Week14-4up - 264 STA 2023 c B.Presnell & D.Wackerly...

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Unformatted text preview: 264 STA 2023 c B.Presnell & D.Wackerly - Lecture 22 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 Thought: Why is “abbreviation” such a long word? dist. :  ¨¦£ ¤   ©§ ¥ ¢ ¢ Hypothesis Tests ¡ Confidence Interval for Small Sample (p. 292): RR p-value ©§ 1 § 5 64 ¡@ 32¡ § © A7 @ § 5 64 ¡ C$ ED¡  I© § 1 §  § © A7 @ § OR For Thursday : Exer. 9.7, 9.15, 9.16, 9.18, 9.20 For Monday : P. 389 – 396 OR 266 smaller score score (tail area) STA 2023 c B.Presnell & D.Wackerly - Lecture 22 or larger G HF Wednesday : P. 378 – 383 (rest of Sec. 9.1) 7 B§ ¡1 32¡ ") 0! For Tuesday: Exer. 9.1, 9.13, 9.19, 9.22, 9.24, 9.25 STA 2023 c B.Presnell & D.Wackerly - Lecture 22 267 ¢ Test statistic : (like before) AND #d.f. p-value Can’t be any more precise using these tables! „ bp X WWibYU „ ¢ ¢ ¢ s X $ 9bp X wvUrgIWWubYU 7 @ § 64 ¢ 5 t X $ 9pq XU 5 $ 9pq X eisrgdeca`Yh1 § 64 gIWca`YU 7 @ § 64 5 From the table, „ „ From the table, „ and „ ) to bX $ Wusr…T tX $ eusr…T pX $ Wusr…T †X $ Wusr…T sX $ ƒvUr…T tX $ wvUr…T ¢ pq X Wcf`YU FX dbca`YU WV$ UU 9eeca`XWUh1 § 564 geecf`YU 7 @ § !4 Fb $ 9Fb X 5 9Fb X eeca`WU 7 @ § !4 $ 5 are Thus, in this case, best we can say is that @t HeusX Closest values in table ( with d.f. -1.732 sX ƒvU‚@ Fb X dca`YU 7 $ § Look at table, , ¢ UU WV$ p-value 9dpWcf`YU qX € x§ y' yA7 7 5 § © S¢ In Everglades example, lower tail test, d.f. 0 9IbWWubYU pX ' uBA7 7 5 x§ ) depends on T  I© § P Q '7 ( ¡  !£ ¤ $ § (new) (looks just like !!) R (and 9 versus Today : P. 374 – 378 (Sec. 9.1, “Large Sample”) 7 8§ Assignments '¡ $ " (&%¡ #! IMMEDIATELY!!! (Exam Conflict) 9 If you are taking BUL 4310 – contact me Ho . Ho . Ho . Ho . Ex. : New case, lower tail test, d.f. ¢ Closest values to is are t$ ¢ bX Wu†F 7 $ § ƒU $ s 271 skills tests. Left-handed (1) ¢ ¢ $ 9qp gIWƒvUX b 7 @ § $ pX g9 ca`F 7 @ § 64 sU 5 ƒV$ 64 ¢ 5 Find a 41 97.5 17.5 Right-handed (2) and s eq bX Wu†F 9b X dWu†F 7 @ § !4 5 then competence in preschool children. Scores on motor sU ƒV$ 9b X dWi†F 1 § 64 5 . Ex. Investigation between “handedness” and motor  ¢ If d.f. STA 2023 c B.Presnell & D.Wackerly - Lecture 22 , that are in table for d.f. and are . 41 98.1 19.2 ¡ ¢ ¢ Closest values to So and £¤ 9X Is YU 1 § 64 ¢ 5 $9p X gdY` YU 1 § 64 ¢ 5 sWsƒvUXr$ 9dpY` Yh1 § !4 XU 5 t$ tail area 270 then  ¢ p-value = . $ 9ps X gIWWubF 7 @ § $ 9sp X gdWeu†YU 7 @ § !4 †5 E$ 64 ¢ 5 s YU X ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 22 Ex. : New case, two-tailed test, d.f. that are in table for d.f. and If d.f. if d.f. So p-value sX ƒvUF that is in table for d.f. p-value = , confidence interval for the true difference in mean motor skills scores for left and right handed preschoolers. ¢ @ @ p-value . . Objective is to compare two means ¢ @ @ And tail area Samples from two populations ¢ ¢ So † E$ 5 !4 9 ¢ Closest value to 269 , ¢ $ p-value = STA 2023 c B.Presnell & D.Wackerly - Lecture 22 sX ƒvUF 7 $ § † $ d.f. s YV$ § E$  XU p Ex. : New case, upper tail test, 268 STA 2023 c B.Presnell & D.Wackerly - Lecture 22 HOW???? ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 22 Large Sample Inferences about Differences 273  £¤ 7  £¤  £¤ 7  £¤ 272 STA 2023 c B.Presnell & D.Wackerly - Lecture 22 ¢ If both populations normal, then  ¢ Independent Samples (Section 9.1) If and  Between Two Population Means, “large”, then normal. approx normal, regardless of shapes of sampled populations. Independent samples from two pop’s. (P. 375) 3 1 5 62 1 3  5 © R ¥ 5 42 1 estimator formula sheet [formula sheet, P. 375] STA 2023 c B.Presnell & D.Wackerly - Lecture 22 ¡ ¢ skills tests. 97.5 £¤ 17.5 41 98.1 19.2  ¡ ¢ What does it mean if ¡ ¢ confidence interval for the true difference in is than by ? equals ? equals by ? How can we interpret the 90% confidence interval, 9` p7 ` X WdusX 8eiF` 7 5 preschoolers. , for the difference in mean motor skills € R  $ y' x E$ I© R scores for left and right handed preschoolers? ¡ ¢ (the true mean dexterity score for left-handers) $ 2T         R 9 £ £   I© ¥  ¤ 7  ¤ 5 could be as much as larger than (the true mean dexterity score for right-handers) . (the true mean dexterity score for right-handers) ¡ ¢ ¥ could be as much as larger than true mean dexterity score for left-handers). or anything in between. ¡ s eq ¡ ¢ mean motor skills scores for left and right handed is equals ¡  41 90% CI is than What does it mean if Right-handed (2) ¢ Find a is 275 `X diFI` 7 What does it mean if competence in preschool children. Scores on motor Left-handed (1) ’s s Ex. Investigation between “handedness” and motor ’s, use them, otherwise use ¡ ¡ 7¡ ¡ ¡ 7¡ ¡ ¡ 7¡      $ ©¡ £ §¥¡ £ ¨ ¦¤    ¡ ¨ ¦ ¤ ¡£  ¡ 7  &$ ©¡ £ §¥¢¡ 274 formula sheet table ` IusX p ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 22 Know [formula sheet, P. 375] standard errors         © R ¥ 9  £ ¤ 7  £ ¤ 5      £¤ 7  £¤ ¡ 7¡ (p. 375) large. 3 62 – Use the difference in sample means, and X 95 .    £¤  Want: Estimate for    £¤  Sample variance Both  Sample mean (p. 376)   ¡  ¡ Sample size $"     %#! Pop. variance Large Sample CI for ' (& Pop. mean  Pop. 2 ) 0& ¢ Pop. 1 (the $ dupX p ¥ ` 276 STA 2023 c B.Presnell & D.Wackerly - Lecture 22 STA 2023 c B.Presnell & D.Wackerly - Lecture 22 277 Consider testing (p. 376) G HF  R1 I© 36R @  © R 7 ¦R ' @¡ 7¡ C$  ¡ 7  ¡ (tail area) hypothesized value standard error ¡ „ W„ NULL HYPOTHESIS         6 7  ¤ 7  ¤ £ ¡ $R ¢ „ Formula Sheet ¢      $R ¦ 7  ¤ 7  ¤ preschoolers by 7 points? 7 estimator ¡ ¢¡ ¢ handed preschoolers is larger that the mean for RH 278 or score TEST STATISTIC Is it plausible that the mean dexterity score for left STA 2023 c B.Presnell & D.Wackerly - Lecture 22 ' RH preschoolers by 5 points? @ © R 7 ¦R ¢ handed preschoolers is smaller that the mean for 5 !4 OR Is it plausible that the mean dexterity score for left smaller score 7R OR larger STA 2023 c B.Presnell & D.Wackerly - Lecture 22 Ex. Investigation between “handedness” and motor 279 competence in preschool children. Scores on motor skills tests. £¤ 17.5 Right-handed (2) 41 98.1 19.2  ¢ Conclusion : motor skills scores for left and right handed Significant difference between mean motor skills scores sX $ ƒvUV2T pre-schoolers at the p-value = „ $ FT 7 $ 2T  ¡ 7  ¡ 0! ")  ¡ 7  ¡ #! " Test Statistic level! ? Cannot claim a difference in mean motor skills scores for left and right handed pre-schoolers for any value of `t eeusX $ $' Rejection Region : level? ¢ for left and right handed preschoolers claim a difference in mean T 97.5 tX $ eusr…T „ 41 sX $ ƒU r2T  Left-handed (1) that is less than !! 9 ' preschoolers by 5 points? p-value R1 © 3¦R s Wq ¡ ¢ handed preschoolers is larger that the mean for RH RR 5 64 ¢ Is it plausible that the mean dexterity score for left 1¡ 7¡ ") 0! confidence level. 7R versus 9 $  ¡ 7  ¡ " ! rejected at the be left and right handers. This claim a fixed particular value of difference ' Claim : no difference in mean dexterity scores for  ¤ $R ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 23 STA 2023 c B.Presnell & D.Wackerly - Lecture 23 280 281 Last Time: Large Sample Inferences about Differences Between Two Population Means : Independent Samples (Section Thought: Diplomacy is the art of saying “Nice doggie” – 9.1) 3 1 5 42 1 3  5 I© R ¥ 5 42 1 estimator IMMEDIATELY!!! (Exam Conflict) formula sheet For Monday : P. 389 – 396 Know ’s STA 2023 c B.Presnell & D.Wackerly - Lecture 23 282 Hypothesis testing (p. 376) ' $  ¡ w ¡ "  7 versus ’s, use them, otherwise use STA 2023 c B.Presnell & D.Wackerly - Lecture 23 formula sheet         I© R ¥ 9  £ ¤ 7  £ ¤ 5    For Thursday : Exer. 9.7, 9.15, 9.16, 9.18, 9.20 table standard errors 3 42 If you are taking BUL 4310 – contact me X 95 9 T 7 U 5 WƒU ¢ ss (p. 376) Today : P. 378 – 383 (rest of Sec. 9.1) Large Sample CI for ¡ Assignments : ¡ 7¡ until you can find a rock. 283 Ex. : #9.23, p. 388 Manufacturing plant discharges a fixed particular value of difference “purified” liquid waste into a river. EPA inspector collected water specimens at the point of discharge and 7R 5 times, average bacteria count for each specimen score reported. Six specimens at each location At Discharge(1) G HF '  R1 I© 36R @  © R 7 ¦R ' ¢ @¡ 7¡ ' C$  ¡ 7  ¡ ¡ ¢¡ 33.4 29.7 30.3 26.4 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? ¡ " ! ) „ W„ NULL HYPOTHESIS ¢  7        7¤ 7¤ £ ¡ $R Formula Sheet 36.2 28.2 hypothesized value standard error (2) 30.1 (tail area) TEST STATISTIC estimator Upstream HOW??? " #! @ © R 7 ¦R 5 !4 or smaller score 9 R1 © 3¦R 5 64 OR larger 7R OR also upstream from the plant. Each specimen analysed p-value 9 1¡ 7¡ ") RR       $R 7¤ 7¤ „ ¢ £¡ ¤¢ Assumptions: (p. 346) 1. Both pop.’s have the Same variance, ¢ Could use ¢ , standard error of d.f. d.f. Will combine or “pool” these values, using BOTH ¢ The “pooled” estimator will have (p. 379) 95 U 9 U 7  5  7   5 F 7       3 62 1  9 U 7   5 $ © 9   QU 7   5  $g9 U 7   5  9 U 7   5 $  § ¨  U   U  ¦ ¥ $ STA 2023 c B.Presnell & D.Wackerly - Lecture 23 286   is a “weighted average” of the individual STA 2023 c B.Presnell & D.Wackerly - Lecture 23 287 ’s  ¢ MORE weight to the estimator based on the LARGER samp size. $ ¥#   "  Small sample (P. 379)   §¨  § U   U  ¥ © I© ¨¥  ¤ 7  ¤ § 9 U 7  5  9 U 7  5 §  ¨   U 9  ¡ 7  ¡5 gives the d.f. for this statistic. Large sample with ¢  5 42 1 3 ibX E$ b aFX $   E$   E$   p b   deg of freedom ¢ has a dist with  §  U   U  ¦ © R ¥  ¤ 7  ¤ ¥ $$      $ ©  pU ƒV$   ¢ ©   U  © ¤ $ 7 9  £¤ 7  £¤ 5 § Note : divisor in 3 and 4.2, closer to ' (&  Note: 3.3 is # d.f. C.I. for  © Ex. : ’s )& Always between the two d.f. # d.f.  ¡ is ¢ Since assuming , respectively.     and ? Could use Both populations Normally distributed with (unknown) means used in Ch. 7-8 9 U 7  5 9 U 7  5 variance, Samples are Independent. 3. -score : New Problem : how do I estimate this common   $ ©¡ £ ¦ ¥¡ £ ¨¤      £¤ 7  £¤ $  $  ¡ 9 £5 9 ¡5 $$    9  5 (unknown). is a R '& Useful when one or both sample size(s) less than 30. 2. 285 Since (1) and (2) are valid  U  ¦ ¥ $R 7 9  £¤ 7  £¤ 5  (Section 9.1, last part) ) 0& Small sample inferences about STA 2023 c B.Presnell & D.Wackerly - Lecture 23 § ¨  U 9  ¡ 7  ¡5 284 STA 2023 c B.Presnell & D.Wackerly - Lecture 23 STA 2023 c B.Presnell & D.Wackerly - Lecture 23 STA 2023 c B.Presnell & D.Wackerly - Lecture 23 288 Ex. : #9.23, p. 388 Manufacturing plant discharges Hypothesis Testing, P. 380 “purified” liquid waste into a river. EPA inspector $  ¡ w ¡ "  7 a fixed particular value of difference collected water specimens at the point of discharge and also upstream from the plant. Each specimen analysed versus 5 times, average bacteria count for each specimen p-value score Upstream At Discharge(1) (2) 30.1 (tail area) 36.2 33.4 29.7 30.3 26.4 28.2 ¢ 7 8§ smaller score 9 G HF ©§ 1 § 5 !4 § © A7 @ § 5 !4 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 location? pE$   p E$   STA 2023 c B.Presnell & D.Wackerly - Lecture 23 9 eF 9U 7   7   5 ¡ 7 ¡" ¢ Rejection Region : $ …T Rejection Region : ¢ Conclusion : At the tX $ eusr…T $§ F 7  $  $ ©    5    9 U 7   5 ¡ 7 ¡" )  ¢ $ level, there is evidence to conclude that the mean bacteria count is greater at the discharge point than it is upstream. ¢ 3 $ 5 eit62 c` 1 $ ` X† Test Statistic d.f. $ ¢ d.f. 291 $ B' x § € 290 FeupX Y‚$  ¤ qF s XF ƒvUeb $  ¤ C$  ¡ 7  ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 23 discharge location exceeds that for the upstream p itt $   X †p sU Wƒ XU ƒV$      © § 1 §  § I© A7 @ § §¨    ©  U  U¥ 6 7  ¤ 7  ¤ @¡ 7¡ Test Statistic : or 7 B§ OR larger reported. Six specimens at each location 9 ¦ 1  ¡ 7  ¡ ") RR OR d.f. 289 P-value  ¨     ©  U U  ¤$§ ¦ 7 9  £ ¤ 7  £ ¤ 5 $ †X $ wU upYU $ ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 23 292 95% CI :    §¨ U   U  ¥ © I© § ¥  ¤ 7  ¤ ¥ ts dWupX b ¥ 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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