Week11 - STA 2023 c D.Wackerly - Review for Exam II...

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Unformatted text preview: STA 2023 c D.Wackerly - Review for Exam II Thought: Never do card tricks for the people you play poker with. Thought: To succeed in politics, it is often necessary to rise above your principles. Monday 11/4/02 : OPTIONAL REVIEW, ask questions about homework, course material, sample exam, etc. Suggestion - print out Sample Exam 2 and bring it to class with you. Help for Exam 2 Monday, periods 3–8, FLO 104 Monday, 6:15 pm – 8:10 pm, McCC 100 ¡ ¡ Tuesday 11/5/02 : EXAM 2–during your regularly scheduled DISCUSSION SECTION Wednesday 11/6/02 : Minitab, Computer Demo, P. 288 – 393 (Sec. 7.2), P. 341 – 345 (Sec. 8.4) 1 STA 2023 c D.Wackerly - Review for Exam II 2 Chapter 4 : Discrete Random Variables Can COUNT the number of distinct values of the ¡ variable. The Binomial Probability Distribution – Some discrete random variables are binomial – NOT ALL   ¨  & "¡ ¨ &¨ 43"¡ § 76 ¢ ¢ 21 5 0 ¢ ¤ ¨ §© © © ¤ £ – Tables : Contain © ¡ – Variance (p. 185) : ¢ ¡ ¢ – Mean (p. 185) : for : (p. 183) ¡!&&&! #! )('''%$" £ number of trials, ¢ § ¦¤ ¥ – If criteria ¨ – Binomial Experiment p. 179 – for & @9'''%$" ¡!888!#! ..... 0 1 2 3 ..... k k+1 n-1 n ¢ 6 ¡ STA 2023 c D.Wackerly - Review for Exam II 3 Chapter 5 : Continuous Random Variables Possible values are all those associated with one or ¡ more line intervals. Probabilities are areas under “density function”. ¡ graph of f(x) P(a<X<b) a is a continuous r.v., ¡ ¡ ¡ ¢ § 5 © ¤ ¡ £ ¢§ 5 © 5 ¤ £ ¡ ¢ © ¢ ¢ ¤ £ ¢§ © 5 ¤ £ ¢ Normal distribution is special case. § © If b ¡ STA 2023 c D.Wackerly - Review for Exam II 4 Areas under normal curves between z-scores of in Table IV, p. 809. ¡ ¡ and , for Key to finding correct areas (probabilities) : draw ¡ pictures Chapter 6: If we plan to take a random sample of size and standard , 1 is a random variable. is called its sampling distribution. © ¢ ¡ © Dist. of ¢ ¡ deviation 0 from a population with mean ¡ (p. 255) © ¢ 0 ¡ ¢ § ©¤ ¢ ¤ is an unbiased estimator . (p. 266, 261) is more  0 ¡ ¡ ¥ 1 ¢ £  1 is called the standard error of .(p. 266) 1 for larger sample sizes. £ 0  concentrated around © (p. 266), so dist. of ¢ 0 £ of ¢ . So ¢ © If the population has a normal dist., then so does , ¢ © §¡ ¥ 1 ! ¤ 0 N . True for any ¡ ¢ ¦ © i.e., . ¡ STA 2023 c D.Wackerly - Review for Exam II 5 ¡ ¥ 1 ! ¤ 0 N §¡ ¢ ¢ ¦ © ¡ approximately normal, i.e., © ), then the sampling distribution of ¢ ¡ ( is large ¡ Central Limit Theorem (CLT): (p. 267) If is , regardless of the shape of the population distribution. Chapter 6: A Parameter is a meaningful number associated , 21 0 ¡ with a Population. , etc. (p. 254) A Statistic is a meaningful number associated with ¡ a Sample. All statistics have sampling distributions ¡ Chapter 7: Point Estimator (p. 261) ¡ Interval Estimator (p. 282) ¡ Confidence Coefficient (p. 282) Confidence Level (p. 282) ¡ ¡ STA 2023 c D.Wackerly - Review for Exam II 6 Estimated Standard Parameter Error of Est. Estimator Standard Error of Est. 1 ¡ ¡ © ¡ 0 ¡  3¨ ¨ ¡ ¡ © ¡ ¡ ¢ ¨ ¨ Both estimators are UNBIASED ¡ ¡ If is “large”, both estimators are approximately ¡ NORMALLY distributed. How large is “large”? ¡ 0 . : ¡ §  ! ¨¤ ¡ § % ! ¨ ¤ smaller of ¡ larger of ¡ ¨ – For valid CI for : ¡ – For valid CI for ¢ ¡ ¡ STA 2023 c D.Wackerly - Review for Exam II 7 £ Confidence Interval for a ¡ ¢§ # # )¤ PARAMETER ¦ ¥ ¨ ¦ §¥ ¤ formula sheet standard errors formula sheet table α/2 & §¨ 2¨ ¥ ¤ ¦ ¤¤   © estimator α/2 1−α     2 2 (P. 283) 0 – Population mean, © 2 & ¡ 1  ¢ ©   ¡ (P. 300) ¡   ¨ 2 ¡ ¨ ¡   2 © ¢ © – Population Proportion, © ¡ ¨ ¡ STA 2023 c D.Wackerly - Review for Exam II 8 Finding the sample size to estimate . Want : Correct to within “ ” units with ¡ ¡ £ confidence. and SOLVE ¢§ ¡ and solve for # )¤ ¡ 2 1 ¢ ¡   ¢ ¤   ¡ ¡ 2 if you have one. Maybe 1 – Use ballpark value for Range . use (p. 307) ¡ ¡ #§ standard error  ¢ 1 Finding the sample size to estimate . £ Want : Correct to within “ ” units with ¡ ¡ £ confidence. ¢§ ¡ and solve for ¨ ¢ ¡ # )¤   ¢ ¤ & ¢   ¨   (p. 333) if you have one, if not to get sample size that will work 2 ¨ for any value of . ¨ ¡ ¡ ¢ ¡ 2 – Use “ballpark” value for use and SOLVE ¡ ¡ #§ standard error STA 2023 c D.Wackerly - Review for Exam II 9 Chapter 8 – Large Sample Hyp. Testing Parts of a statistical test. (p. 322) The hypothesis of MAIN INTEREST is the ¡ ALTERNATIVE or RESEARCH hypothesis, – § £ ¤ ¡ ¡ # 0 ¢¡ ¡ , ¤ light bulb ex. . (p. 322) What we are “trying to prove” in an objective, fair manner The “other” hypothesis is called the NULL ¥ § ¡ HYPOTHESIS, – , light bulb ex. (p. 322) £ ¤ ¡ # ¢ 0 ¢¥ ¤ Errors: p. 325 Reality Decision Ho true Ha true Accept Ho Correct Type II error Reject Ho Type I error Correct STA 2023 c D.Wackerly - Review for Exam II (p. 323), SIGNIFICANCE § Type I error 10 ¤ £ ¡ ¡ ¢ LEVEL of the test. ¡ ¢ ¡ ! when true ¤ ¡ £ ¢ ¢ ¡ ¡ ¡ ! accepting ¥ ¡ ¤ and/or (p. 325) § Type II error £ £ ¡ ¡ ¢ ¡ ¡ £ £ £ ¤ ¡ # ¢ 0 when £ ¤ ¡ ¡ # 0 saying ¤ In our lightbulb example, ¤ ¢ saying what we “want” to say when we should not £ £ ¢ ¡ ¡ STA 2023 c D.Wackerly - Review for Exam II 11 0 Chapter 8 : Large Sample Tests about calculated value of . ¡ ¤¡ ¢ £¡ ¢ p-value ¡ ¤¡ ¢ £¡ § ¡ ¤¡ ¢ £¡ § ¢ 0 ¢ ¡  ¤ £ 0 ¢¥ ¡ ¢¡ RR ¡ ¡ 0 0 OR ¢  ¤ £ ¢ ¢ 0 0 OR ¢ £   or (tail area) 2   ¡ 2 ¦ ¦ ¨¦ 0 ¦ §© ¡ ¥ ¢ 0 ¥ ¢ 0 Test Statistic 1 ¡ ¢ Estimator and Standard Error from Formula © Sheet Hypothesized Value from NULL HYPOTHESIS ¡ ¨ rejected. for which © © value = smallest value for can be STA 2023 c D.Wackerly - Review for Exam II Large Sample Tests about 12 ¨ calculated value of . ¢ ¡ ¤¡ ¡ £ p-value ¡ ¤¡ ¡ £ § ¡ ¤¡ ¡ £ § ¢ ¨ ¢ ¡  ¤ £ ¨ ¢¥ ¡ ¢¡ RR ¡ ¡ ¨ ¨ OR ¢  ¤ £ ¢ ¢ ¨ ¨ OR £ ¡ ¥£¤¡¢ ¨ ¦¨ ¦¨¦ ¢ or (tail area)   2   ¡ ¢ ¨ ¥ ¢ ¨ 2 Test Statistic ¡ ¦ ¨¦ ¢  Estimator and Standard Error from Formula © Sheet Hypothesized Value from NULL HYPOTHESIS © © ¡ STA 2023 c D.Wackerly - Lecture 19 255 Thought: Eagles may soar, but weasels aren’t sucked into jet engines. Assignments Today : Minitab, Computer Demo, P. 288 – 393 (Sec. 7.2), P. 341 – 345 (Sec. 8.4) For Thursday: Exer. 7.27, 7.30, 7.33, 7.80, 7.81, 8.49, 8.50, 8.53, 8.54, 8.56, 8.57, 8.105 – 108, 8.111, 8.117 STA 2023 c D.Wackerly - Lecture 19 256 Summary: Large Sample Hypothesis Tests calc. value of . ¢ p-value ¡ ¤¡ ¢ £¡ § ¡ ¤¡ ¢ £¡ § ¢ ¢ ¡  ¤ £ ¢¥ ¡ ¢¡ RR ¡ value param. ¡ ¤¡ ¡ £ value param. ¡ OR ¢  ¤ £ ¢ value param. ¢ OR or ¢   2   ¡ ¢ ¥ value £ ¢ param. (tail area) 2 Test Statistic hypothesized value ¦ ¦ standard error ¦ ¨¦ estimator ¢ Estimator and Standard Error from Formula © Sheet Hypothesized Value from NULL HYPOTHESIS © © ¡ STA 2023 c D.Wackerly - Lecture 19 257 Ex. : #8.68, p. 352 In a “Pepsi Challenge”, 100 Diet Coke drinkers were given unmarked cups of both Diet ¡ Coke and Diet Pepsi. indicated that they preferred the taste of Diet Pepsi. Is there sufficient evidence to indicate that a majority of the Diet Coke drinkers will select Diet Pepsi in a blind taste test? true proportion of Diet Coke drinkers who ¡ ¢ ¨ would select Diet Pepsi in a blind taste test. (1) ¢¡ ¡ & (2) ¨ & ¢ ¨ ¢¥ ¢ &# ¢ ¢ £  ¢ ¡ level test, RR : & ¢ ¡ ¡ ¢ Assumptions : the 100 individuals participating in the the Pepsi Challenge are a RANDOM SAMPLE ¡ of all Diet Coke drinkers. Note: is “large” ¡ STA 2023 c D.Wackerly - Lecture 19 258 & £ & ¢ & %# ¡ ¢ # & ¢ ¢ & ¨ ¡ & & ¢ & ¡ ¡ ¢ & # ¢ ? ¡ &# in favor of ¡ ¢ £ & %# ¥ ¢ ¡ NO!! - Cannot reject %# Conclusion : is ¡ Data : AT THE LEVEL!! & ¢ ¡ In terms of this problem: ¡ “ CANNOT claim that there is sufficient evidence at £ ¢ & 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. ¨ value? ¨ value = ¢ &# ¢ # %# & ¢ ¡ & & ¢§ £ ¡ & %# ¤ £ ¡ ¡ STA 2023 c D.Wackerly - Lecture 19 259 Minitab? Basic Statistics Stat 1 Proportion ¡ Click radio button “Summarized Data”, type in ¡ Number of trials, Number of Successes Click Options, Select Alternative, Type in Null Value Click Box “Use test and interval based on normal ¡ ¡ distribution”, OK, OK Test and Confidence Interval for One Proportion Test of p = 0.5 vs p > 0.5 Sample 1 X 56 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 19 260 Computer Study: ¢ & &# ¢ & ¢ ¢¡ ¨ ¢ or ¥ ¡ &# & ¢ ; RR : ¨ ¢¥ #& ¢ ¡ ¡ ¡ ¡ ¨ & ¢ ¢ & ¡ ¡ ¥ not reject ¡ ¥ reject ¢ Sample size for each test is tests Prop. rejects ¨ .5 4 46 50 .08 .6 12 38 50 .24 .7 32 18 50 .64 .8 48 2 50 .96 .2 47 3 50 .94 .1 50 0 50 1.00 STA 2023 c D.Wackerly - Lecture 19 261 ¡ ¥ ¥ not reject ¢ reject Sample size for each test is tests Prop. rejects ¨ .5 6 44 50 .12 .6 21 29 50 .42 .7 48 2 50 .96 .8 50 0 50 1.00 What do we see? moves ¨ For each fixed sample size, as the value of ¡ away from .5, ( and the null becomes “less true”) ¥ we REJECT a greater percentage of the time. — GOOD! & ¥ ¢ ¨ ¤ ¨ £ ¡ # is “better”. ¡ ¢ ¢ greater percentage of the time for larger . Big a ¡ , we REJECT ¥ #& & For each fixed value of approx. . § ¡ of the time. we reject ¡ , for each ¥ When ¡ STA 2023 c D.Wackerly - Lecture 19 262 Ex. Phosphorus content is a water quality index that is of concern to the EPA in the Everglades. In one section # ¢ ¡ ¢ & £ ¡ ¢ and £ of park, EPA makes measurements, obtaining (measurements in parts per & ¢ # ¢ © billion [ppb]). Can the EPA support the claim that the # . ¢¥ & ¡ ¢ § ¢¡ £ # )¤ Sample size is small! How??? mean level of phosphorus is less than ppb? Use ¡ ¡ STA 2023 c D.Wackerly - Lecture 19 263 Small Sample Inferences about ¡ can’t use CLT to get NORMALITY ¢ ¡ scores can’t use © of the sampling distribution of © ¢ 0 does not have a standard normal dist. ¡ ¥ However: If the POPULATION is approximately NORMALLY distributed © ¢ 0 (looks a lot like !!!) ¡ ¡ ¢ ¥ has a sampling distribution called the ¡ # ¡ d. f. distribution with “degrees of freedom”, ¡ STA 2023 c D.Wackerly - Lecture 19 264 ¡ Properties of the -distribution: Symmetric about 0. (like the -distribution) ¡ Bell-shaped . (like the -distribution) ¡ More variable (heavy-tailed) than the -distribution – Variability depends on degrees of freedom. ¢ as d.f. . ¡ – Variability – Becomes more and more like the -distribution as d.f. . ¡ t with 2 df t with 8 df Std Normal -4 -2 0 2 4 ¡ STA 2023 c D.Wackerly - Lecture 19 265 2 Note: d.f. = denominator in calculating : ¢ 2§ © # ¡ © ¤ ¡ 2 ¢ Thus © ¢ ¡ 0 ¡ ¢ ¥ used in its calculation. ¢§  ¡ ¤ ¤ ) £ % & and & -values for # " & '! £  &! $%#   ¡ ¡ ¡ Table VI (p. 811) gives ¡ ¡ ¡ so that ¢§ ¡ (Remember:  so that ¡ Define 1 ¡ has the same number of d.f. at the estimator for & $! £ & '! & '! #& ¢ ¡ ¡ STA 2023 c D.Wackerly - Lecture 19 ¢ ¡ ¢ ¡ ¡ ¢ ¢ 2 ¡ ¢ £ ¢ ¡ ¢ ¡ d.f. 266 ¢ 1 3.078 6.314 12.706 31.821 63.657 2 1.886 2.920 4.303 6.965 9.925 3 1.638 2.353 3.182 4.541 5.841 4 1.533 2.132 2.776 3.747 4.604 5 1.476 2.015 2.571 3.365 4.032 6 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 9 1.383 1.833 2.262 2.821 3.250 10 1.372 1.812 2.228 2.764 3.169 11 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 STA 2023 c D.Wackerly - Lecture 19 267 ¢ £ £ ¢ 16 1.337 1.746 2.120 2.583 2.921 17 1.333 1.740 2.110 2.567 2.898 18 1.330 1.734 2.101 2.552 2.878 19 1.328 1.729 2.093 2.539 2.861 20 1.325 1.725 2.086 2.528 2.845 21 1.323 1.721 2.080 2.518 2.831 22 1.321 1.717 2.074 2.508 2.819 23 1.319 1.714 2.069 2.500 2.807 24 1.318 1.711 2.064 2.492 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 ¡ ¢ ¡ ¢ d.f. ¡ ¢ 2 ¡ ¢ ¡ ¢ ¡ ¢ STA 2023 c D.Wackerly - Lecture 19 268 .025 t ¢ 2 ¡ ¢ ¢ 2 & ¢ ¢ ¢ 2 £ ¢  ¢ ¢ ¢ ¢ & ¡ ¡ 2 ¢ ¡ 2 ¢  ¢ ¡ ! ¢ Note : When d.f. £ 2 df= ¢ ¡ df=30 ¢ df=20 £ ¢ ¡ df=10 ¢ ¢ df=5 STA 2023 c D.Wackerly - Lecture 19 269 Small Sample Inferences About Assumption : POPULATION approx. NORMALLY dist. ¡ Small sample situation similar to large, except use d.f. instead of dist. # ¡ dist. with Confidence Interval : 1 ¡   © ¢ Large Sample: © 2 ¡   2 ¡ © ¢ © Small Sample (p. 292): ¡ STA 2023 c D.Wackerly - Lecture 19 270 Hypothesis Tests (p. 342) 0 ¢ 0 ¢¥ ¡ versus ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¤£  ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ 0 0 ¡ ¡ ¡ ¡ ¡ OR  RR ¡ ¡ ¡ ¢ ¡ ¡ ¢ 0 ¡ ¡ ¡ 0 ¡ ¡ ¡ OR ¡ ¡ ¡ ¡ ¡ ¡   ¡ or ¡ 2   ¥ ¡ ¡ ¡ ¢ ¥ 0 ¡ ¡ ¡ ¡ ¡ ¢ ¡ ¡ 0 ¡ ¡ ¢ ¡ 2 Test statistic : © ¡ ¢ ¡ ¥   ) depends on ¡ ¡ ¢ 0 2 ¡  ¡ (new) (and (looks just like !!) (like before) AND #d.f. ¡ STA 2023 c D.Wackerly - Lecture 19 271 Ex. Phosphorus content is a water quality index that is of concern to the EPA in the Everglades. In one section # ¢ ¡ ¢ measurements, obtaining (measurements in parts per & £ ¡ ¢ and £ of park, EPA makes & ¢ # ¢ © billion [ppb]). Can the EPA support the claim that the ppb? Use # . 0 ¢¡ ¢ # ¡ ¢ 0 ¡ ¢ . ¢  ¡ ¢ ¡ ¡ ¥ ¢ ¡ ¢ reject ¡ ¢ , , ¢ # ¢¥ & Rejection Region: Lower tail test. d.f. mean level of phosphorus is less than if ¢ & ¡ ¢ £ ¤ & £ Test statistic: ¢ # ¢ © ¡ &%# ¢ ¢ £ ¤ © ¡ ¢ 0 ¡ ¡ ¢ ¥ ¡ ¡ &# ¢ Conclusion: Since is in the ppb. ¡ # phosphorus is less than ¢ level of significance, that the mean level of & evidence to conclude, at the rejection region, CANNOT reject Ho . There is ¡ STA 2023 c D.Wackerly - Lecture 19 272 Ex. Give a 95% CI for the mean phosphorus index in the section of the Everglades ¢ & ¡ ¢ ¢ & ¢ 2 £ ¢ # ¢   ¡ ¢ ¢ © £ # ¢ ¡ ¡ ¡ ¢ 2 95% CI is ¢   ¡ 2 ¡ © ¢ © ¢ Note: In last example (both test and CI), we are assuming that population from which the sample is taken is (approx) normally distributed That is, that are (approx) normally distributed ...
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