Week13-4up_001 - STA 2023 c B.Presnell & D.Wackerly -...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: STA 2023 c B.Presnell & D.Wackerly - Lecture 20 231 Last Time: Large Sample Hypothesis Testing about # $ ¦ ¡¡  £ §¥    ¦ ¡¡   ¦ ¡£¤¡ ¨©§¤¢ ¦¥ £ ¤¡ ¢ UNKNOWN population mean OR ¢ Test Statistic 6 74 £ 8 2% 53 1 £ ¦¡ Wednesday: P. 288 – 294 (Sec. 7.2), P. 341 – 345 (Sec. 8.4) B Thursday: Exer. 7.27, 7.30, 7.33, 8.49, 8.50, 8.53, 8.55 – 8.57 standard error Sheet Hypothesized Value from NULL hypothesis 232 hypothesized value Estimator and Standard Error from Formula B CB STA 2023 c B.Presnell & D.Wackerly - Lecture 20 estimator % @9 8.61, 8.67–69 RR 9 Tuesday : Exercises 8.29, 8.33, 8.34, 8.38–41, 8.59, % 0 ! &# Today: P. 334–338, 347–351  Assignments ! ()' " readily if you tell them that Benjamin Franklin said it first. 9 A9 Thought: People will accept your ideas much more ¡ 230 STA 2023 c B.Presnell & D.Wackerly - Lecture 20 STA 2023 c B.Presnell & D.Wackerly - Lecture 20 233 Ex. : pH of 7 is neutral, over 7 is alkaline, under 7 water specimens ? QH £ RPEIGF ¨¦¥  §¥ ¨S§¥ ¥ ¦ ¤¢ and level? QRPEIF ¢ H£ ED Cu£ 8 ¢ pH is NOT that of neutral water at the Back to pH example: level test: RR : the Q RPEH ¢ v 0 z α/2 “Two Tailed test” (p. 331) level of significance. In terms of this application: “There not .” α/2 ¦¥ α/2 at the Q REH α/2 Reject £ sq t$!  sq % tr! &# `¨ pi i 8 6 4 h fg8 6 453 a 1 d¡ e2% a 1 e¡c% b£ `¨ YW d X ¡ U£ ¨  ¦ V¡ S§¥ ¡£ ¦ T¡ ¨ ¦ ¥ or £ How? -z PD$w£ a 1 Hv from a recreational lake. Can we claim that the mean H HE r€yx£ h E CD indicates acidity. Randomly select enough evidence at level to indicate that the mean pH reading is 234 STA 2023 c B.Presnell & D.Wackerly - Lecture 20 STA 2023 c B.Presnell & D.Wackerly - Lecture 20 235 Hypothesis Testing v If the mean pH is NOT , what is it? to reject CONFIDENCE in our SO). h ¤ could © ¨ ¢ The p-value or observed significance level (P. 335) Recall the lightbulb example the 99% v v&T¡ £ PEH © PD$v H    HQ £  ¡€`D Q ¡ S§¥  ¨ €£ `D Q ¤¡ ¨ ¦ ¥  ¡ ¢ 236 STA 2023 c B.Presnell & D.Wackerly - Lecture 20 237 F rejection region CANNOT reject " Q EH £ #RPIGF " € EH £ $`PIGF " D EH £ %CPIGF "$PIGF  EH £ Q IGF H£   ¢ ECCPDy€  DH E PEy€  H £  Q ¡€`D w¡ ¨ §¥  € £ ¨¦ `D Q ¤¡ ©§¥  ¡ – " # QRPEH €`PEH D CEPH EPH PEH    . ¢  ¢ true. is . REJECT . . . Instead of “imposing” YOUR CHOICE of indicative that ¥ p-value = " HQ Cw   – Larger z-values are "   #2F ¢ !F ¢ – – the one observed p-value REJECT In our case, p-value = .0256 – Probability of a z-value p-value . . F ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 20 ¦§¥ ¦ §¥ 1 Agrees with two-tailed test!! ¦¥ ¦§¥ ¦¥ H£ I GF % Q confidence interval p-value ( IF we DO for which ¦¥ ¤ ¥¡ — the value “ ” is ? ¦¥ ¢ £¡ £ F 86 What is the SMALLEST value of ¥ £ €F ¤ ? in favor of F ¡ £ ¨d d ¡ §¦ decision to reject be rejected in favor of Do you think that . ¦¥ – Provides ¦¥ – ¥ Smaller ¦¥ ¢ . F confidence interval for F¢ Construct a is chosen BEFORE the test is performed on a person who might be interested in your conclusions, the p-value allows him/her to assess the “rareness” of the observed event.    £ 238 STA 2023 c B.Presnell & D.Wackerly - Lecture 20 £ DCPDHr€ I£ % E wG¡ ¨ §¥ Q#  E Q T¡ ¨ ¦ ¥ £ ¢ Find the area in whichever “tail” the -value is in and DOUBLE IT. EX. : Have done a two-tailed test: Probability of a z-value  Hv Q £ v Q U£ ¡ ¨  ¥ v Q ¤¡ ©§¥ £ ¨¦ the one observed is true. £ ? – . STA 2023 c B.Presnell & D.Wackerly - Lecture 20 241 F ¦ §¥ ¦¥ ¢ ¢ Large Sample Tests About ¢ EH `PE&# F ¢ 240 with ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 20 claim that v Q V¡ U£ . . PEIF H£ that is value = . ¡ EH CPEw for any ¡ that is PHI£F ¢ E % ¡¢ for any  £ ¥ Smaller z-values are more indicative that p-value 239 TWO - Tailed Test Ex. : #8.24, P. 333 p-value STA 2023 c B.Presnell & D.Wackerly - Lecture 20 (Section 8.5) Ex. : #8.68, p. 352 In a “Pepsi Challenge”, 100 Diet Interested in a POPULATION that contains an Coke drinkers were given unmarked cups of both Diet UNKNOWN but FIXED PROPORTION of items with a ¢   the taste of Diet Pepsi. Is there sufficient evidence to ¥£ ¦X ¤ particular attribute indicated that they preferred  Coke and Diet Pepsi. . Recall the BINOMIAL EXPERIMENT. select Diet Pepsi in a blind taste test? £ B indicate that a majority of the Diet Coke drinkers will the proportion of Diet Coke drinkers who select Diet Pepsi in a blind taste test. £ B true proportion of Diet Coke drinkers who guarantee expires. # of ¥ ¦X £ ¨ £8B Estimate for £1 # of trials ; based on a “large” in the in the sample sample size trials H number of trials ¥ ¦X £ £ 18 £ § ¢ ¨¦¥ (2) GOAL : Test hypotheses about 8 ¨¥ (1) ¢ How??? the proportion of batteries that fail before £ ¡¢ would select Diet Pepsi in a blind taste test. ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 20 if STATISTIC £ ¤ ¢ standard error Estimator and Standard Error from Formula B CB Hypothesized Value from NULL hypothesis Sheet ¥ ¦¥ ¢ has a STANDARD ¢ Rejection Regions (RR): s tq$!  sq % t$! &# 0 244   77 ¦ U £    £   7 ¦ #  §¥ % ! &#  STA 2023 c B.Presnell & D.Wackerly - Lecture 20 or RR    OR   7$ ¦ U £   7 ¦ #   £  ¥  ¦    ¦   ! " ¦ £ ©§¥ ¨¦ OR OR ) NORMAL distribution ,  B OR is true (' 3£ a fixed particular value of ¦ £  ¡ If  % § £ ¡ ¡ B £ ¡ 3 Consider testing 8 ¦ §¦ ¦ ¢ £ £ ¦ % 8 distribution. versus hypothesized value B has an approximate estimator £ § B distribution 9 has an That is is the null hypothesis, TEST %9 8¢ is “large” 243 9 A9 If 242 ¦ £ ¨©¦§¥ ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 20 STA 2023 c B.Presnell & D.Wackerly - Lecture 20 Ex. : #8.68, p. 352 In a “Pepsi Challenge”, 100 Diet 245 Coke drinkers were given unmarked cups of both Diet indicated that they preferred ¢ the taste of Diet Pepsi. Is there sufficient evidence to Conclusion : reject   ¢ AT THE In terms of this problem: “ claim that there is sufficient evidence at E  I£ ¨ ¦ ¥ H E  & S ¥ H¨ (3) the (4) confidence ) to indicate that the majority of Diet  level of significance” ( or with care reform is the leading priority LEVEL!! in favor of ¢ Coke drinkers will select Diet Pepsi in a blind taste level test, RR : test. ¢ the the Pepsi Challenge are a 8 SAMPLE of all Diet Coke drinkers. Note: is “large” % ¡¢ PEIF ¢ H£   Assumptions : the 100 individuals participating in % ¡¢ £ ¡¢ true proportion of all voters who think health £ § £ £ PEH   £ E CE Q £ 8 Data : PEIF H£ select Diet Pepsi in a blind taste test? ¦ §¥ indicate that a majority of the Diet Coke drinkers will ¥   Coke and Diet Pepsi. value? value = £ ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 8.55 – 8.57 ¢ For Tuesday: Exer. 9.1, 9.7, 9.13, 9.15, 9.16, 9.19, B 9.22, 9.24, 9.25 Sheet Hypothesized Value from NULL HYPOTHESIS STA 2023 c B.Presnell & D.Wackerly - Lecture 21 249 Sample size for each test is Computer Study: .8 .12 29 50 .42 48 2 50 .96 50 0 50 1.00 For each fixed sample size, as the value of Prop. rejects .64 .8 48 2 50 .96 .2 47 3 50 .94 .1 50 0 50 1.00 When For each fixed value of , for each of the time. we reject . approx. , we REJECT greater percentage of the time for larger is “better”. 8 50 . Big a 8 18 a greater percentage of the time. ¢ 32 — GOOD! EQ .7 we REJECT ¦¥ .24 ¢ 50 ¢ 38 H I£ 12 moves away from .5, ( and the null becomes “less true”) .08  50  HIU £ £E Q HIGF  £ 8 46 ¦¥ .6 ¦¥ 4 tests 50 ¦¥ E Q IF ¢ H£ H I£ ¨ ¦ ¤¢ ¥ .5 not reject Prop. rejects What do we see? ¢  reject 21 .7 44 tests ¨ ¦¥ ECu£ 8 D 8 H H  % H ¢£ §   H QS%&#  H w Q   H ¨ IU £ S§¥  .6 Sample size for each test is 6 ¦ §¥ .5 or not reject ¨ ¦¥ ; RR : reject E C £ 8 248 (tail area) hypothesized value standard error score Estimator and Standard Error from Formula B CB STA 2023 c B.Presnell & D.Wackerly - Lecture 21 estimator £ Monday : P. 374 – 383 (Sec. 9.1, “Large Sample”) Test Statistic or 9 For Thursday: Exer. 7.27, 7.30, 7.33, 8.49, 8.50, 8.53, % @9 U£ param. s tqr!  sq % t$! &# P. 341 – 345 (Sec. 8.4) € # value OR smaller score 9 A9 % ! &# Today: P. 288 – 294 (Sec. 7.2) value   param. OR larger % value  ! Assignments p-value  param. RR % knows it all is the person who argues with him. value   ¨ §¥ ¨¦¥ ©§¤¢ param. Thought: The only fool bigger than the person who 247 values for Large Sample Tests £ Last Time: % 246 STA 2023 c B.Presnell & D.Wackerly - Lecture 21 STA 2023 c B.Presnell & D.Wackerly - Lecture 21 251 Small Sample Inferences about 8 64h ¡ c% a 1 ¢  Q . However: ppb? Use If the POPULATION is approximately NORMALLY distributed ¨ §¥ ¦ ¡ is small! How??? (looks a lot like !!!)   has a sampling distribution called the % €Q ¢ ¨  ¤¢ ¥ PEIGF H£   Sample size distribution with ¡ d. f. 252 “degrees of freedom”, STA 2023 c B.Presnell & D.Wackerly - Lecture 21 Q%8 PDy€ Q £ a 1 H mean level of phosphorus is less than scores does not have a standard normal dist. 8 74 h 6 ¡ a1 c% b£ H r £ h € Q £ 8 (measurements in parts per billion [ppb]). Can the EPA support the claim that the can’t use " and % measurements, obtaining 1 E CD # 8 of the sampling distribution of of concern to the EPA in the Everglades. In one section of park, EPA makes can’t use CLT to get NORMALITY " Ex. Phosphorus content is a water quality index that is 250 STA 2023 c B.Presnell & D.Wackerly - Lecture 21 STA 2023 c B.Presnell & D.Wackerly - Lecture 21 253 ¡ ¢ Note: d.f. = denominator in calculating 8 64h ¡ a1 c% b£ a 1 Q% % 1 8 £ h s ¤ as d.f. . Thus £ ¢ – Becomes more and more like the -distribution as d.f. . has the same number of d.f. at the estimator for £ used in its calculation. ¡ ¡ Define ¡ Std Normal HyECCPEH Q ECEH CEH RPEH ¡`PEH PEH E Q IF E E Q €  H £ ¤ ¤ ¤ ¤ ¤ ! ¢ Fx£  !    ! F x£  !    ¢ ! t with 2 df t with 8 df so that (Remember: so that ) ¡ Table VI (p. 811) gives -values for and -4 -2 0 2 4 3 ¢ – Variability : ¡ – Variability depends on degrees of freedom.  More variable (heavy-tailed) than the -distribution s ¢ ¢ Bell-shaped . (like the -distribution) sh Properties of the -distribution: Symmetric about 0. (like the -distribution) ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 255 s d¦ s d¦ 3.078 6.314 d ¨d ¦ § 12.706 d d¦ § ¨¦ 1 31.821 63.657 16 1.337 1.746 d ¨d ¦ § 2.120 2.583 d d¦ 2.921 2 1.886 2.920 4.303 6.965 9.925 17 1.333 1.740 2.110 2.567 2.898 3 1.638 2.353 3.182 4.541 5.841 18 1.330 1.734 2.101 2.552 2.878 4 1.533 2.132 2.776 3.747 4.604 19 1.328 1.729 2.093 2.539 2.861 5 1.476 2.015 2.571 3.365 4.032 20 1.325 1.725 2.086 2.528 2.845 6 1.440 1.943 2.447 3.143 3.707 21 1.323 1.721 2.080 2.518 2.831 7 1.415 1.895 2.365 2.998 3.499 22 1.321 1.717 2.074 2.508 2.819 8 1.397 1.860 2.306 2.896 3.355 23 1.319 1.714 2.069 2.500 2.807 9 1.383 1.833 2.262 2.821 3.250 24 1.318 1.711 2.064 2.492 2.797 10 1.372 1.812 2.228 2.764 3.169 25 1.316 1.708 2.060 2.485 2.787 11 1.363 1.796 2.201 2.718 3.106 26 1.315 1.706 2.056 2.479 2.779 12 1.356 1.782 2.179 2.681 3.055 27 1.314 1.703 2.052 2.473 2.771 13 1.350 1.771 2.160 2.650 3.012 28 1.313 1.701 2.048 2.467 2.763 14 1.345 1.761 2.145 2.624 2.977 29 1.311 1.699 2.045 2.462 2.756 15 1.341 1.753 2.131 2.602 2.947 1.282 1.645 1.960 2.326 2.576 ¡ dd ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 257 Assumption : POPULATION approx. NORMALLY dist. Small sample situation similar to large, except use d.f. instead of dist. Confidence Interval : ¡ © a1 a1 Small Sample (p. 292): © Large Sample: 8 6 sq h tr! 8 6 sq 3 tr! dist. with Q%8 ¢ § ¡ ¡ d td ¦ ¡ t ¡ ¡ Small Sample Inferences About ¡ § s d¦ ¡ E £ § s d¦ € PEr€x£ § s d ¦ H   EPr€x£ § s d ¦ H £ § s d¦ £ § s d¦ ¡ ¡ ¢ ! £! ¡ ¡ ¡ ¡ ¤ ¡ ¡ £ Note : When d.f. d td df= ¡ df=30 ¡ df=20 § df=10 ¡ df=5 .025 ¡ 256 d td STA 2023 c B.Presnell & D.Wackerly - Lecture 21 ¦ d.f. d.f. § ¨¦ 254 STA 2023 c B.Presnell & D.Wackerly - Lecture 21 258 STA 2023 c B.Presnell & D.Wackerly - Lecture 21 STA 2023 c B.Presnell & D.Wackerly - Lecture 21 259 Ex. Phosphorus content is a water quality index that is of concern to the EPA in the Everglades. In one section d T¡ ©§¥ ¡ £ ¨¦ (' ppb? Use €Q `D Hv S% £ ¢ 6 €% `D Hv QSI£ £ e8 ¡2% 4 h a 1 £ d Hr £ h PDy€ Q £ a 1 H ¢  ¦¥ § £ £ ¨d ¦ £ ! £8 ¢  Q £¤¡ ©¦§¥  Q ¡ ¨  ¤¢ # ¨ ¥ PEIGF H£ . % ! w# ¡    Rejection Region: Lower tail test. ) ¡   ¡ d.f. , ¡ ! , .  ¡ 0 reject if Test statistic: ¡ Conclusion: Since in the rejection region, CANNOT reject Ho . There is evidence to conclude, at the ¡ F level of significance, that the mean level of ¡ !¢ ¡ ¡ 260 Q sq t$! 8 64h d¡ e2% a 1 £ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 ppb.  ¢ phosphorus is less than PEH F £ ¡ (like before) AND #d.f. is   ¡ s tqr!  sq % t$! &# ¡ ¡  deV¡  ¡ U£    d G¡   §¥ ¡#  d¡  eG¡  ¡ ¡ or RR Q (new) mean level of phosphorus is less than (looks just like !!) ) depends on (measurements in parts per  OR (and and billion [ppb]). Can the EPA support the claim that the OR Test statistic : measurements, obtaining of park, EPA makes PDy€ Q £ a 1 H versus H r £ h € Q £ 8 ¢ Hypothesis Tests (p. 342) STA 2023 c B.Presnell & D.Wackerly - Lecture 21 261 Ex. Give a 95% CI for the mean phosphorus index in the section of the Everglades ¢ % 0   ¡ (tail area) p-value   % score      € Q% `D Hv SI£ ¢ ¢ d uU £ ¡ ¡  de¡ ¡  #  ¨  ¥   d ¡ ¡   OR ) ¡ OR score ('  € H r £ h £ § s d ¦ £ t$! sq Dr€ Q £ a 1 H ¡ ¡ €Q £ 8 £ GF 95% CI is How about -values? £ 8 6 tr! h sq £ ¡ © a1 Table does not allow us to get exact p-values. assuming that population from which the sample is In Everglades example, lower tail test, d.f. ¡   £ are (approx) normally distributed ¡ That is, that € Q% `D Hv S&# p-value   taken is (approx) normally distributed Q `Q £ Note: In last example (both test and CI), we are , STA 2023 c B.Presnell & D.Wackerly - Lecture 21 263 ¢ ¡ ¢ ¡  and ¢ Ho . Ho . ¢ £  £ ¢ From the table, ¢ From the table, Can’t be any more precise using these tables! DH £ CPEIGF B  EH £ PIGF B  EH £ ¡PIGF B PEIGF B H£ E Q I£GF B H  Q HI£GF B €7D Hv Q CQ £ Q € HQ  € Q % `D v Cw  £ `D Hv S&#   are ) to # PEH  Closest values in table ( with d.f. p-value E Q &# H Look at table, Thus, in this case, best we can say is that ¢ 262 STA 2023 c B.Presnell & D.Wackerly - Lecture 21 Ho . Ho . £  C¡PDH S&#  D Q %  Q Hv w  £   Hv Sw# Q% ¡ ¡   ¡ ...
View Full Document

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.

Ask a homework question - tutors are online