Week15 - STA 2023 c B.Presnell & D.Wackerly - Lecture...

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Unformatted text preview: STA 2023 c B.Presnell & D.Wackerly - Lecture 24 293 Assignments Today : P. 389 – 396 For Tuesday : Exer. 9.29, 9.33, 9.35, 9.38, 9.39, 3.42 Wednesday : p. 402 – 406 For Thursday : Exer. 9.46, 9.52–54, 9.56, 9.59, 9.60, and rest of Chpt 9 problems on syllabus! Assumptions: (p. 379) 1. : Same variances,  © ¥ ¦ © : Independent samples.  © § 3. § ¨¦ ¥ 2. : Normal distributions. ¤¡ £¢¡ Last Time : Small sample inferences about STA 2023 c B.Presnell & D.Wackerly - Lecture 24 294 Pooled estimator for common variance  ¤ ¤  ¥ ¥ £ ¤ ¤© £ £ ¤ ¤© £ ¦  © ¦ ¨ ¥ ¢ ¤  ¥ £ ¢ ¤© § £ £ ¢ ¤© £ ¢ ¤¡ # d.f. $ " ¥ ¤ # ¥ £ ¦ ¢ ¤¡ " £  !   ¤ 0( 1)' % ¤ & (table value)(standard error) & £¢     % estimator  C.I. for Test Statistic 3 ¢ 7 4 % £¤ 2 ¢ 5 64 ¦ £ % £¢ ¤¡ ¢ ' ¢ test stat estimator hyp. value (standard error) STA 2023 c B.Presnell & D.Wackerly - Lecture 24 295 Ex. : #9.26, p. 389 Sea urchins were starved for 48 hours, then fed a 5 cm blade of turtle grass. 10 urchins were randomly selected and fed fresh grass, another independently selected 10 were fed decaying grass. The measurements were the time (in hours) necessary to ingest the grass blades. Green Blades Decayed Blades (1) (2) Number of urchins 10 10 Mean Ingestion Time 3.35 2.36 Standard Deviation 0.79 0.47 ¤¢ ¥£¡ Construct a confidence interval for the difference in mean ingestion times for urchins fed green and decaying grass. STA 2023 c B.Presnell & D.Wackerly - Lecture 24 296 90% CI : ¥ ¤ ¤   £¤ £ ¤ ¤© £ ¦ ¢ ¤© £ ¢ £ ¤  ¥ £ ¢ ¤© ¦ £ ¤¡ ¢ ¢ ¦ ¤¢ §¢ ¥£¡ ¢ ¢  © ¨ § d.f.  ' ¥ ¤ ¥ £ ¦ ¢ £ ¤¡ ¤ 0( 1)' ¤ & % £¢ ¨ ©¥ , ¢ ¢ d.f. % & or ¦ §¢ ¦ ¢ & (A bunch of “scholarly” words about sea urchins, green grass and decayed grass) Is the average time it takes sea urchins to consume green grass larger than the average time to consume decayed grass? STA 2023 c B.Presnell & D.Wackerly - Lecture 24 ¤¢¡ , at the 297 confidence level. All “plausible values” for the difference in means (green vs. decayed) are . In terms of this example, what are the assumptions necessary for the above test and CI to be valid? § ¨¦ ¥ – ¥ ¥ ame for are approximately the grass. : the sea urchins in the study were ¦ – : the POPULATION variances of assigned to green and decayed grass. § – : the a 5 cm blade of sea grass is approximately for both green and decayed grass. distributed STA 2023 c B.Presnell & D.Wackerly - Lecture 24 298 Ex. : #9.101, P. 430 Study to assess the effect of biofeedback exercises on blood pressure. Five 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 ¥ © © Before After ¡ ¢ ¤ ¢¥ ¡ ¢ £¢¥ ¢¡ £ ¤¢ ¢¨ ¡ ¢ £©¥ ¨¢ ¤ ¤ ¥ ¤¢ ¡ £ ¥ ¢ ¥¦ ¡ ¥ ¤¢ ¤ ¥ ¢¢¤ ¥ Subject ¥ ¢ ¥ ¡ ¥ ¢¢ ¢ ¦ £ ¥ ¢ ¡ ¡ ¡¢ ¡ ¤ ¦ ¥ ¥ ¤ STA 2023 c B.Presnell & D.Wackerly - Lecture 24 299 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 B.Presnell & D.Wackerly - Lecture 24 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?? 300 STA 2023 c B.Presnell & D.Wackerly - Lecture 24 301 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 B.Presnell & D.Wackerly - Lecture 24 Popn. 2 £ ¤ ¢ ¤ £ ¡ £ £ depen. . . . 4 ¡ . . . ¤ £ £ ¢ . . . ¢ £ Indep. £ depen. £ depen. Difference ¤ Popn. 1 302 ¢ ¡ ¤¡ ¢¡ differences ¢ Pairs ¢ mean of the population of DIFFERENCES ¢ ¥ ¡ § 3 ¦ 3 ¤¡ £¢¡ ¢ 2 ¡ ¢ ¥ ¢ ¡ § ¥ 3 ¦ NORMALLY distributed ¨¨¨ ©¨  is NOT required 2 ¤¡ £¢¡ £ ¥ ASSUMPTION : the DIFFERENCES are approx. 3 same as “2”. £ if differences are taken “1” . ¤ ¤  ¢ ¤ ¢ ©  ¨¨¨ ©¨ STA 2023 c B.Presnell & D.Wackerly - Lecture 24 303 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. Five 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 . ¥ © © Subject Before After ¥ ¡¢ ¤ ¡ ¢ ¢ ¢¥ ¡ ¡ £¨ ¢ £ £ © ¤ ¨ ¡ ¢ ¢ ©¥ ¡¢ ¢ ¡ £¢ ¨ £ ¢ ¡ £ ¨¢ ¦ ¡ Diff. ¤ ¡¢ ¤ ¥¢ ¡ £ £ ¤¢ ¥ ¥ ¥ ¤¢ ¢ ¢ ¡ £¤ ¥¥¢ £ ¨ ¢ ¥ ¤ ¤ ¤¢ ¤ ¡ ¡ £ ¥ ¢¢¤ ¥ ¥ ¥ ¥ ¥ ¡¢ ¡ ¤ ¥ ¢ ¡ ¡ ¥ ¤ ¥¢ ¤ ¥ ¢ ¥ ¢¢ ¢ ¦ ¢ ¡ ¦ ¥ ¥ ¨¢ ¤ ¤ ¥ Totals £ ¢ ¢ Diff. ¡ £ ¦ ¤ STA 2023 c B.Presnell & D.Wackerly - Lecture 24 304 % ¢ ¤ ¤ © £ £ £¢ ¢ ¡ £ £ ¥ ¤ ¥ ¥ ¤ ¥ £ ¢ £ ¥ ¢ ¥ ¤ ¢ ¢ ¥ d.f. ¢ Mean blood pressure larger before than after? ¢ ¢ § ¤¡ £¢¡ § ¡ ¥ ¢ ¦ 3 3 ¦ ¡ § ¤¡ £¢¡ § ¡ ¦ ¥ 3 ¦ £ 2 ¥ % ¥  '  ' £ ¥ ¢ ¦ ¦ ¢¢ ¢ ¢ ¦ §¢ ¢ ¢ Conclusion : At the ¢ ¢ ¢ ¢ d.f. level there that the mean blood pressure reading is higher before learning biofeedback. STA 2023 c B.Presnell & D.Wackerly - Lecture 24 305 P-value? 0  ¤ '  (2.571) £ $ " # (table value)(standard error) ¥ ¢ £ 0 6( ' & ¤  ¤ '  ¢ & or ¥ ¤ & " d.f. " ¡ ¢ estimator (3.365) ¤  C.I. for ¢ '  2.977 ¨ ¡¢ ¨ & ¢ ¢ ¥ STA 2023 c B.Presnell & D.Wackerly - Lecture 24 306 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 B.Presnell & D.Wackerly - Lecture 24 307 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!!! STA 2023 c B.Presnell & D.Wackerly - Lecture 25 Assignments Today : p. 402 – 406 Tomorrow : Exer. 9.46, 9.52–54, 9.56, 9.59, 9.60, and rest of Chpt 9 problems on syllabus! Last Time : Paired-Difference Experiments Assumptions: Differences appr. Normally dist. Method of Analysis : do a one-sample “t” ON THE DIFFERENCES Test Statistic: 3 2 ¢ ¤ £ £ ¥ ¥ ¢ ' Confidence Interval: (table value)(standard error) ¥ ¥ £ ¤ 0 6( ' & ¤ & estimator 308 STA 2023 c B.Presnell & D.Wackerly - Lecture 25 309 Ex. # 9.54, p. 407 Managerial careers of men and ¦¡¡ from Fortune 500 corporations. ¤¦£ managers married, female managers ¤¤¨ male managers, ¡ ¥ women. of the male of females managers married. Find a 95% CI for difference in proportions of male and female managers who are married. (Male = “1”, Female = “2”). ¢ ¢ ¢ ¢ £ ¤ £ % ¢ ¡ ¢ ¢ ¤ ¤ ¢ ¢ % £ £ ¡ ¢ ¤ STA 2023 c B.Presnell & D.Wackerly - Lecture 25 310 Comparing Two Population Proportions Independent Samples (p.402) Have: Two populations ¢  ¡ ¤ ¢  © attribute ¢ © Pop 2 attribute ¡ Pop 1 Independent samples: # with attribute ¢ # with attribute ¡ ¢ ¡ ¤ ¡ £¢ ¤ (p. 402) ¤ £ ¤ ¢ ¢ £ ¤ ¡ £ ¤ ¢ ¡ ¡ £¢ ¡ ¡ £¢ ¡ ¤£ ¤ ¥¤ £ £ % ¡ ¤ ¢ ¡ ¢ estimates (p. 402) ¢ ¡ ¢5 ¡ 7¡ ¦ ¡ ¢£ ¢ ¤¢ £ is approx. normally dist’d when both ¢ 7¡  ¡ ¤ ¡5¡ ¡ £¢ are large. (p. 402) £ % ¢ estimates ¤ % ¤ estimates ¡ % ¢ from pop 2, ¢ from pop 1, ’s STA 2023 c B.Presnell & D.Wackerly - Lecture 25    CI for (P. 403) standard errors ¢  © ¨ formula sheet ¡  ©§ ¤ $ # & © ¨ § 0 6( ¢  © ¨ § ¡ ¤£ ¤£¤  ¡ table   formula sheet estimator  !  Large Sample 311 ¡ ¦ ¢£ ¢£¢ ¡ ¤ 0 6( ¡ ¤ & ¢ £¢© Ex. # 9.54, p. 407 Managerial careers of men and from Fortune 500 corporations. ¤¦£ managers married, female managers ¤¤¨ ¦¡¡ male managers, ¡ ¥ women. of the male of females managers married. Find a 95% CI for difference in proportions of male and female managers who are married. (Male = “1”, Female = “2”). ¢ ¦ §¡ ¡ ¢ ¡ ¢ ¤ £ £ ¢ ¤ ¨¢ ¢ ¦ £¢ % ¢ ¢ ¡ ¤ ¤ ¢ ¢ % £ £ ¡ ¢ ¤ STA 2023 c B.Presnell & D.Wackerly - Lecture 25 312 ¢  ¤  ¢ ¢ ¤ 0 1( ¤ §¡ ¦ Confidence interval: ¦ §¢ ¢ ¦ §¢ ¢ ¢ ¡ ¤£ ¤£¤ ¢ ¢ ¦ ¡ ¢£ ¢£¢ ¡ ¤ 0 6( & ¢ ¤ ¡ £¢© ¢ & ¦ ¢ ¢ ¡ ¢¢ ¢ & ¤ §¡ ¦ At the confidence level , the proportion of male managers who are married exceeds the proportion of female managers who are married by between and STA 2023 c B.Presnell & D.Wackerly - Lecture 25 313 Consider testing (p. 404) a fixed particular value of difference ¡  ¦ ¡ £¢ ¤ ¢ 2 § 3 versus p-value score £ ¢ larger  RR © § ¦ ¡ ¡ ¡ ( ¢  ¡ ¤ £¢ 2 ¢ £ ¢ smaller score © ( £ ¢  2 ¢ ¡ ¡ ¤ £¢ OR ¢ or (tail area) ¤ 0 1( £ ¢ ¡ ¢  ¤ ¡¢ 2 ¡ £¢ ¡ ¤ 0 1( ¢ ¢ TEST STATISTIC hypothesized value £ £ standard error £ ¤£ £ estimator  OR ¢ ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 25 ¡ ¤ £ ¢ ¤ ¢ ¡ ¤ ¦ £ ¢ ¢ ¡ £ ¢ ¢ ¡ £ ¢ ¢ ¡ ¡ ¡ ¡ ¤ ¢ ¢ , estimate this with ¡ ¢ total # “S” in experiment ¢ total sample size % ¡ ¤ ¤ ¥ £ ¥ ¦ ¢ ¢ £ ¦ £ £ ¢ ¦ ¢ % ¢ ¢ £ ¢ ¢ ¢ ¡ ¢ ¢ 2  2 ¡ ¡ on formula sheet. ¤  ¡¢ standard error ¡ ¢ ¢ , then  ¥¦ © ¤§ £¢ , use the individual ’s COMMON value of £ £¤ 2 If  ¢ ¡ If NULL HYPOTHESIS ¢ Formula Sheet 314 ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 25 315 Ex. : # 9.107, p. 431 Does inositol (found in breast milk) reduce the risk of eye damage in premature infants? ¢¢ . ¢ ¥ ¥ ¢ £ of ¥ Use premature infants given inositol had an ¥ eye injury to to high oxygen levels used to compensate for poorly developed lungs ¢ ¥ ¥ ¡ of ¢ true prop. of premi’s given inositol with eye on standard diet had eye injuries. ¡ ¢ injuries true prop. of premi’s not given inositol with ¡ ¤ ¢ eye injuries ¤ (2) ¡ ¦ ¡ £¢ ¡ § ¦ ¡ £¢ § 3 ¢ RR : ¢ ? ¦ §¡ ¥ ¢ ¢ ¢ ¢ ¢ ¤ ¢ % ¤ ¤ £ % ¢ ¦ ¦  ¢ ¢ £ 2 ¢ ¢ ¡ ¡ ¡ ¢ ¢ Is ¤ tail test, (1) ¢ ¢ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 25 ¥  ¤ £ £¤ 2 ¥ ¦ ¢ ¢ £¢ ¢ £ ¢ ¡ ¡ ¢ ¢ £ Test Statistic Decision : at the 316 ¢ ¢ ¡ ¢ £ ¡ ¦ ¢¢ ¢ level, there ¢ ¢ ¢ evidence to claim a difference in the proportion of premature infants with eye injury due to high oxygen levels for infants given inositol and those not given inositol. P-value : Lower tail test – ¢ p-value STA 2023 c B.Presnell & D.Wackerly - Lecture 25 ¦¢ §£¢ ¢ -value larger than ¥ For any 317 , claim there is a difference in the proportions with breathing irregs. ¦ §¢ ¢ ¢ ¥ ¢¢ ¢ – Claim ¦ ¢ ¢¢ ¢ – ? ? – Claim ¦ ¢ ¢¢ ? ¥ No significant difference for any ...
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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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