW do we analyse the data??? for) (2) – (6). HOW?? Want: information about (1), compensating (controlling 6. Etc. – – 5. Driving habits of drivers 4. Road conditions and Surface 3. Speed driven 2. Weight of cars 1. Quality of the tires Things that influence mileage: cars – drive around – record mileages. Randomly select some tires of each brand – install on belted radial tires. – – What “factors” can impact BP? Why take before and after BP’s on same subject? Samples are will be low. Is likely that both before and after BP measurement not overweight? What happens if someone is careful about diet and after learning biofeedback. Each individual subject had BP taken before AND Objective : Compare durability of two brands of steel STA 2023 c D.Wackerly - Lecture 22 Sample sizes are small. 309 STA 2023 c D.Wackerly - Lecture 22 ¡ ¡ ¡ ¡ ¡ 310 ¡ ¡ Can’t analyse data using method of Section 9.1 Indep. Before/After Experiments, Biofeedback #9.101 – By design to control for other factors manner. – Someone else collected the data in a paired Why Pair? Two brands of solar collectors : one better? Treated wood last longer than untreated? PAIRED DIFFERENCE EXPERIMENT ¡ Problem : Samples are NOT independent This strategy will control for (2) – (6). How about left and right sides? installed on the front of each car. Popn. 1 . . . depen. . . . differences Difference . . . same as if differences are taken “1” NORMALLY distributed is NOT required . “2”. mean of the population of DIFFERENCES Pairs depen. depen. Popn. 2 ASSUMPTION : the DIFFERENCES are approx. ¡ ¡ Randomly select one tire of each brand to be § ©© ¡ ¡ §& ¡ § ¡ ¤ ¤ © ©© Method to consider : © © ¤ %  ¡ STA 2023 c D.Wackerly - Lecture 22 ¤ 311 %  ¡ ¥ ¥ © © © ¡¢ ¤ ¡¥ STA 2023 c D.Wackerly - Lecture 22 ©© § ©© ¡ ¡ ¤ ¡¢ £ © ¨ £ £ £ © £ £ £ ¡¢ ¡¢ ¤ ¨ £ £ £ ¤  ¤  ¥ ¦¤ £ §© ¥ ¤ ¡¥ 312 Subject less than before? Use Before After Totals . Diff. training. Mean blood pressure after learing biofeedback (millimeters of mercury) were taken before and after the were taught biofeedback. Blood pressure measured biofeedback exercises on blood pressure. Six subjects Ex. : #9.101, P. 430 Study to assess the effect of § Diff. $ §  ¢ Mean blood pressure larger before than after? d.f. §( d.f. level there learning biofeedback. that the mean blood pressure reading is higher before Conclusion : At the ¢ DIFFERENCES ¡ ¡  ¥ Method of Analysis : do a one-sample “t” ON THE ! ! ¢ STA 2023 c D.Wackerly - Lecture 22 ¥ 313 © ¢  ¡  ¢ ¡ ¤ ¢ £ ¢ ¢ ¢ £ ¢ ! ! ¢ £ ¡ " £ ! ! ¦ ¤ £ ¡ ! ¡ ¢ !  !  ¢ ¢ ¢ ¢ ¡ § ¤ £! ¢ " ! !  !  £ ¡ ! ! ¢£ ¡ ! ¢ £  !  ¢ ! " " ! !  !  ! ¤£ STA 2023 c D.Wackerly - Lecture 22 ¡ $ ¤ ¤ ¤ ¡ ¢ ¡ ¥ §( ¡ ¢ § ¦ ! ¤ §( ¤ ¨¢ £ ¥ §& § ¡ ¢ ¥ © ¨¢ £ §© ¤ £!  ! ¢£ ¢ ! ! " ¡ ¡¥ ! © ¦ ! ¤ ¡¢ £ ¦  ¡ §& ¤ ¤ §& ¤ ¤ ¤§ © ¡¢ £ §© ¤ ©£ ¤! £ ¤! ! ¢ ! ! !  ¤ ¤ £ " ! 314 d.f.  ¢£  ¡ ¢ £ 0 (2.571) 2.977 (3.365) (table value)(standard error) § estimator  ¢ or 315 (and others) level. confidence. and dictated the method of analysis The manner in which the data was collected test? Why did we analyse the data using the paired difference 316 milliliters of – MORE information in CI, no more work!!! mercury with – Between CAN tell how big the difference is from the CI difference at the Can’t tell from hypothesis test. CAN say there is A before and after learning biofeedback? Is there a “big” difference in the mean BP readings STA 2023 c D.Wackerly - Lecture 22 ! ¢ C.I. for    $ ¤  ¦ £ P-value?   ¡ ¢  ¦ ¡ ¢ ¢ ¦ ¤ ! ¢  ¡ ¦ ! ¡ ¡ ¡ ¤! ! ¨ STA 2023 c D.Wackerly - Lecture 22 ¡ ¥ £ ¤ §( §& ¤ £ . 317 , controlling for the others by collecting the data in this manner!!! Can assess the impact of 9. Etc. 8. Ethnic background 7. Gender 6. Stress level 5. Lifestyle 4. Weight of patients ¤ 3. General physical condition 2. The age of the patients 1. Learing biofeedback measurements? What “factors” could have an impact on the BP Why were the data collected this way? STA 2023 c D.Wackerly - Lecture 22 § ¢ ...
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This note was uploaded on 07/28/2011 for the course STA 2023 taught by Professor Ripol during the Fall '08 term at University of Florida.

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