week13-2up_001 - ¡ ¡ ¡ ¡ Offers any amount Offers $5?...

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Unformatted text preview: ¡ ¡ ¡ ¡ Offers any amount Offers $5? Offers any amount Offers $7? Offers $15? $6? $6? ¡ ¡ ¡ ¡ ¡ ¡ ¡ $10? Offers any amount Offers $4? $5? Offers any amount between $5 and $10? Offers $7? Offers any amount Offers $10? Offers $15? between $5 and $10” Choices: Sell, Not Sell, Can’t tell. Same scenario, but I say that I want at least “something ¡ ¢ £ ¢ £ 252 re-ordered. Pts. earned added to exam score. each, no computation. Answers same, but contain 5 EXTRA questions from Exam 2, 4 pts OPTIONAL Up to 20 BONUS POINTS - will Tuesday, April 9, regular 20 point quiz LAST QUIZ 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, (Sec. 8.4) Wednesday : P. 288 – 393 (Sec. 7.2), P. 341 – 345 Tuesday: Exer. 8.59, 8.61, 8.67–69, 8.116, 8.119 Today : P. 347–351 (Sec. 8.5), Assignments ¡ ¡ leave. What do you do if someone: into jet engines. smallest amount that I will take for this jacket is $6”. If STA 2023 c D.Wackerly - Lecture 19 Thought: Eagles may soar, but weasels aren’t sucked 1 Garage sale, I want to go for coffee and tell you that “the STA 2023 c D.Wackerly - P-value Analogy OR OR RR or larger score (tail area) smaller score p-value Hypothesized Value from NULL HYPOTHESIS Sheet Estimator and Standard Error from Formula Test Statistic ¥  0 00 ¡ £ ¡ Last Time: Large Sample Tests about § ¨¡ ¤ © ¦ ¦ © ¢£¡ ¤ ¥   ¦ ¦   £ £    253 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 Coke and Diet Pepsi. indicated that they preferred Coke drinkers were given unmarked cups of both Diet Ex. : #8.68, p. 352 In a “Pepsi Challenge”, 100 Diet STA 2023 c D.Wackerly - Lecture 19 STA 2023 c D.Wackerly - Lecture 19 #"!         ¡ UNKNOWN but FIXED PROPORTION of items with a . guarantee expires. the proportion of batteries that fail before select Diet Pepsi in a blind taste test. the proportion of Diet Coke drinkers who Recall the BINOMIAL EXPERIMENT. particular attribute 254 (Section 8.5) Interested in a POPULATION that contains an Large Sample Tests About ¡ © §¡ ¤3 8 @9   ¢ £¡ ¤3  12 4 5© 16    ¥ 4 3 ¥ 30 30 ¥ ¥ & ' $ % )( % ¦ %% ¥ 7 16 )0 # of trials ; If Estimate for ) in the sample size in the sample # of trials based on a “large” has an APPROXIMATE NORMAL distribution is “large” distribution. has an approximate STANDARD NORMAL That is ¥ £ ¤¢ 0 ¡ number of trials 3 ¡ ¡ GOAL : Test hypotheses about ¥ 30 ¥& £¢ 0 ¥$ 8 3 ) 3 ¥  @9 3 & ¡ 3 8 @9 ¥ 3 3 ¥ ¥$ ) ) Consider testing STATISTIC if versus a fixed particular value of STA 2023 c D.Wackerly - Lecture 19 ¡ 255 0 00  £¢ ¢£¡ ¤3 OR OR estimator is the null hypothesis, TEST hypothesized value standard error Hypothesized Value from NULL hypothesis Sheet Estimator and Standard Error from Formula ¥ 3 3 STA 2023 c D.Wackerly - Lecture 19 ¥   3 ¢ ¢ ¢ ) 4 ¢¡ ¤3 ¥ ¥ 3 3 ¢ §¥ £¡ 3 §§§§§§§§ ©©©©©©©©¨¦ 3 §§§§§§§ ©©©©©©©©§ 3 ¥ % ¥  3© 3 3   ¢ £ ¢ ¢    % ¢ 3 %% 256 §¥ ¡ 3 3 OR OR ¥ 3 3 £ ¢ 3 §§§§§§ ©©©©©©©§ ©©©¨¦ §§§§ §§§§§§§§§ ©©©©©©©©©©§ § ¢ 3© Rejection Regions (RR): ,  or has a STANDARD ¢¡ NORMAL distribution is true 3 ¥ 3 ¡    ¡     If   RR 257 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 Coke and Diet Pepsi. indicated that they preferred SAMPLE of all Diet Coke drinkers. Note: the the Pepsi Challenge are a (2) is “large” Assumptions : the 100 individuals participating in level test, RR : 258 (1) Coke drinkers were given unmarked cups of both Diet Ex. : #8.68, p. 352 In a “Pepsi Challenge”, 100 Diet STA 2023 c D.Wackerly - Lecture 19 ¡ ¢ ¢   £ £   © ¥¤ ©        1 §§§ ©©©§ ¡§§§§§§ ¢©©©©©©§ 12 § £¡ ¤3 ¢¡ ¤3 ¥ 4 3 ¥ ¡ ¡ § £§§§§§§§§§ ¢©©©©©©©©©§ 6 4 4© 16 16 STA 2023 c D.Wackerly - Lecture 19 ) ¡ Data : 3 reject AT THE Basic Statistics 1 Proportion value = Sample 1 X 56 N 100 Sample p 0.560000 90% CI (0.462710, 0.657290) Test of p = 0.5 vs p > 0.5 distribution”, OK, OK Z-Value 1.20 Click Box “Use test and interval based on normal Coke drinkers will select Diet Pepsi in a blind taste test. 260 P-Value 0.115 Click Options, Select Alternative, Type in Null Value Number of trials, Number of Successes Click radio button “Summarized Data”, type in Stat Minitab? STA 2023 c D.Wackerly - Lecture 19 Test and Confidence Interval for One Proportion level of significance” ( or with claim that there is sufficient evidence at in favor of 259 confidence ) to indicate that the majority of Diet the “ In terms of this problem: LEVEL!! Conclusion : value? ¡ 3 STA 2023 c D.Wackerly - Lecture 19 ¥¤ £ 6 4 ¡ ¥ ) ¥  ¡ ¡ ¡ 66  3 ¢ 1 4 6 1 ¥ ¥ ¢¡ ¥ ¥ § £¡ 1 ¡ ¡ ¡ ¡ £ param. param. OR OR param. param. ¥ value RR standard error Hypothesized Value from NULL HYPOTHESIS Sheet score (tail area) smaller hypothesized value or larger score p-value Estimator and Standard Error from Formula estimator Test Statistic value value value Summary: Large Sample Hypothesis Tests 261 Computer Study: STA 2023 c D.Wackerly - Lecture 19 3 STA 2023 c D.Wackerly - Lecture 19 ¡ reject 4 47 48 50 .2 .8 32 12 .6 .7 ; RR : or .1 0 3 2 18 38 46 not reject Sample size for each test is .5 ¡ ¡ ¡ ¤ ©  ¢ £¡ ¤3 ¥¤ ¢£¡ ¤ §¡ 0  £  ¥   £  ©   % ¥ 00 % £ ©           !" 4 1 ¢¡  ¥ 4 2 3 1  4 1" 6 §¡ ¤3 ¥ 4 4 4 )  4© ¥ 1  1 ¢¡   %% 50 50 50 50 50 50 tests 6 £  1 ) ¥ ¡ 4 2 ¡ 262 1.00 .94 .96 .64 .24 .08 Prop. rejects 1 3 50 48 21 6 0 2 29 44 50 50 50 50 tests When — GOOD! we REJECT moves 1.00 .96 .42 .12 is “better”. . Big , we REJECT approx. greater percentage of the time for larger For each fixed value of . we reject a a greater percentage of the time. , for each of the time. 263 Prop. rejects away from .5, ( and the null becomes “less true”) For each fixed sample size, as the value of What do we see? .8 .7 .6 1 4 ¥ ¢¡  ¥¤  ¡ not reject ¢¡ .5 reject ¢¡ Sample size for each test is 3 ¢ ) 4 6 3 ¥ 4 ¡ ) 1 6 ¡ ¢¡ 62 ¥ ¡ STA 2023 c D.Wackerly - Lecture 19 264 explanation on intro page for Lecture 19. OPTIONAL Up to 20 BONUS POINTS- see Tuesday, April 9, regular 20 point quiz LAST QUIZ 9.24, 9.25 For Tuesday: Exer. 9.1, 9.7, 9.13, 9.15–17, 9.19, 9.22, Monday : P. 374 – 383 (Sec. 9.1, “Large Sample”) 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, Today : P. 288 – 393 (Sec. 7.2), P. 341 – 345 (Sec. 8.4) Assignments readily if you tell them that Benjamin Franklin said it first. Thought: People will accept your ideas much more STA 2023 c D.Wackerly - Lecture 20 ¡ ¡ 3 ) ¢¡ ) param. ¡ param. OR OR param. param. ¡ ¥ value RR standard error Hypothesized Value from NULL HYPOTHESIS Sheet score (tail area) smaller hypothesized value or larger 265 score p-value Estimator and Standard Error from Formula estimator Test Statistic value value value Summary: Large Sample Hypothesis Tests 0  ¢£¡ ¤ ¤ © £  §¡ 00   ¥   £ £ © 266 (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 20 ©      % ¥¤ 4 ¡ ¡   ¡  4! 6 §¡ 1 Sample size ¤  ¥  STA 2023 c D.Wackerly - Lecture 20     % is small! How???  !" ¡ ) ¥ ¥ 41 !  %% ¥$ ! ¢¡ ¤ ppb? Use 1 ) ¡ can’t use scores of the sampling distribution of distributed d. f. distribution with “degrees of freedom”, has a sampling distribution called the (looks a lot like !!!) If the POPULATION is approximately NORMALLY ¢  ¡ does not have a standard normal dist. ¢ ¥ 267 Bell-shaped . (like the -distribution) Properties of the -distribution: Symmetric about 0. (like the -distribution) ¢ STA 2023 c D.Wackerly - Lecture 20  However: ¡ ¡ ¡ 6 £ ' $ )( can’t use CLT to get NORMALITY ¡ ¡ ¡ ¡ Small Sample Inferences about ' $ )( ) as d.f. . -4 as d.f. . -2 0 2 268 4 Std Normal t with 2 df t with 8 df – Becomes more and more like the -distribution – Variability – Variability depends on degrees of freedom. More variable (heavy-tailed) than the -distribution £ $  ¤ STA 2023 c D.Wackerly - Lecture 20 ¤      : Define ¢  (Remember: Table VI (p. 811) gives so that so that   ¢ used in its calculation. 4 -values for 6 ) has the same number of d.f. at the estimator for ¥¤ ¢  ¢ ¡6 41 ¡ ¡ and 269 1.638 1.533 1.476 1.440 1.415 3 4 5 6 7 1.397 1.886 2 ¢ 1.363 1.356 10 11 12 1.350 1.372 9 1.345 1.341 13 1.383 8 3.078 1 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 £¦ ¤ ¦ £¤ ¦¥¦ d.f. £¤ ¦¦ ¦ STA 2023 c D.Wackerly - Lecture 20 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 ¢ Thus ¡ ¡ ¥  $  ) ¡ ¢6 4 ¡6 41 6 ¡! 41 ¡6  4 ¡ ¡ 6 ©   ¥ ©  ¤ ¥ 66 ¤ 66 4 ¡ ¢ ¥ ' $ )( $ Note: d.f. = denominator in calculating 41 £¦ ¢ ¥¦ STA 2023 c D.Wackerly - Lecture 20 & £¦ ¤ ¦¦ 270 1.699 1.645 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.282 16 17 18 19 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 1.746 1.337 d.f. £¦ ¤ ¦ ¢ 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 2.921 271 ¢ Note : When d.f. df= df=30 df=20 df=10 df=5 STA 2023 c D.Wackerly - Lecture 20 ¢ £¤ ¦¥¦ STA 2023 c D.Wackerly - Lecture 20 ¥ ¢ ¢ £ ¡ 4 4 ¢ ¢ ¢ 6! 6 ¢ ¡ 6!  £¤ ¤ ¤ ¤ £ £ £ ¦¦ ¦¦ ¦¦ ¦¦ ¥ ¥ ¥ ¥ ¥ ¦¦ ¥  £¦ ¤ ¦ 2 .025 ! t 272 ¡ )  Small Sample (p. 292): Large Sample: Confidence Interval : d.f. instead of dist. dist. with  Small sample situation similar to large, except use ¢ Assumption : POPULATION approx. NORMALLY dist. ¡ versus Hypothesis Tests (p. 342) OR OR Test statistic : (new) (and or ) depends on RR 274 (like before) AND #d.f. (looks just like !!) ¢ Small Sample Inferences About ¡ STA 2023 c D.Wackerly - Lecture 20 ¢ $ $ ¢   £ 273 ¤ ¥ STA 2023 c D.Wackerly - Lecture 20 ¢   )( & ¡ )( ¥ ¢ § £¡  ¡ §§§§§§§§§§§ ©©©©©©©©©©©¨¦ §§§§§§ ©©©©©©©§ ©©©§ §§§ © ¦  ¥ ¢ £  ¦ ¡ ¦ ¤ ¦   ¢¡ $ ' ) ( ¦ £ ¢ ©  © ¢ ¢   ¢ ¢   ¢    ¡§§§§§§§§§§ ¢©©©©©©©©©©§ §§§ ©©©§ £§§§§§§ ¢©©©©©©§ 275 ¡ ¥$ (measurements in parts per measurements, obtaining ¡ ¥¤ 4 6 §¡ . £ d.f. if Test statistic: reject , Rejection Region: Lower tail test. ¥ ¡ 1 mean level of phosphorus is less than Conclusion: Since evidence to conclude, at the ppb. level of significance, that the mean level of phosphorus is less than , in the rejection region, CANNOT reject Ho . There is is . ppb? Use the section of the Everglades Ex. Give a 95% CI for the mean phosphorus index in STA 2023 c D.Wackerly - Lecture 20 4! ¢ ¥ ¡ 1 ¤ ¢¡ billion [ppb]). Can the EPA support the claim that the and ¡ of park, EPA makes ¡ 95% CI is ¡ of concern to the EPA in the Everglades. In one section ¡ 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 ) ¢ ¥ ¥ ¢  ¥ ¢ ¢ ¥¤ ¢ ¢ ) ¥ ¥$ ¥ ¥ 41 $ ' ) ( ¦ ¡ Ex. Phosphorus content is a water quality index that is ¢ £¤ ¦¦ ¥ ! ¢¡ ¥ ¡ ¤ ¥ ¥$  STA 2023 c D.Wackerly - Lecture 20 ¡ £ ¦¦ ¥ 4 4! 1 ¡ 1 ) ¥  $ ¡ ¥ )(   ! ¥ 1 ¥ ! 4 ! 6 4 ¥¤ ¡ ¥ 41 1 4!   ¥ 41 276 ¢ score (tail area) score In Everglades example, lower tail test, d.f. ¡ are and Closest values in table ( with d.f. Look at table, p-value ! ¡ §¡ ¤ Table does not allow us to get exact p-values. ¥ ¡ OR ¥ ) to §§ ©©§ , p-value 277 £ ¢ ¢ Ho . Ho . Ho . Ho . Can’t be any more precise using these tables! Thus, in this case, best we can say is that From the table, ¢ ¡ ¡ From the table, STA 2023 c D.Wackerly - Lecture 20 §§§§ ©©©©©§ ©©¨¦ §§§ §§§§§ ©©©©©©§ ©§ § © OR How about -values? ¡ 7   #"! ¢  ¢   ¥  £ STA 2023 c D.Wackerly - Lecture 20 ¦ £ ¦ ¦  ¥  ¥ ¡ ¡ 0 0 0 ¥¤ ¥¤ 4 0 ¥¤ ¥¤ 4 4 4 4 ¡§§§§ ¢©©©©§ ¡  0 0 ¥¤ ¥¤ 4 4 6  4  4 6 ¡ 26 6 § ©§  4© ¡ 1 6 £§§§§§ ¢©©©©©§ ¡ ! 2 ¥ 1 ¡ ¡ ¡ ¡ 2 ¥ 4  2 ¥ 278 2559 2543 ¡ ) ¥ §¡ 2562 2541 2620 2552 2544 2553 measurements Basic Statistics 2553 2560 1-Sample . Variable C1 N 10 vs Mean 2558.70 Test of mu = 2550.00 T-Test of the Mean StDev 22.75 SE Mean 7.19 mu not = 2550.00 T 1.21 Select Alternative, Type in Null Value, OK P 0.26 Select variable and click Radio button “Test mean” Stat Punch in data values Minitab? Want : two tailed test !  161 6 temperature different that target? Use Target setting is 2550 degrees. Actual mean pouring 279 claim that the mean pour ? Basic Statistics 1-Sample Minitab CI? Variable C1 N 10 Mean 2558.70 T Confidence Intervals Test of mu = 2550.00 vs StDev 22.75 SE Mean 7.19 mu not = 2550.00 Type in Confidence Level OK interval” 99% CI (2535.32, 2582.08) Select variable and click Radio button “Confidence Stat significance. temperature differs from 2550 at the .01 level of Is Conclusion at .01 level? STA 2023 c D.Wackerly - Lecture 20 Exercise 10.52 Pouring temperature of molten iron. ¢¡ ¥¤ ! 6 4 161 ¢ ¡ ¡ STA 2023 c D.Wackerly - Lecture 20 ¤ ¥ £ ¡ ¡ ¡ ¡ ¡ £ ¤ ¥ £ £ ¡ ¡ ¡ ¡ ¢ 280 ...
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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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