Week10 - STA 2023 c D.Wackerly - Lecture 17 224 Thought: A...

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Unformatted text preview: STA 2023 c D.Wackerly - Lecture 17 224 Thought: A committee is a group of people who, individually, can do nothing, but collectively can meet, and decide that nothing can be done. Assignments Today : P. 328 – 332 For Tomorrow : Exercises 8.18, 8.21–23, 8.25, 8.27 Wednesday : P. 334 – 338, 347–351, Thursday : Exercises 8.29, 8.33, 8.34, 8.38–41, 8.59, 8.61, 8.67–69 Monday 11/4: OPTIONAL review Tuesday 11/5: EXAM 2 – During your DISCUSSION SECTION STA 2023 c D.Wackerly - Lecture 17 225 Last Time: to within “B” units with ¨¨¢ ©§¦ ¥ ¤ £¡ ¢ Estimate confidence. if you have one, . 1) 20¢ $ # ! "    ' $ ( $ otherwise use % Use “ballpark” value for . (p. 309) & and solve for Parts of a statistical test. (p. 322) ' The hypothesis of MAIN INTEREST is the ALTERNATIVE or RESEARCH hypothesis, – 9 7 86 4 , ¦B1@ ACA"¢ 3 5¡ 4 3 light bulb ex. . (p. 322) What we are “trying to prove” in an objective, fair manner The “other” hypothesis is called the NULL D 3 HYPOTHESIS, – , light bulb ex. (p. 322) ¦B1@ ACA©¢ $ 7 86 D 3 5¡ ' STA 2023 c D.Wackerly - Lecture 17 226 Errors: p. 325 Reality Decision Ho true Ha true Accept Ho Correct Type II error Reject Ho Type I error Correct (p. 323), SIGNIFICANCE ¦ Type I error ¡ ' ¥ $ LEVEL of the test. ¡£¤ true D 3 when ¦ 4 ¥ ¤ 3 accepting ¢ and/or (p. 325) ¦ Type II error ¡ ¥ ¡£¢ ' ' ¥ $ ¥ B1@ CA©¢ 9 7 $ ' ¥ The test that we will discuss have the SMALLEST ¥ for the ’s that we pick. ¡ 7 when $ ¥ ' ¥ $ saying ¦B1@ §A©¢ ' ¥ In our lightbulb example, ’s ¦ ¡ $ saying what we “want” to say when we should not STA 2023 c D.Wackerly - Lecture 17 227 Parts of a Statistical Test (p. 326) . 3 3 2. Alternative Hypothesis : 4 D 1. Null Hypothesis : . 3. Test Statistic : (TS) ' computed from the sample data using a formula ' forms the basis for our decision. 4. Rejection Region : (RR) ' ' ' ' Get data ' Compute value of TS ' Make decision ¥ Do experiment Then 3 depends on the choice of D gives values of TS for which is REJECTED STA 2023 c D.Wackerly - Lecture 17 228 Decision : ' If the value of the TS is in the REJECTION and 3 4 REGION, we . 3 D If the value of the TS is NOT in the REJECTION 3 . because we usually do D 3 – We do not ACCEPT D REGION, we DO NOT REJECT not know the probability of making an error if we do so. D 3 – If we accept , what kind of error could we make? – What is the probability of a TYPE II error? usually 3 judgement D Don’t want to accept , so we reserve 3 that is really true 4 depends on the value of the parameter in ' STA 2023 c D.Wackerly - Lecture 17 229 Courtroom Analogy ' Prosecutor : Experimenter ' Innocent : D 3 ' Guilty : ' Put burden of proof on Prosecutor : Experimenter 4 ¥ 3 Proof “beyond a reasonable doubt” : small. ' STA 2023 c D.Wackerly - Lecture 17 230 Large Sample Hypothesis Testing D7 D7 $ 9 7 86 7 86 D 4 3 3 B1@ 8"¢ 9 7 6 7 6 D 4 3 3 $D7 ' ' Rejection Region, RR $ Test Statistic, TS B1@ 8"¢ Need . 7 a fixed particular value of Lightbulb Example : In this case, about a Population Mean, STA 2023 c D.Wackerly - Lecture 17 231 ¦¨ @ Recall : Large Sample &¡ ' has an APPROXIMATE distribution ¡ ' £ $ & ¤ ) $ ¢ 7 ' ¡ ¤ & 7 £ ¤ ¡ 7 ¢ That is 7 £ $ ¤ ¥ ¢ £ is a is true ¤ D7 $ ) 7 C6 D 3 ' ¡ ¤ & $ ¤ ¡ D7 D7 if ¤ ¥ £ ¢  $ £ is a TEST STATISTIC is true, ¤ ¤ ¡ & ¤ ) £ 3 $ ¡ D7 ¥ ¤ ¢ £ $  has a D7 D If distribution ' STA 2023 c D.Wackerly - Lecture 17 is close to the true value of 7 ¡ FACT: 232 , whatever that true value is. is POSITIVE and LARGE than 7 ' ' Should is probably than 7 7 86 $ D ' ¡ The true value of by a “lot” of standard errors. in favor of 3 7 7 6 zα ¦  α $ 9 3 ¥ Rejection region : 0 “something” ¥ 3 4 type I error 9 4 ' ' Want 7 7 86 Rejection region : 9 If we are interested in D7 is D7  If . ' STA 2023 c D.Wackerly - Lecture 17 233 D7 D7 9 7 86 ¤ ¤ ¡ 7 86 $ & £ ) 3 4 ¡ ¤ 3 ¡ 7 D 7 ¢ ¤ & )  6 $ £  9  6 ¤¤ “Upper Tail test”, “One Tail Test” (p. 329) Ex. : Lightbulb Example true mean lifelength of ALL BULBS B1@ CA©¢ 7 86 $ B1@ CA©¢ 9 7 86 D $ @@ 21 ¦ $ 1 £ 3 3 4 @ @ B A¨ ¡ $ ¨ ¦ @ "¢ $ ' 7 $ ¦ §¥ $ & ' ¥ $ Data : level test, RR : $  ' STA 2023 c D.Wackerly - Lecture 17 234 ' Conclusion : 3 in favor of AT THE 4 D 3 LEVEL!! B A¨ ¥ $ ' In terms of this problem: B1@ CA©¢ ¦ confidence ).  AB ¢¨ ¥ $ If we wanted , is level of B 8¨ significance” ( or with at the 7 “ Claim that the mean lifelength of all bulbs, – RR : 1 9 B ¦ ¢ $  ¢¨ at the – @ C@ – Conclusion : Is ? level 3 D – In terms of this problem claim that the mean lifelength of all B1@ A©¢ confidence ).  ¦¦ with at the bulbs is larger than ¢¨ “ – NOTE: This DOES NOT mean that is true!! level” ( or B1@ CA©¢ $ 7 86 3 ' STA 2023 c D.Wackerly - Lecture 17 235 Ex. : #8.24, p. 333 Manufacturer inspection equip. for printed circuit boards claims that “product can inspect, on average, at least 10 boards per second”. Evidence to & B 8¨ ¥ $ . ? 6 D 3 4 and $ one-second runs? Use ¥ “refute the claim” based on data for 3 6 D 4 ' 3 3 If we are interested in : D7 D7 ¤ 7 86 $ 7 86 ¡ ¤ ¡ & ¤ ) £ 3 3 4 ¡ 7 D ¢ & 7 ¤ ) £ $  6 ¡ ¤¤ 6 α − Zα 0 “Lower Tail test”, (p. 329) STA 2023 c D.Wackerly - Lecture 17 236 Back to #8.24: ' Data: 10 9 10 10 11 9 12 8 7 10 11 9 9 13 9 10 9 9 9 7 12 6 9 10 11 12 10 0 10 11 12 9 8 9 6 10 11 10 12 8 10 8 7 9 7 9 9 10 @¨ C"¢ 1 £ $ ¡ 1 ¦ 1 ¦ $ £ level test & $ B A¨ ' ¥ $ ' RR : @ C@ 1¤ ¨¤  $ $ level of significance. In terms of this ¥ ¨ @ B 8¨ application: “There $ ¨ at the enough evidence at the level to indicate that the mean number of circuit B A¨ boards inspected per second is less than 10 .” ' STA 2023 c D.Wackerly - Lecture 17 237 Minitab? Hypothesis test for a mean. £ ¡ Must have actual data (not just and ). ' Ex. # 8.24 Number of solder joints inspected in 48 1-second runs. Data : 48 actual numbers ¡¡ ¨ ©¢ ¢ ¦ ¨ "¢ ¢ Minitab File Open Worksheet; find M8-24.mtw in MiniData (double click). Stat Basic Statistics Display Descriptive Statistics, select (double click) variable, Click OK ¢ ¢ Descriptive Statistics Variable JntsInsp N 48 Mean 9.292 Variable JntsInsp Minimum 0.000 Median 9.000 Maximum 13.000 TrMean 9.432 Q1 9.000 StDev 2.103 SE Mean 0.304 Q3 10.000 ¢ ¢ Stat Basic Statistics 1 Sample Z, select (double click) variable; Click radio button Test Mean; Type in null value for mean; Select Alternative; Type StDev in box labelled Sigma, Click OK Z-Test Test of mu = 10.000 vs mu < 10.000 The assumed sigma = 2.10 Variable JntsInsp N 48 Mean 9.292 StDev 2.103 SE Mean 0.304 Z -2.33 P 0.0099 STA 2023 c D.Wackerly - Lecture 18 238 Thought: When a man is wrapped up in himself, he makes a pretty small package – (John Ruskin) Assignments Today: P. 334–338, 347–351 Thursday : Exercises 8.29, 8.33, 8.34, 8.38–41, 8.59, 8.61, 8.67–69 Monday 11/4: OPTIONAL review Tuesday 11/5: EXAM 2 – During your DISCUSSION SECTION STA 2023 c D.Wackerly - Lecture 18 239 7 Last Time: Large Sample Hypothesis Testing about UNKNOWN population mean D7 ¡¡ ¦¥ ¡¡  9  £ D7 9 §  ¤ ¡  £ ¤£ D7 7 ¡ 7 86 $ ' 7 $ ' 3 D ¡¡ ¡¡ OR RR 3 4 $ ¢ 7 ' Test Statistic hypothesized value ¨ ¤ standard error ¨ $ ¡ D7 & ¤ ) £  $ Estimator and Standard Error from Formula Sheet Hypothesized Value from NULL hypothesis ¨ ©¨ estimator STA 2023 c D.Wackerly - Lecture 18 240 Ex. : pH of 7 is neutral, over 7 is alkaline, under 7 water specimens ¨ @ indicates acidity. Randomly select from a recreational lake. Can we claim that the mean ¥ $ level? ? 6 D 3 4 and ¢¨ pH is NOT that of neutral water at the 3 6 3 ' D 3 4 How? D7 7 86 $ D 3 D7 ¤ 7 86 $ ¤ ¡ ¡ ¡ 7 &    ¤ ¢ £ ) 9 3 4  & 7 ¤ )     6 $ £ ¤ ¡  6 ¤¤ α/2 or α/2 -z α/2 0 z α/2 “Two Tailed test” (p. 331) STA 2023 c D.Wackerly - Lecture 18 241 Back to pH example: ¤ 1 ¨ $ £ ' ¡ @ ¨ C@ $ level test: ¢¨ & $ $ ¥ ' RR : $ ¢¨ at the $  D 3 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 D.Wackerly - Lecture 18 242 If the mean pH is NOT , what is it? confidence interval for . 7 $¢ ¡  £ $ £ ¥ £ 1 £¦¦ ¥ $ £ ¦¦  Construct a $ ¥ C¢ ¤ ¡ & ¥¦ 6 ¨ £ @ ? $ 7 Do you think that ' — the value “ ” is confidence interval Agrees with two-tailed test!! the 99% ' STA 2023 c D.Wackerly - Lecture 18 243 Hypothesis Testing is chosen BEFORE the test is performed ¥ to reject 3 CONFIDENCE in our in favor of D decision to reject 3 – Provides . 4 ' – D ' ¥ Smaller ( IF we DO 3 SO). 3 ¥ 4 be rejected in favor of for which D What is the SMALLEST value of could ? 3 The p-value or observed significance level ' (P. 335) Recall the lightbulb example B1@ 8"¢ B1@ 8"¢ B ¦ 7 6 $ ¢ 9 $ 7 6  3 D 4 3 STA 2023 c D.Wackerly - Lecture 18 rejection region  ¥ B ¢ 9 B 8¨   9 ¨@ C@ 9 1 9 ¨ @ C¨  1 ¥ 9 ¨ A¨ B ¥ 1 8¨  ¢¨ B1@ CA©¢ B1@ CA©¢ 7 86 $ 9 D $ 7 86 3 3 4 p-value 244 Probability of a z-value the one observed ' 9  ¡ p-value = ¦ true. 3 indicative that 4 Larger z-values are is $ ' STA 2023 c D.Wackerly - Lecture 18 245 ' . 3 CANNOT reject D 3 ¥ ¥ p-value REJECT D p-value . ¡ In our case, p-value = .0256 3 . . . D 3 B A¨ ¥ $ @ ¨ ¥ $ 3 1 A¨ ¥ $ 3 ¢¨ $ ¥ Instead of “imposing” YOUR CHOICE of ¥ ¥ $ – D REJECT D ¢ – . D 3 ' – – . D ' – 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 D.Wackerly - Lecture 18 246 Ex. : #8.24, P. 333 ¨ "¢ ¨ "¢ @ @ D 7 6 ¡ 1¤ 7 6 $ 3 3 4  $ Probability of a z-value $ p-value the one observed 4 ' ' p-value 3 Smaller z-values are more indicative that $ ' . ¨ C¨ ¡ ¥ D D 3 ' 3 Z-Test Test of mu = 10.000 vs mu < 10.000 The assumed sigma = 2.10 N 48 Mean 9.292 StDev 2.103 SE Mean 0.304 Z -2.33 P 0.0099 ¦¦ that is ¦¦ ¥ for any that is ¨ ¨ for any See page 234 of notes: Variable JntsInsp is true. . STA 2023 c D.Wackerly - Lecture 18 247 TWO - Tailed Test '  Find the area in whichever “tail” the -value is in and DOUBLE IT. EX. : Have done a two-tailed test: ¢ 7 6 $ D 3 ¢ 7 6 $ B ¢ $ 4 3  value = . ? B A¨ ¥ $ $ ¥ with ¢ $ 7 claim that B A¨ ¤ – . ' ' STA 2023 c D.Wackerly - Lecture 18 248 Ex. : #8.68, p. 352 In a “Pepsi Challenge”, 100 Diet Coke drinkers were given unmarked cups of both Diet B 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. 6 6 (2) 3 4 D 3 How??? (1) ' STA 2023 c D.Wackerly - Lecture 18 249 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 ¡ ¡ $ & $ & Estimate for # of £¡ # of trials ; sample size $ ¤ ' STA 2023 c D.Wackerly - Lecture 18 is “large” & ¤ If 250 has an ' distribution ( & ¤ ¤ $ £ 7 $  $ £ That is has an approximate distribution. a fixed particular value of £ D ¡¡ ¡¡ 9 D ££ ¤¤£ D ¡¡ ¡¡ $ ¡¡ ¡ ¡¡ ¡¡ OR $ $ ¡¡ OR ££ D 6 D 3 versus Consider testing ¢ 4 3 STA 2023 c D.Wackerly - Lecture 18 is the null hypothesis, TEST D $ 6 D 3 ' if 251 STATISTIC hypothesized value ¨ ¤ standard error ¨ ©¨ ¨ estimator  $ Estimator and Standard Error from Formula Sheet Hypothesized Value from NULL hypothesis ¡ D ¤ D(D  $ & ¦D $ ¡ is true ,  D 3 If has a STANDARD ' NORMAL distribution Rejection Regions (RR): ¡¡ ¥ ¡¡  ¡¡ ¡¡  9 £ ¡¡ D ¡¡ ¡¡ ¡¡ 9 ¡¡ ¡¡ OR ¡  ¡¡ RR ¤  ¡ £ ¤£ ¡¡ D ¡ ¡ ¡¡ $ ¡¡ OR ¡¡ ¡¡ ¡ ¡¡ § or        9 ¤  ¡  £ ¤£ £ D $ ¡¡ ¡¡ ¡¡ ¢ ¡ 4 3 ' STA 2023 c D.Wackerly - Lecture 18 252 Ex. : #8.68, p. 352 In a “Pepsi Challenge”, 100 Diet Coke drinkers were given unmarked cups of both Diet indicated that they preferred B Coke and Diet Pepsi. 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 ¨B (3) ¨B (4) 9 $ 6 3 4 6 D 3 level test, RR : B A¨ ' ¥ $ ' Assumptions : the 100 individuals participating in the the Pepsi Challenge are a $ $ $ B 8¨ $ ¤ ¨¨ C"¢ $ & Data : & SAMPLE of all Diet Coke drinkers. Note: is “large” $  ' STA 2023 c D.Wackerly - Lecture 18 253 ' Conclusion : 3 in favor of 4 D 3 reject AT THE LEVEL!! B A¨ ¥ $ ' In terms of this problem: claim that there is sufficient evidence at level of significance” ( or with ¦ the  8B “ 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 18 254 Minitab? ¢ ¢ ' Stat ' Click radio button “Summarized Data”, type in Basic Statistics 1 Proportion 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 ...
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This note was uploaded on 07/28/2011 for the course STA 2023 taught by Professor Ripol during the Fall '08 term at University of Florida.

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