Week11-2up - Monday, 6:15 pm – 8:10 pm, McCC 100 P. 288...

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