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

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Unformatted text preview: STA 2023 c B.Presnell & D.Wackerly - Lecture 20 230 Thought: People will accept your ideas much more readily if you tell them that Benjamin Franklin said it first. Assignments Today: P. 334–338, 347–351 Tuesday : Exercises 8.29, 8.33, 8.34, 8.38–41, 8.59, 8.61, 8.67–69 Wednesday: P. 288 – 294 (Sec. 7.2), P. 341 – 345 (Sec. 8.4) Thursday: Exer. 7.27, 7.30, 7.33, 8.49, 8.50, 8.53, 8.55 – 8.57 STA 2023 c B.Presnell & D.Wackerly - Lecture 20 231 Last Time: Large Sample Hypothesis Testing about UNKNOWN population mean ¤ ¨¨      ¨¨ ¤  RR      ¤  ¢ ¡ ¢ ¥ ¤ £ ¡ §¨¨ ¢¦£ ¨¨ OR © Test Statistic hypothesized value % % standard error $ ¢ ¤  #" ! Estimator and Standard Error from Formula ¢ ' Sheet Hypothesized Value from NULL hypothesis % &% estimator ' (' ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 20 232 Ex. : pH of 7 is neutral, over 7 is alkaline, under 7 ¡ ¢ indicates acidity. Randomly select water specimens from a recreational lake. Can we claim that the mean ¢ £ ¦£ and ¥ ¦¡ ¤ pH is NOT that of neutral water at the level? ? ¥¤£ ¤£¡ ¥¦£ How?       !     #  $ $ ¢ ¥ ¤ £ ¤ §¢ ¥ ¦ £ ¤ ©¨ ¢ ¥ !   ¥     #" α/2 or α/2 -z α/2 0 z α/2 “Two Tailed test” (p. 331) STA 2023 c B.Presnell & D.Wackerly - Lecture 20 233 Back to pH example: ¤ ¢¤ ¡ ¡ ¤ ¢ ¢ ¢£¡ ¥ ¦¡ ¤ ! ¢$¡ ¡ level test: RR : ¢ ¥ ¦¡ ¤ at the ¢ ¤£ Reject level of significance. In terms of this application: “There ¥ ¦¡ ¤ the not .” enough evidence at level to indicate that the mean pH reading is ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 20 234 If the mean pH is NOT , what is it? ¢ £¡ confidence interval for . ¢ ¨  ¡ ¤ §¦ ¢ ¡ £¤ Construct a ¢ £ ¥¡ ¤ ¤ ¢ £ ¥  ©! $ ¥ ¡¡ ¤  ¢ © ¤ Do you think that ? ¡ — the value “ ” is confidence interval Agrees with two-tailed test!! the 99% ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 20 235 Hypothesis Testing £¡ is chosen BEFORE the test is performed to reject CONFIDENCE in our decision to reject ¤£ – Provides . in favor of ¦£ ¡ – ¤£ £ Smaller ( IF we DO SO). ? £ ¦£ be rejected in favor of for which ¤£ What is the SMALLEST value of could The p-value or observed significance level ¡ (P. 335) Recall the lightbulb example ¢ ¥ ¤ £ ¡ ¥  ¥ ¦ £ ¡ ¥ ¢ ¡ ¤ ¥ STA 2023 c B.Presnell & D.Wackerly - Lecture 20 rejection region ¤ ¥  £  ¡¤ ¡¤  ¡ ¡¤ ¡¤   ¡ ¥ ¡¤  ¡  ¡ ¢ ¤ ¡ ¡¡ ¤ ¡ ¥ ¡ ¥ ¢ ¥ ¤ £  ¥ ¦ £ ¢ p-value 236 Probability of a z-value the one observed indicative that ¡  ¢ ¡ p-value = £ true. ¦£ Larger z-values are is ¢ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 20 . . ¤£ CANNOT reject . ¤£ ¡ ¡ p-value ¤£ £¡ REJECT ¤£ p-value 237 . £¡  In our case, p-value = .0256 ¤£ ¡ ¢¡ ¤ REJECT . . . ¡ ¡ ¡ ¡ ¡¤ ¥ ¦¡ ¤ ¢£ ¢£ ¢£ ¢£ Instead of “imposing” YOUR CHOICE of £ ¡¡ ¤ ¤£ – ¢£ – ¡ – ¥¤ – ¤£ ¡ – on a person who might be interested in your conclusions, the p-value allows him/her to assess the “rareness” of the observed event. STA 2023 c B.Presnell & D.Wackerly - Lecture 20 238 Ex. : #8.24, P. 333 ¡¥ ¢ ¥ ¤ £ ¡¥ ¢ ¤  ¥ ¦ £ ¢  ¡ ¢ p-value Probability of a z-value the one observed ¡ ¡ p-value ¦£ Smaller z-values are more indicative that is true. ¢ that is  £ £ for any . ¡ ¡ ¡ ¤ that is ¡¢¡ ¤ ¡ for any . ¤£ ¤£ ¡ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 20 239 TWO - Tailed Test  ¡ Find the area in whichever “tail” the -value is in and DOUBLE IT. EX. : Have done a two-tailed test: ¢ ¥ ¤ £ ¥ §¢ ¥ ¦ £ ¥ ¤ ¥ ¢ . ? with ¢£ ¥ §¢ claim that ¡¡ ¤ ¡¡ ¤  ¡ ¢£¡ – value = . STA 2023 c B.Presnell & D.Wackerly - Lecture 20 240 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) How??? ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 20 241 Large Sample Tests About (Section 8.5) Interested in a POPULATION that contains an UNKNOWN but FIXED PROPORTION of items with a £© £ ¡ ¢ ¢ particular attribute . ¡ Recall the BINOMIAL EXPERIMENT. ' ¢ the proportion of Diet Coke drinkers who select Diet Pepsi in a blind taste test. the proportion of batteries that fail before ¢ ' guarantee expires. ¡ GOAL : Test hypotheses about based on a “large” number of trials £© ¡ ¢ ! in the trials in the sample sample size ¤ £© ¡ ¢$' ¢ ! $ Estimate for # of $ # of trials ; ¢¤ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 20 $ is “large” ¤ If 242 has an ¡ ' distribution ¢ £ ¢ $ ¤ ¡ " ¢ ¡ ' ' That is ¢ ¡ " has an approximate distribution. a fixed particular value of ¤  ¤ ¨¨ ¨¨ ¤    ¢¦£ ¨¨ ¨¨ ¨¨ OR © §¢ ¢ ¨¨ OR  §¨¨  ¤ ¨¨ ¥¤£ versus Consider testing STA 2023 c B.Presnell & D.Wackerly - Lecture 20 ¤ ¢ ¥¤£ ¡ if 243 is the null hypothesis, TEST STATISTIC % hypothesized value % ¢ standard error % &% estimator Estimator and Standard Error from Formula ' Sheet Hypothesized Value from NULL hypothesis ¤  $ ¤¢¤ £ ¤ ¢ ¡ ¢ ¢ is true ,  ' (' ¤£ If has a STANDARD ¡ NORMAL distribution Rejection Regions (RR): ¨¨  ¨¨  ¨¨ ¨¨  ¤  ¨¨ §¨¨ ¨¨  ¨¨ ¨¨ ¨¨ OR ¨   RR ¨¨  ¤   ¨ ¨¨ ¨¨ or ¨¨ ¨¨  ¨ ¢¦£ ¨¨              ¤ ¨¨ OR ¨¨ §¢ ¨¨ ¨¨ © ¨ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 20 244 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 all voters who think health ¢ ¡ care reform is the leading priority ¡ ¤ ¡ ¤ (3) (4) ¥¦£  ¢ ¥¤£ level test, RR : ¡¡ ¤ ¢£¡ ¡ Assumptions : the 100 individuals participating in the the Pepsi Challenge are a ¢¤ ¢ ¢ ¡¤ ¢ ¡¡¥ ¢$ Data : $ SAMPLE of all Diet Coke drinkers. Note: is “large” ¢ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 20 245 ¡ Conclusion : in favor of ¦£ ¤£ reject AT THE LEVEL!! ¡¡ ¤ ¢£ ¡ In terms of this problem: 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 B.Presnell & D.Wackerly - Lecture 21 246 Thought: The only fool bigger than the person who knows it all is the person who argues with him. Assignments Today: P. 288 – 294 (Sec. 7.2) P. 341 – 345 (Sec. 8.4) For Thursday: Exer. 7.27, 7.30, 7.33, 8.49, 8.50, 8.53, 8.55 – 8.57 Monday : P. 374 – 383 (Sec. 9.1, “Large Sample”) For Tuesday: Exer. 9.1, 9.7, 9.13, 9.15, 9.16, 9.19, 9.22, 9.24, 9.25 STA 2023 c B.Presnell & D.Wackerly - Lecture 21 values for Large Sample Tests value ¥¦£ larger ¢  ¡   value p-value  ¥¤£¡ param. RR score £ param. ¢  Last Time: 247 ¢   ¡  value smaller   param. score OR or ¡      (tail area)      value §¢ param.  Test Statistic hypothesized value % % standard error % &% estimator ¢ Estimator and Standard Error from Formula ' Sheet Hypothesized Value from NULL HYPOTHESIS £ OR ' (' ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 248 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 B.Presnell & D.Wackerly - Lecture 21 249 ¢$ ¤£ ¤£ 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! approx. . a . Big $ , we REJECT $ ¤ §¢ £ ¢ ¡ ¢¡¥ greater percentage of the time for larger is “better”. ¤£ ¢ ¢ For each fixed value of ¡ ¥¤ ¤ of the time. we reject £ $ , for each ¤£ When ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 250 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 ¥ mean level of phosphorus is less than . ¥¤£ ¡¡ ¤ ¢£ £ ¥¦£¡ ¡ ¥ ¢ Sample size is small! How??? ppb? Use ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 251 Small Sample Inferences about ¡ ¢ can’t use CLT to get NORMALITY  can’t use ! ¡  $ of the sampling distribution of scores ¡ does not have a standard normal dist.  $ # ! However: If the POPULATION is approximately NORMALLY distributed  (looks a lot like !!!) $  # ! ¢ ¡ has a sampling distribution called the $  ¡ d. f. ¥ distribution with “degrees of freedom”, ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 252 ¡ 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 B.Presnell & D.Wackerly - Lecture 21 253  :   £ ! ¥ ! $  ¢ ¢  ¡  # ! $ ¢ ¡ has the same number of d.f. at the estimator for used in its calculation. £ ¢ £ £   ) ¢ -values for and ¤ ¡¡ ¢¡ ¤ ¡  ¡ ¡ ¢ ¡ ¡ so that    ¡ ¥ ¢¡ ¤ ¤ ¢¡ ¤ ¡¦¡ ¤ ¤ ¡ ¡ ¤¥ ¡ ¡ ¡ Table VI (p. 811) gives  (Remember: £ so that ¢ Define " ¡ Thus  Note: d.f. = denominator in calculating ¡¤ ¤ ¡¤ ¤ ¡ ¥¤ ¢£ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 § ¨  ¦ ¡  ¦ ¡ §  ¦ ¡  § ¦ ¡   ¦ ¡ d.f. 254 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 B.Presnell & D.Wackerly - Lecture 21 255 §  ¦   ¦ 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. ¡ ¡  § ¦ ¡ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 256 .025 t ¡ ¡ ¢ §  ¦ ¢ §  ¦ ¡ ¢ §  ¦ ¤ ¡¡ ¡ ¢ §  ¦ ¤  ¡ ¡ df= ¢ §  ¦  ¢¤ ¡ ¡ ¡ ¡ ¢ Note : When d.f. ¡ ¡ df=30 df=20 df=10 §  ¦ df=5 STA 2023 c B.Presnell & D.Wackerly - Lecture 21 257 Small Sample Inferences About Assumption : POPULATION approx. NORMALLY dist. ¡ Small sample situation similar to large, except use dist. ¥ $  © ! ¡    © ! Small Sample (p. 292): $     Large Sample: " Confidence Interval : $ d.f. instead of  dist. with ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 258 ¡ Hypothesis Tests (p. 342)   ¢ ¥ ¤ £ versus ¨¨  ¨¨  ¨¨ ¨¨  ¡   ¡ ¨¨ §¨¨ ¨¨  ¨¨ ¨¨ ¨¨ OR ¨  RR ¨¨ ¡    ¡ ¨¨ ¨¨ ¨ or ¨¨ ¨¨  ¨ ¦£ ¨¨     ¡  ¡   ¨¨  §¢ ¡ ¨¨ OR ¨¨ ¨¨ ¨ © ¡ ¡ Test statistic :    ¢ !    ) depends on (looks just like !!) $ # ¡ ¡  ¡ (new) £ (and (like before) AND #d.f. ¡ 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 ¥ ¢$ ¤ ¢ ¤ ¡ ¥ and ¡ of park, EPA makes measurements, obtaining (measurements in parts per ¢ ! billion [ppb]). Can the EPA support the claim that the ¥ mean level of phosphorus is less than ¡¡ ¤  ¥ ¦ £ ¡ ¢£ . ¥ . ¢ ¡ ¢ ¡ ¤£ ¡ if ¤ ¡ ¥ ! ¢ ¢ is ¤ ¥  ¢ ¡ ¤ ¥  ¢   $ # ! ¢ in the ¡ rejection region, CANNOT reject Ho . There is level of significance, that the mean level of ¥ phosphorus is less than ppb. ¢£ evidence to conclude, at the ¡¡ ¤ ¡ ¡ Conclusion: Since ¡ ¢ Test statistic: ¤ ¡ reject , ¢ ¨ ¦ § , ¢$ ¥ ¢ ¥ ¤ £ Rejection Region: Lower tail test. d.f. ppb? Use ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 260 Ex. Give a 95% CI for the mean phosphorus index in the section of the Everglades ¤ ¤ ¡ ¥ ¢ ¢ §  ¦ ¡ ¢  ¢   ! ¥ ¡ ¢$ ¢£ ¡ 95% CI is ¢  $ ¡   © ! ¢ 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 STA 2023 c B.Presnell & D.Wackerly - Lecture 21 261 How about -values? ¨¨  ¨¨ £ ¨¨  score ¨¨  ¡ ¢ ¡ §¨¨ ¨¨  ¨¨ ¨¨ OR p-value ¨¨ £ ¨¨  score   ¡ ¢ ¡ ¨¨ ¨¨ ¥ ¦ £  ¨¨ ¡ ¨¨ ©  §¢ ¨¨ OR (tail area) ¨¨ ¡ In Everglades example, lower tail test, d.f. ¢ £ ¡ ¡ ¤ ¥   ¡ ¢ ¡ ¢ p-value ¥¥ ¡ Table does not allow us to get exact p-values. , ¤ ¥  ¢ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 ¤ ¥ ¢ ¡ ¥ ¢¥ ¢ ¡ £ ¤ ¥   ¡ ¡ ¢ ¡ ¡ ¢ ¤ ¥ £ ¢ ¢ ¡ ¤ ¥  £  ¢ ¡  £ ¡ ¡ From the table, ¡ ¢ ¡ From the table,  and ¤ ¥ are ) to ¡ Closest values in table ( with d.f. £ Look at table, 262 ¤ ¥   ¡ ¢ ¡ ¡ ¢ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 21 263 Thus, in this case, best we can say is that ¡ ¥¤  p-value  ¡¡ ¤ Can’t be any more precise using these tables! Ho . Ho . ¡ ¥¤ ¥¤ ¡¤ ¡¤ ¡¡ ¤ Ho . ¢¡ ¤ Ho . ¢£' ¢£' ¢£' ¢£' ¢£' ¢£' ¡ ¡ ...
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