Week11-4up - 1 Thought: Never do card tricks for the people...

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Unformatted text preview: 1 Thought: Never do card tricks for the people you play STA 2023 c D.Wackerly - Review for Exam II STA 2023 c D.Wackerly - Review for Exam II 2 Chapter 4 : Discrete Random Variables poker with. ¡ Can COUNT the number of distinct values of the ¡ The Binomial Probability Distribution variable. Thought: To succeed in politics, it is often necessary to rise above your principles. – Some discrete random variables are binomial – NOT ALL 9 £5YYY565 3 ¤ [email protected]@8704¨V V U § XWET¦ 9%£ ¤ P S1GFRQI 9£ ¤ HGFED [email protected]@8670342 10(&$ "£  5955 ¤ #)'% # ! ¤  § ¤©¨¦ § ¤ ¥£ ¢ – Binomial Experiment p. 179 – Monday 11/4/02 : OPTIONAL REVIEW, ask questions number of trials, – If about homework, course material, sample exam, etc. Suggestion - print out Sample Exam 2 and criteria : (p. 183) for bring it to class with you. – Mean (p. 185) : Help for Exam 2 ¡ Monday, periods 3–8, FLO 104 – Variance (p. 185) : Monday, 6:15 pm – 8:10 pm, McCC 100 – Tables : Contain ¡ Tuesday 11/5/02 : EXAM 2–during your regularly scheduled for DISCUSSION SECTION ..... 0 1 2 3 ..... k k+1 n-1 n Wednesday 11/6/02 : Minitab, Computer Demo, P. 288 – 393 (Sec. 7.2), P. 341 – 345 (Sec. 8.4) STA 2023 c D.Wackerly - Review for Exam II 4 ¡ Areas under normal curves between z-scores of ¡ Key to finding correct areas (probabilities) : draw r ¡ Possible values are all those associated with one or ¡ Probabilities are areas under “density function”. 3s e¨r and , for Chapter 5 : Continuous Random Variables in Table IV, p. 809. 3 3 STA 2023 c D.Wackerly - Review for Exam II pictures more line intervals. Chapter 6: If we plan to take a random sample of size is called its sampling distribution.  @fFhEihqaTg@WEiqa¨p¤ § ¦ ¤ f h  U § ¦ f U  h § ¦ ¤ f U  U § AFEicba¨g@Fedcba¨¦ (p. 266), so dist. of for larger sample sizes. t D concentrated around is more is called the standard error of .(p. 266) t . (p. 266, 261) is an unbiased estimator v# I £ €Q4¤ I ¡ y x I v# D D ¤  § w v# 4t  q4¤ D ¡ of . So t ¡ (p. 255) b is a continuous r.v., If the population has a normal dist., then so does ,  y x I5  £ €Q„ƒD§ ‚t  i.e., N . True for any £ ¡ ¡ Normal distribution is special case. is a random variable. t t u¡ If , Dist. of P(a<X<b) a I deviation and standard t £ from a population with mean D graph of f(x) . ¡ £ is large £ €Q„ƒD§ ¤t  y x I 5 £ t 3 ¢ £ ¡ ), then the sampling distribution of ¡ approximately normal, i.e., STA 2023 c D.Wackerly - Review for Exam II 5 Central Limit Theorem (CLT): (p. 267) If ( STA 2023 c D.Wackerly - Review for Exam II 6 Estimated is N , Standard regardless of the shape of the population Parameter Error of Est. Estimator Standard Error of Est. distribution.  £y £ % 1 § £ ¡ ¡ ¡ , etc. (p. 254) £  ¤ ¦  ¡ Both estimators are UNBIASED £¡ P QI D a Sample. ¨¦%¦  § D All statistics have sampling distributions , £y A Statistic is a meaningful number associated with with a Population. I A Parameter is a meaningful number associated ¥ Chapter 6: If is “large”, both estimators are approximately NORMALLY distributed. Chapter 7: ¡ How large is “large”? © ¢ £ ¡ Want : Correct to within “ ” units with and solve for (p. 307) if you have one. Maybe 7 – Use ballpark value for Range use . I 6 0I Finding the sample size to estimate . 4 ¡ Want : Correct to within “ ” units with confidence. and solve for  – Use “ballpark” value for (p. 333) if you have one, if not to get sample size that will work  9¢ ¤ 4¤  % use and SOLVE £ 45¤ % £ P ' % 1 § (&r ¡ §  P (&r ¡ '%  3   (3 A6 ƒ¨C6§ standard error 4¤ 5 $ ! §  P# (&r  # "  ' "%! ! " r P )% r ' 3 ¥ D P )% R 'r $ 9 £ P' 2¦%¦  § )% r ¦   $ £ y P' $ 0 )% r t  1£ I y P )% r t  ' (P. 300) and SOLVE £ 9 # – Population Proportion, standard error α/2 (P. 283) confidence. 4¤ 5 ¡  3   3 A6 ƒ¨C6§ ¡ – Population mean, 45¤ £ I y P (% r ¡ ' §  P (&r ¡ '%  3   (3 A6 ƒ¨C6§ standard errors 1−α 8 4 ¡ α/2 smaller of Finding the sample size to estimate . PARAMETER formula sheet larger of STA 2023 c D.Wackerly - Review for Exam II 7 Confidence Interval for a table : 3 ¡ STA 2023 c D.Wackerly - Review for Exam II . !  H¦ § ¦% 5  H¦ § ¦% 5 ¡ Confidence Level (p. 282) formula sheet  – For valid CI for Confidence Coefficient (p. 282) estimator : – For valid CI for Interval Estimator (p. 282) 3 ¢ £ D ¡ Point Estimator (p. 261) for any value of . 9 STA 2023 c D.Wackerly - Review for Exam II STA 2023 c D.Wackerly - Review for Exam II 10 Chapter 8 – Large Sample Hyp. Testing Parts of a statistical test. (p. 322) ¡ The hypothesis of MAIN INTEREST is the Type II error accepting , light bulb ex. (p. 322) when true saying what we “want” to say when we should not  ¢ §¡ 6 ¤ D ©£ § ¦ ¤¨ Errors: p. 325 ¡ Ha true Type II error Correct OR rh % R ¨r  ! $¡ ! e¨br¨¦ #rh§ OR (tail area) P )% es r 'r P )% R ¨r  2 ' r h  %¤ ¦  is D D D  ih D x ' £ ySHI ' (' D ¨'  ¤ r ¡ P )% e¨r 'rs P (% R ¨r  W&D ' r h D %¤ Test Statistic ¡ ) Sheet Hypothesized Value from NULL HYPOTHESIS can be ) 0)   ) ) 0)  rejected. for which p-value or (tail area) Test Statistic Estimator and Standard Error from Formula value = smallest value for RR ' rs % e¨r  ! $¡ "e¨br¨¦ # !r s § ¤¡ or calculated value of . '(' '7 (' 95 865 4 ' ¦ 3  ¤ r p-value ¦ RR Large Sample Tests about h   E calculated value of . 12 rh % R ¨r ¤ ! $¡ "r ¤  WED ©£ ¡ D ¤ ¤¨ #! OR r OR D Chapter 8 : Large Sample Tests about STA 2023 c D.Wackerly - Review for Exam II  ! $¡ ! eh r ¨¦ #r § 11 rs % e¨r STA 2023 c D.Wackerly - Review for Exam II  ! $¡ "es r ¨¦ # !r § Type I error 1 E s ¤¡ ¥£ Reject Ho ¤ ! $¡ ! r ¤  ¤  ¤ ¨ ¡  # Correct when r Accept Ho saying  Ho true Decision In our lightbulb example, ¦ ¥g¤  Reality ¢ §¡ 6 s D ¦ ¨ ¢ HYPOTHESIS, – and/or ¦ ¤ ¢ §¡ 6 ED ¡ The “other” hypothesis is called the NULL (p. 325) ¡ ¢ ¨  5  manner  What we are “trying to prove” in an objective, fair  LEVEL of the test. . (p. 322)  ¥¦ ¤¡ ¦ ¥g¤  ¡ 5  ¨ ¡ §¦  Tg¤ ¡  ¢ §¡ 6 s D ¥£ § ¢ ¦ ¤¡ ¡ , (p. 323), SIGNIFICANCE  §¦ ¨g¤  ¡ light bulb ex. Type I error  ALTERNATIVE or RESEARCH hypothesis, – Estimator and Standard Error from Formula Sheet Hypothesized Value from NULL HYPOTHESIS 255 STA 2023 c D.Wackerly - Lecture 19 STA 2023 c D.Wackerly - Lecture 19 256 Summary: Large Sample Hypothesis Tests RR rh % R ¨r %¤ value P )% e¨r 'rs P (% R ¨r 'r h P. 288 – 393 (Sec. 7.2), P. 341 – 345 (Sec. 8.4) ¡ Test Statistic 8.50, 8.53, 8.54, 8.56, 8.57, 8.105 – 108, 8.111, 8.117 ) Sheet Hypothesized Value from NULL HYPOTHESIS ¡ ¢ 9 6 s 3 ¢ ¨9 ¦ 86 ¤ r 6 3 A6 3 33 9 03 9 ¢ 93 83 9¢ 9  8 9¢ ¦ 86 ¤ 93 ¤ 8¢ i 9¢ § ¤ r ¢ 04¤ 3 7¢3 6 ¤ ¦  3 A6 ¤ £ 93 3 ¡ ¡ NO!! - Cannot reject the level of significance” ( or with confidence ) to indicate that the majority of Diet Coke drinkers will select Diet Pepsi in a blind taste test.  ¡ the the Pepsi Challenge are a RANDOM SAMPLE LEVEL!! value? value = 9  9¢ S3 ¦ 86 ¨br¨¦ ¤ 9 s § ¢ 93 ¤  ¡ Assumptions : the 100 individuals participating in AT THE “ CANNOT claim that there is sufficient evidence at  ¡ level test, RR : in favor of In terms of this problem: ¢ 93 ¢ 986 ¤ ¡¢ £ 4¤ % e¨r r rs 6 38 9¢ ¤  ¤ ¨ 39 8¢ ps  ¤ ¢ ¡ (2) ¡ ¤ q ¡ (1) ¢ 93 ¤  true proportion of Diet Coke drinkers who ? 96¢ 8076 96 ¤ ¢ select Diet Pepsi in a blind taste test? © 6 ¤¡ £ indicate that a majority of the Diet Coke drinkers will ¢© the taste of Diet Pepsi. Is there sufficient evidence to is “large” Conclusion : is indicated that they preferred would select Diet Pepsi in a blind taste test. 258 Data : Coke drinkers were given unmarked cups of both Diet £ of all Diet Coke drinkers. Note: standard error STA 2023 c D.Wackerly - Lecture 19 Ex. : #8.68, p. 352 In a “Pepsi Challenge”, 100 Diet Coke and Diet Pepsi. hypothesized value 257 (tail area) Estimator and Standard Error from Formula ) 0) STA 2023 c D.Wackerly - Lecture 19 estimator ¤r For Thursday: Exer. 7.27, 7.30, 7.33, 7.80, 7.81, 8.49, or ' param. Today : Minitab, Computer Demo, ' OR ¦ h value  ! $¡ ! e¨br¨¦ #rh§ s rs % e¨r param. p-value ' (' ¤¡ ¤¨ ©£ ¡ value OR Assignments calc. value of .  ! $¡ "e¨br¨¦ # !r s § ¤ param. ¤ ! $¡ ! r ¤ # into jet engines. value r Thought: Eagles may soar, but weasels aren’t sucked param. ¡ 259 STA 2023 c D.Wackerly - Lecture 19 STA 2023 c D.Wackerly - Lecture 19 260 Computer Study:  ¨ ¨ 3¤ ¡ ¥£ £  9¢  9¢ ¢ i ¦  § ¤ r 9 3 A 96 ¤  ¡ ¢ 86  h¨r ¢ 86 Tr 9 9s 6 9¢ % ¤  ¤ 6 £ 9¢ ¤  ¤ ¨ ¡ ¡ Minitab? Basic Statistics 1 Proportion ¡ Stat ; RR : ¡ Click radio button “Summarized Data”, type in Number of trials, Number of Successes or Sample size for each test is ¡ Click Options, Select Alternative, Type in Null Value ¡ reject ¡ Click Box “Use test and interval based on normal not reject tests Prop. rejects .5 N 100 Sample p 0.560000 90% CI (0.462710, 0.657290) Z-Value 1.20 P-Value 0.115 STA 2023 c D.Wackerly - Lecture 19 261 ¨ ¢ ¨ £ not reject 50 .24 32 18 50 .64 .8 48 2 50 .96 47 3 50 .94 50 0 50 1.00 STA 2023 c D.Wackerly - Lecture 19 262 ¡ reject 38 3 ¥£ ¤ Sample size for each test is 12 X 56 .08 .1 Sample 1 50 .2 Test of p = 0.5 vs p > 0.5 46 .7 Test and Confidence Interval for One Proportion 4 .6 distribution”, OK, OK tests Prop. rejects  .5 6 44 50 .12 .6 21 29 50 .42 Ex. Phosphorus content is a water quality index that is .7 48 2 50 .96 of concern to the EPA in the Everglades. In one section .8 50 0 50 1.00 of park, EPA makes 9 g¤ ¥ ¢ ¤ ¡ 9 ¦ 6 2t  and ¤6 ¦ 6 ¥£ What do we see? measurements, obtaining (measurements in parts per billion [ppb]). Can the EPA support the claim that the , we REJECT ¡ 9¢ % ¤  3 €A 96 ¤  § 3 A6 is “better”. £ ¡ ¤ 9¢ 2 greater percentage of the time for larger . Big a £ . approx. ¨ £ of the time. we reject ¨ For each fixed value of , for each ¡ When . Sample size  ¦ C§ 6 ¨ — GOOD! a greater percentage of the time. ¢ 93 ¤  we REJECT mean level of phosphorus is less than ¤¡ ¡ ¡ away from .5, ( and the null becomes “less true”) ¤ £ ¨  moves ¢ 76 For each fixed sample size, as the value of is small! How??? ppb? Use 263 STA 2023 c D.Wackerly - Lecture 19 STA 2023 c D.Wackerly - Lecture 19 264 ¡ Properties of the -distribution: Symmetric about 0. (like the -distribution) 3 ¡ ¡ More variable (heavy-tailed) than the -distribution – Variability depends on degrees of freedom.  as d.f. . r  – Variability r y £ Sx D  t ¥ does not have a standard normal dist.  ¡ scores r 3¡ h £ can’t use r Bell-shaped . (like the -distribution) can’t use CLT to get NORMALITY of the sampling distribution of r Small Sample Inferences about – Becomes more and more like the -distribution However:  as d.f. . ¡ If the POPULATION is approximately NORMALLY distributed t with 2 df t with 8 df £ yx¥ D  t ¤ r (looks a lot like !!!) Std Normal ¡ has a sampling distribution called the “degrees of freedom”, -4 2 4 STA 2023 c D.Wackerly - Lecture 19 ¡  £ £ ¡ ¡ P £ ¡  ¢ £ ¡ ¡  8 £ ¡ £ ¤ d.f. 266  £ 265 0 ¡ ¢ ¡ STA 2023 c D.Wackerly - Lecture 19 -2  d. f. 6 e £ distribution with 1 ¡ £ €x y D  t ¥ ¤ 6 i £ P Ct   § ¤ P ¥ ¢ ¡ ) 4.303 6.965 9.925 1.638 2.353 3.182 4.541 5.841 1.533 2.132 2.776 3.747 4.604 1.476 2.015 2.571 3.365 4.032 1.440 1.943 2.447 3.143 3.707 7 1.415 1.895 2.365 2.998 3.499 8 1.397 1.860 2.306 2.896 3.355 1.383 1.833 2.262 2.821 3.250 10 1.372 1.812 2.228 2.764 3.169 11 ¡ ¤ rs§ 5 % e¨brT¦ ¡ ¡ % so that 1.363 1.796 2.201 2.718 3.106 12 1.356 1.782 2.179 2.681 3.055 13 1.350 1.771 2.160 2.650 3.012 14 1.345 1.761 2.145 2.624 2.977 15 1.341 1.753 2.131 2.602 2.947 -values for and 9 0¢ 3 3 93 ¡ ¤ 5 % s ¨¦ § ¡ %r % ¡ (Remember: I Table VI (p. 811) gives so that 2.920 9 ¡ Define 1.886 6 used in its calculation. 63.657 3 has the same number of d.f. at the estimator for 31.821 5 P¥ Thus 12.706 2 : 6.314 4 Note: d.f. = denominator in calculating 3.078 6X3 3 @50¢ 3 3 [email protected] 6G3 A5 ¢ ¦ 3 A5 ¢ 3 A0A 96 ¤  9 9 9 953 STA 2023 c D.Wackerly - Lecture 19 268  8 £ ¡ ¢ ¡ ¡  £ £ P £ ¡ ¡  ¢ £ ¡ ¡  8 £ ¡ d.f. 267 STA 2023 c D.Wackerly - Lecture 19 £ ¤ 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 25 1.316 1.708 2.060 2.485 2.787 26 1.315 1.706 2.056 2.479 2.779 27 1.314 1.703 2.052 2.473 2.771 28 1.313 1.701 2.048 2.467 2.763 29 1.311 1.699 2.045 2.462 2.756 1.282 1.645 1.960 2.326 2.576 df=30 df= Note : When d.f. 269 STA 2023 c D.Wackerly - Lecture 19 ¡ Hypothesis Tests (p. 342) versus 3 ¡¢ OR  ih D D ¢ D  is D ¢ ¢ ¢ ¡ ¢ ¢ ¢ OR ¢ ¢ ¡ ¢ D %¤  W&D ¡ ¢ ¢ ¢¤ RR or r (looks just like !!) ) depends on  £ €x ¥ y  D  t ¤ ¡ P (% ' ¡ ¡ %¡ ¡ (new) ¡ ¢ ¡ £ Test statistic : (and P ')% s P (%  h ' ¢ dist. £ £ y P' $ )% t  ¥ £ y P' $ I )% r t  ¡ r 6 e £ Small Sample (p. 292): ¢ Large Sample: ¢ Confidence Interval : ¢ d.f. instead of ¡ ¢ ¢ dist. with ¡ Assumption : POPULATION approx. NORMALLY dist. Small sample situation similar to large, except use %h ¢ Small Sample Inferences About 270 %s STA 2023 c D.Wackerly - Lecture 19 df=20 ¡ 2.492 ¨ 2.064 ¢ 1.711 ¢ 1.318 ¢ 24 df=10 ¢ 2.500 ¢ 2.069 ¢ 1.714 ¢ 1.319 ¢ 23 df=5 ¢ 2.508 ¢ 2.074 ¢ 1.717 ¡ 1.321 ¢§ 22 ¢ 2.518 ¢ 2.080 ¢ 1.721 ¡ 1.323 ¢ 21 t ¢ 2.528 ¢ 2.086 ¢ 1.725 ¢ 1.325 ¢ 20 ¢ ¦¥ 2.539  W¤ D ©£ D ¤¨ 2.093 1.729 ¤ 1.328 19 .025 ¡ 2.552 r % 4¤ % 5 2.101 1.734 ¡ 1.330 ¡ 18 ¡ 2.567 ¦ 93 6 93 ¦¦ 2.110 £ 1.740 ¡ 1.333 ¡ 17 ¤R¡ P  £ ¤R¡ P  £ ¤ ¡ P £ ¤ ¡ P £ ¤ R¡ P  £ 2.583 3 2.120 ¡ 1.746 P £ 1.337 ¡ 16 (like before) AND #d.f. ¡ STA 2023 c D.Wackerly - Lecture 19 Ex. Phosphorus content is a water quality index that is of concern to the EPA in the Everglades. In one section Ex. Give a 95% CI for the mean phosphorus index in the section of the Everglades ¡ Note: In last example (both test and CI), we are ¡ ¦9 §¡ 86  ¤ ¡ ¡ ¡ ¨ ¢ ¤ ¤  £D y €x  ¥ t ¤ is assuming that population from which the sample is taken is (approx) normally distributed in the rejection region, CANNOT reject Ho . There is evidence to conclude, at the level of significance, that the mean level of ¢ 76 phosphorus is less than ppb. ¢ 93 ¤  ¡ ¡ ¡ Conclusion: Since . ¦9 §¡ 86  ¤ 9 g¤ ¥ ¢ ¤ ¡ 9 ¦ 6 2t  ¤¡ R¢ £ ¤ % Test statistic: , 6 ¢ 93 ¤  ¢ 76 h D ¤ ¡ ¡ if ¤£ ¢ B6 ¤ED ©£ ¤¨ reject , 95% CI is ¤ $ ¤ £ y P )% t  ' ¥ ¢ 76 Rejection Region: Lower tail test. d.f. ¤ . ppb? Use ¡ ¤ ¡ 9 ¦ 6 2t  mean level of phosphorus is less than ¡ 9 g¤ ¥ ¢ billion [ppb]). Can the EPA support the claim that the ¤ ¦ 6 ¥£ (measurements in parts per ¤R¡ P  £ ¤ P (% ' ¤ ¡ 9 ¦ 6 2t  measurements, obtaining 9 g¤ ¥ ¢ and ¤6 ¦ 6 ¥£ of park, EPA makes 272 6 271 STA 2023 c D.Wackerly - Lecture 19 That is, that are (approx) normally distributed ¡ ...
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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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