Week13-2up - 230 8.55 – 8.57 Thursday Exer 7.27 7.30 7.33...

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Unformatted text preview: 230 8.55 – 8.57 Thursday: Exer. 7.27, 7.30, 7.33, 8.49, 8.50, 8.53, P. 341 – 345 (Sec. 8.4) Wednesday: P. 288 – 294 (Sec. 7.2), 8.61, 8.67–69 Tuesday : Exercises 8.29, 8.33, 8.34, 8.38–41, 8.59, Today: P. 334–338, 347–351 Assignments readily if you tell them that Benjamin Franklin said it first. Thought: People will accept your ideas much more STA 2023 c B.Presnell & D.Wackerly - Lecture 20 UNKNOWN population mean Last Time: Large Sample Hypothesis Testing about STA 2023 c B.Presnell & D.Wackerly - Lecture 20 ¢ ¢ Test Statistic  ¡ £ ¡ ¨¦¤ £   OR ¡ estimator RR standard error Hypothesized Value from NULL hypothesis Sheet 231 hypothesized value Estimator and Standard Error from Formula £ ¢ ¥ ¦¤ § 2  ¥¡ ! ¥¡   £ ¥¡ &  " ©  ¡ '( " 0) ¥¡ £ 22 1 1  ! $  # %    "  ¡ 11 water specimens -z α/2 ¨¤ ¥ ¦¤ §  0 α/2 z α/2 §  ¢ ¨¦¤ ¦¤ ¥ § α/2 ¡ ! or level? “Two Tailed test” (p. 331) "  How? £ ¥¡ § ¨ ¦¤ § §¨  ¡ £ £¦ &© ¥ ¡ '( " 0 ) ¡  ?  and ¥¦¤ § pH is NOT that of neutral water at the ¥ from a recreational lake. Can we claim that the mean indicates acidity. Randomly select Ex. : pH of 7 is neutral, over 7 is alkaline, under 7 ¡  &©  ¢ £  Back to pH example: STA 2023 c B.Presnell & D.Wackerly - Lecture 20 0¢ 232 at the the not .” 233 level to indicate that the mean pH reading is enough evidence at level of significance. In terms of this application: “There Reject RR : level test: ¥ ¡ £ ¤ (" 0) ¡  STA 2023 c B.Presnell & D.Wackerly - Lecture 20 £ ¢ ¢ ¢ £ ¡  £¡ £ £ ¤ £  £ ¡ £ ¡ ¥   ¥ ¥¤  £  £ &© ¡ £  £ If the mean pH is NOT , what is it? ¢ ¥ Construct a ££ ¢ £  confidence interval for ? . to reject ? Recall the lightbulb example (P. 335) The p-value or observed significance level be rejected in favor of could ( IF we DO for which in favor of CONFIDENCE in our . What is the SMALLEST value of SO). decision to reject – Provides – Smaller is chosen BEFORE the test is performed Hypothesis Testing STA 2023 c B.Presnell & D.Wackerly - Lecture 20 Agrees with two-tailed test!! the 99% 234 confidence interval — the value “ ” is Do you think that £ " ¡ & ¢ £ ¡ ¥ ¦¤ § ¨¦¤ § ¢ £ ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 20 £ ¢  ¢ ¡ § £  0) § ¡ ¨¤ ¢ ¢ ¢ ¢  §¨  ¨¤ ¢ ¤ ¥  ¦ ¥¤ ¥¤ ¡ ¡  ¢ ¥ £ ¥  ¥ £ £ £ ¢ © © ©  ¥¤ 235 p-value £ p-value = true. Larger z-values are the one observed indicative that Probability of a z-value ¢ ¢ rejection region STA 2023 c B.Presnell & D.Wackerly - Lecture 20 ¨¤ ¥¦¤ § ¡ ¨¦¤ § ¡ £ ¡  ¢ ¥ £ £ £ £ £ £ © © ¥  ¢ ©¡ ¡ ¡ ¡ ¡   ¨  ¥ £ is 236 p-value p-value . . . . . . p-value allows him/her to assess the “rareness” of the observed event. 237 on a person who might be interested in your conclusions, the Instead of “imposing” YOUR CHOICE of – – – – REJECT In our case, p-value = .0256 – . CANNOT reject REJECT STA 2023 c B.Presnell & D.Wackerly - Lecture 20 ¢ ¢  £    £  ¥    ©¨ ©¡ ¢ ¢ ¢ ¢    ¥ ¡ ¡ £ ¥ ¥ ¤ ! ¢ ¢ £ £ £ £ £ £ £ £ £ ¢ ¥ ¡ ¡ ¡ £ ¢ ¥¤ ¥¤ ¥¤ ¥ ¦¤ ©¡  ¥ ¥ ¥ ¥ ¥ ¥¤ ¥ ¦¤ ¥ ¦¤ ¢ ¥ ¦¤ § p-value for any that is for any ¢ Smaller z-values are more indicative that ¡ ¢ the one observed that is ¡ . is true. . – ? value = EX. : Have done a two-tailed test: and DOUBLE IT. claim that with . . Find the area in whichever “tail” the -value is in TWO - Tailed Test STA 2023 c B.Presnell & D.Wackerly - Lecture 20 ¢ ¡  £ £ ¥ ¦¤ ¢ Probability of a z-value ¥ ¦¤ ¢ ¥¡ ! ¥¡ £ ¡ ¨¦¤ § ¢  £ " £ p-value ¢ ¤ £ ! £ 238 ¡ Ex. : #8.24, P. 333 ¡ ¡ ¢ £ ¢ ¢ " £ ¤ ¨¤ ©¡ ¥ ¦¤ § ¡  £ ¥ £ ¦ ¨ ¦¤ § ¡ £ £ ¦  ¥  ¥ ©  STA 2023 c B.Presnell & D.Wackerly - Lecture 20  ¢ £ £ ¥ £ ©¡ 239 true proportion of Diet Coke drinkers who How??? would select Diet Pepsi in a blind taste test. select Diet Pepsi in a blind taste test? indicate that a majority of the Diet Coke drinkers will £ (2) (1) . Estimate for # of trials ; number of trials in the sample size in the sample # of GOAL : Test hypotheses about trials based on a “large” the proportion of batteries that fail before select Diet Pepsi in a blind taste test. the proportion of Diet Coke drinkers who guarantee expires. 241 (Section 8.5) Recall the BINOMIAL EXPERIMENT. particular attribute 2 2 the taste of Diet Pepsi. Is there sufficient evidence to ¨¤ ¥¤ © § § £ £ £ indicated that they preferred £ Coke and Diet Pepsi. ¢ ¢ 0£& UNKNOWN but FIXED PROPORTION of items with a ¡ ¢ ¡ ¢ Coke drinkers were given unmarked cups of both Diet £ ¨ ¢ ¢ £ Interested in a POPULATION that contains an Large Sample Tests About STA 2023 c B.Presnell & D.Wackerly - Lecture 20 ¤¡ & £ ¨ ¢ 02 ¢ ¢ ¨ Ex. : #8.68, p. 352 In a “Pepsi Challenge”, 100 Diet 240 STA 2023 c B.Presnell & D.Wackerly - Lecture 20 0 £ ¢ 0 2 £ distribution has an ¨¤ ¥¤ § versus £ Consider testing distribution. a fixed particular value of has an approximate That is OR OR ¡ 2 is “large” If £ STA 2023 c B.Presnell & D.Wackerly - Lecture 20  ©   2 £ ¡ ¢ ' ¡ £  ¥  ! ¦ £ £ £ 0 £ ' ¡ 242 STATISTIC if hypothesized value standard error is true Rejection Regions (RR): , OR OR or has a STANDARD Hypothesized Value from NULL hypothesis Sheet Estimator and Standard Error from Formula estimator NORMAL distribution If is the null hypothesis, TEST STA 2023 c B.Presnell & D.Wackerly - Lecture 20 ¢ ¢ ¨¦¤ £ ¥ ¥  ©   ¥ ¥¤ " ¥ 2  £  ! ¢ ¥¦¤ §  ¥ " ¥ £ ¥ ¢ 0 ¥ £   £   ¥ ¦    22 " 1 ¥ £ ! ¥ ¥      ! "      "     ¤ 1 £ £ 11 243 RR $  # %  indicated that they preferred true proportion of all voters who think health ¢ care reform is the leading priority Data : ¡ ¡¥ SAMPLE of all Diet Coke drinkers. Note: the the Pepsi Challenge are a £ (4) (3) 244 is “large” Assumptions : the 100 individuals participating in level test, RR : £ ¢ ¢ 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. © Coke drinkers were given unmarked cups of both Diet Ex. : #8.68, p. 352 In a “Pepsi Challenge”, 100 Diet ¢ £ ¤ £  ¢ £ ©¡ 0 £ £ reject AT THE level of significance” ( or with Coke drinkers will select Diet Pepsi in a blind taste value? test. value = 245 claim that there is sufficient evidence at in favor of confidence ) to indicate that the majority of Diet the “ In terms of this problem: LEVEL!! Conclusion : STA 2023 c B.Presnell & D.Wackerly - Lecture 20 ¢ ¢ ¨¤ ¥¤ © £  § § ¡ £¤£ © £ ¡ ©¡ £ £ ¢ £ STA 2023 c B.Presnell & D.Wackerly - Lecture 20 0 £ ¥¤ ©¡ £ ¤ " " ¢ ¢ ¨¤ © ¡ 9.22, 9.24, 9.25 For Tuesday: Exer. 9.1, 9.7, 9.13, 9.15, 9.16, 9.19, Monday : P. 374 – 383 (Sec. 9.1, “Large Sample) 8.55 – 8.57 For Thursday: Exer. 7.27, 7.30, 7.33, 8.49, 8.50, 8.53, P. 341 – 345 (Sec. 8.4) Today: P. 288 – 294 (Sec. 7.2) Assignments knows it all is the person who argues with him. param. OR param. OR value value value value RR standard error score (tail area) smaller hypothesized value or Hypothesized Value from NULL HYPOTHESIS Sheet 247 score larger p-value Estimator and Standard Error from Formula estimator Test Statistic param. ¢ " param.  Thought: The only fool bigger than the person who ¡ values for Large Sample Tests ¢ Last Time: ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 £ ¢ ¥ ¦¤ § 2  ! £  £   1 " ¦   !  "   ¢   "    246 11 ! £ " ¨¦¤ § 22 1   " STA 2023 c B.Presnell & D.Wackerly - Lecture 21 £   ¢ ¢ 4 12 32 48 47 50 .5 .6 .7 .8 .2 .1 reject 0 3 2 18 38 46 not reject 50 50 50 50 50 50 tests 1.00 .94 .96 .64 .24 .08 Prop. rejects ©¨ £ Sample size for each test is ¥¤ £ ¤ © £ £ ¥¡ ¥  £ ¥¤ ¢ ¥¦¤ § £ © £ £ 0 £ ¨¦¤ § ©¨ ¦ © £ !  £" © £ ¥ " 50 48 21 6 0 2 29 44 not reject 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. 249 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 .5 ¥¡ or ; RR : ¡ £ £ © ¥¤ © ¥¤ 0 ¤ reject ¢ Computer Study: ¢ £ £ Sample size for each test is £ ¤ £ 0 ¤ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 0 248 ¥¤ ¡ £ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 £ ¢ ¥¤ ¥¡ ¦ © £ £ ¢ ¢ ¡ 0 ¥¤ 0 ¥ &© £  £ and (measurements in parts per measurements, obtaining £ ¤ £ ¢ ©¡ ¢ ¢ ¨¦¤ § Sample size  ¢ ¥ . £ mean level of phosphorus is less than is small! How??? ppb? Use billion [ppb]). Can the EPA support the claim that the 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  0 £© ¥ £ ¨ £  ¥¤ § 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: scores does not have a standard normal dist. can’t use of the sampling distribution of can’t use CLT to get NORMALITY Small Sample Inferences about STA 2023 c B.Presnell & D.Wackerly - Lecture 21 ¡ 250 ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 £  ( &© 0) " ¡ 0  ¥© 0 ¡ !  ( &© 0) " ¡ " ¥ " ¢  " ¥ ¥ & 251  ¢ ¢   ¡ -2 0 2 4 ¢ ¡ ¢ -4 £ ¢ Table VI (p. 811) gives so that so that (Remember: £ ¤ Std Normal Define used in its calculation. ¢ t with 2 df t with 8 df : -values for ) has the same number of d.f. at the estimator for ¡  ¥¡ £ as d.f. . ¡ ¢© – Becomes more and more like the -distribution £ ¢ Thus ¡ as d.f. . ¢ ¡ – Variability ¥ – Variability depends on degrees of freedom. ¢ © ¡ ¢  £  ¡  ¡ More variable (heavy-tailed) than the -distribution  Note: d.f. = denominator in calculating ¡ ¢© ¡ £ ¡  ¢  ¡ £  £ Bell-shaped . (like the -distribution) £ Properties of the -distribution: Symmetric about 0. (like the -distribution) £ ¡ ¢ £  ( &© 0) " ¡ £ ¡ £ ¡ " ¥ ¢ 0 & " &©  ¢  ¡ and £ ¥ ¢ ¢  ¢ ¡ £  STA 2023 c B.Presnell & D.Wackerly - Lecture 21 ¡ 252 ©¡ £ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 253 ' 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372 1.363 1.356 1.350 1.345 1.341 3 4 5 6 7 8 9 10 11 12 13 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 ¡ 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  ¡ 1.699 1.645 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 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 1.333 17 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 16 1.746 ¡ 1.337 d.f.  STA 2023 c B.Presnell & D.Wackerly - Lecture 21  1.886  6.314 254 ¤ ¦ ¡ 2 3.078  1 d.f.  ¦ ¤ ¡  ¤ ¡  ¦ ¤  ¤  ¤ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 ¡  ¦ ¤ ¡  ¤ ¡  ¦ 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 ¤ 255 ¡ ¡ Note : When d.f. df= df=30 df=20 df=10 ¡ £ t 257 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. Small Sample Inferences About STA 2023 c B.Presnell & D.Wackerly - Lecture 21 0 ¡  ¦ ¤ ¡  ¦ ¤ ¡  ¦ ¤ ¡ £ £ ¡ £ £ £ £  £  ¦ ¤ ¡  ¡ df=5 .025 256 ¢ ¥ "  ¦ ¤ £  ¡ ¤ ¢ ¡  &© § § &©  ¦ ¡      ¨  0) '  0) STA 2023 c B.Presnell & D.Wackerly - Lecture 21 ¡ versus OR OR (new) (and Test statistic : ¡ Hypothesis Tests (p. 342) ¡ £ ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 ¡ ¢ ©  ¡ ¡ ¡ ¡ ¡  ¢       ¡ ¨¤  ¡ ! ¡   ¡ ¦  ¡ or ) depends on (like before) AND #d.f. (looks just like !!) RR 258 259 (measurements in parts per measurements, obtaining . mean level of phosphorus is less than if Test statistic: reject d.f. , Rejection Region: Lower tail test. is 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 . ppb? Use 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 B.Presnell & D.Wackerly - Lecture 21 &© ¢ ¡  ¡ ! ©¡ £   £  ¡ ¥¤ § £ &©  (" 0 ) ¡ ¡  ¡ " ¡ ¥¤ ¡ ¡ "   ¥© ! £ ¤ ¢ ¢ ¥¤ £ £ ¡ ¡  ¥ £  ¡ § ¡ ¤ ¦ &©  £ &©  (" 0 ) ¡ ! ¢  £ ¡ ¥ £ ¥ £ " £ £  £     £ 0 £© ¥ £ ¨ £ ¢ ¨¦¤ § ¢ ¥© 0 $  # %  ¥© £ £ £ ¥©  ¢ £  £ £  " ¥ £ £© ¨ ©¡  0 &© § ¡ £ are (approx) normally distributed That is, that ¢ ¢ ¡ ¡ ¥ £ 0) £    £ £ &©   ¥ £  £  ¦ ¨  £© £ taken is (approx) normally distributed ¡ score score (tail area) In Everglades example, lower tail test, d.f. ¡ assuming that population from which the sample is ¡ Table does not allow us to get exact p-values. £ " ¥ £  p-value Note: In last example (both test and CI), we are OR ¦ ¨¦¤ § OR  95% CI is £ ¤ How about -values? ¢ ¡ the section of the Everglades ¢ £ STA 2023 c B.Presnell & D.Wackerly - Lecture 21  ¡ Ex. Give a 95% CI for the mean phosphorus index in 260 £ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 £ ¥ £ ¢ ¢ ! " ¥ £  £ ©  ¡  ! ¡ ¡ £ ¡   ¡  ¡   ¡ " ¡ "  , 261 p-value $  # %  ¥ ¡ ! ¥ ¢ From the table, ¡ From the table, ¡ " £  ¡ ¢ and ¢ £ ¢ ¡ ! £ ¡ are ¢ Closest values in table ( with d.f. ¡ ) to £   ¢ Look at table, ¡ !    £ ¢ £ ¢ £ " ¥ ¡ " ¢ £  £  ¥ £ £  ¥ ¥ £ ¡ ¥ £ £ ¥  £ ¥ £ 262 Ho . Ho . Ho . Ho . Can’t be any more precise using these tables! p-value Thus, in this case, best we can say is that STA 2023 c B.Presnell & D.Wackerly - Lecture 21 £ ©¡ 2 2 2 2 2 STA 2023 c B.Presnell & D.Wackerly - Lecture 21 ¢ ¢ 2 ¢ ¢ ¢ ¢ £ £ £ £ £ £ £ £ ¢ ! £ ¢ ! £ £ £ £ ¥© ¡ ¥¡ ¡ ¡ ©¡ ¥¡ 263 ...
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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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