week11-2up_001 - Standard Error of Est. NORMALLY...

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Unformatted text preview: Standard Error of Est. NORMALLY distributed. For “large” sample size, estimators approximately Estimator Parameter Last Time: SEE FORMULA SHEET For Thursday : Exercises 8.18, 8.21–23, 8.25, 8.27 For Wednesday: P. 328 – 332 8.7, 8.9, 8.10, 8.13 For Tuesday: Exercises 7.54, 7.60–7.62, 7.66 Today: P. 308 – 310, 322 – 326, 328 – 332 estimator α/2 formula sheet  ©  LARGE SAMPLE Confidence – Population mean, $ Assignments :  Interval for a PARAMETER    Last Time:  STA 2023 c D.Wackerly - Lecture 16  207 & stays bought.—(Simon Cameron) ¡ ¦  " ¢ ¦§  ! $ %# 1−α table  ¡ formula sheet α/2 208 (P. 283), LARGE sample CI: – Population Proportion, sample CI: standard errors  Thought: An honest politician is one who, when bought, £ ¥¤ ¦¨ ¥ (P. 300), LARGE ¢0 '( ¢0  ! '( & ! £ ¥¤ $# 2 ¥¤ STA 2023 c D.Wackerly - Lecture 16 ¦¨ § ¥ $# & " )  '( " 1 ¦ '( & $# § & $ '( & ) '( ¦§ $ ¡ ( ¦ ¤ (when ¥ ¢ ¦ ¢ © ¦© § ¦ §¥ ¡© , ) then . is binomial) has an approx For an approx. norm. dist. var., ALMOST ALL ¦ §¥ £ normal (symmetric) distn for LARGE ¢ 209 SKEWED RIGHT (SYMMETRIC, SKEWED LEFT). ¦© ¨ © ¥ ¦§ © Easier to check:  If  smaller of larger of and is approx normally dist., expect that KNOW:  values of the var. are within 3 std. dev. of the mean. ¦¨ § ¥ © ¦§  ¦ §  © ? Can we validly use the formula to construct a CI for p? Ex. Took sample of size 40, obtained STA 2023 c D.Wackerly - Lecture 16 #$ (respectively) the binomial distribution histogram is ( ¦ ¥ (Project) If ¦ LARGE SAMPLE CI for ¦ ¢ ( ¡ ¢ ¢    for C.I. – Need : at least a “ballpark value” for and solve for (p. 307) that is within “ ” units with confidence level Estimate of Want: Choosing the Sample Size to estimate © ©    ¡ ¥ ¡   £   & ¢ ( ¦§ §  ¦  ¦§ ¨ § § ¨ ¢  “large enough” to Validly construct a  ¦§ & ¥    ! ¥ "  $   & '  '( ! §   £ ¥¤ " ¦¨ § ¥ ¡ "! £ When is  ¢ ) STA 2023 c D.Wackerly - Lecture 16 % . 210  © © $ © $ and solve for (p. 309) © # & & '(  ¢ Parameter : ,  ¦ $ ©  ¢ “kill rate” to within to within , Confidence : with probability . within 1 hour. How large sample size to estimate true Ex.: Insecticide designed to kill approx. of insects and SOLVE confidence. Need : at least a “ballpark value” for   ¦  ¢ ¦ Don’t know but What is ? STA 2023 c D.Wackerly - Lecture 16 If you have a “ballpark” value of . ¢ ¥£ – USE IT !!!  for C.I. ¥ '(  standard error ¢  !   & & '( & ' ¦¨ " &    ) # © ©  units with &  to be correct to within “ ” $ 211 £ ) Want : Estimate for  ¦  £  ¢ ¦¨ ¥ $  Finding the sample size to estimate .  " ¡  ¦   ¥ £   STA 2023 c D.Wackerly - Lecture 16 ¢ 1 ) )   ¥ ¢ ¢ ¤ & ¢ ) ( #  £  # ) ) ) )  ¥¤  #  ) ¢ ¢ ¢ ¢ ¢ ( ¤ ¥ ¦ # ) # ¥¤ ¥¤ ) ¥ ¢ ¤ ) 212 '( '( ¦¨ ¥ ¢ bigger .25 .24 .21 .16 .09 .4 or .6 .3 or .7 .2 or .8 .1 or .9 bigger smaller .5 $ © &  ¢ $ & ¦ ¡ ¥ & Higher confidence:   Max value of bigger bigger bigger is at bigger . smaller for , I get a sample size Use . How about if I had no information about Recall that : about 214 FOR EVERY VALUE OF . Ex.: Insecticide example – Had some preliminary info $ © " © ( ¡ ¡ that will work FOR ANY value of . (P. 309) ¦  If I evaluate ¢ WHAT? & – THEN ¢ ¡ ¦©  ©  ¨ That is ¦ ¢ ¥ ¥ STA 2023 c D.Wackerly - Lecture 16 #  ¢ # )    How about if we have no idea of the value of ¦¨ #$ )   ¦  ) $ 213 &  & ¦  ¢ ?  ¦ ¡  # ¨ ¢  £  ¥ $ )  ¦  ¢ #  ) # ¥ ¨ ¦¨ ¥£ ¢ ¢ ¥ ¢  1 ) $  ¢ ) # ¢$ ) ¦ ¢ ) ¦ # ¥¤  ) ¤  STA 2023 c D.Wackerly - Lecture 16 ¦ ¥ Put a random sample of hours. ¤ © Data :  ¢  ¢ # 2     ¢ ¢ © bulbs on test, Can the claim be made? (VALIDLY) ) record time to burnout for each.  that the average lifetime of our bulbs exceeds 1325 Ex. : A manufacturer of light bulbs would like to claim Tests of Hypotheses: Section 8.1 #$ #$   ¢ £ ¡¢ £ ¤¢ £ © © © Decision is to be made using sample data. UNKNOWN ) lifelength of the manufacturer’s bulbs. ( FIXED, but true mean (ALTERNATIVE hypothesis) (NULL hypothesis) We do not know the true value of ¡ ¡ Choose between two HYPOTHESES. ¡ © 215 216 Sample DOES NOT contain ALL of the information STA 2023 c D.Wackerly - Lecture 16 Ho true Type I error Reject Ho – Claim Type II Error – Claim Type I Error Correct Type II error when really when really Consider the lightbulb example Correct Accept Ho Ha true Reality Decision p. 325) possibility of making an ERROR. (See Table 8.1, Using sample data to make decision – always the about the POPULATION. © © © © STA 2023 c D.Wackerly - Lecture 16 ¢ £ Suppose (p. 323) (p. 325) How about both very small? and © £ ” Type II Error – Leading to an – Thus, “accept must decide to “reject – If we want to REDUCE ¢ ” ¡ in 217 OFTEN. Type I Error we . OFTEN. to be is a fixed sample size. Ideally, want BOTH © ¢ are PROBABILITIES ¡ ( numbers between 0 and 1 ) and £ © Note: ¢ ¢  ©  beta ¡ ¢ " " ¥ ¢  ! !  ¡ Type II Error  ¡ Type I Error ¢ For fixed directions. small enough to be “CONVINCING” move in . SMALL we confidence in our conclusion to ACCEPT – By making is our . The PROBABILITY of making a TYPE I error ERROR. The ONLY type of error possible is a CORRECT – If we decide that the ALTERNATIVE IS Choose Our strategy and . STA 2023 c D.Wackerly - Lecture 16 © ) £  ¡ £ ) ¢ © © © ¥ ¤  ¥  £ £  alpha  STA 2023 c D.Wackerly - Lecture 16 ¤ 218 . (p. 322) The NULL hypothesis is always that a In our set-ups – Denote by HYPOTHESIS . (p. 322) The “other” hypothesis is called the NULL – Denote by manner – What we are “trying to prove” in an objective, fair Hypothesis. hypothesis. (NULL hypothesis) (2) (ALTERNATIVE hypothesis) (1) specifying the ALTERNATIVE or “RESEARCH” PARAMETER EQUALS the value used in ¡ accepting Type I error when true when ’s that we pick. The test that we will discuss have the SMALLEST ’s SIGNIFICANCE LEVEL of the test, LEVEL of saying the test. for the 220 saying what we would like to say when we should not ¡ ALTERNATIVE HYPOTHESIS, or the Research ¡ ¢ ¢ © © © ¢ ¤ ¡ ¡ ¡ £ £  £ ¢ ¤ The hypothesis of MAIN INTEREST is the ¤  ¢ © © © © © ¡ ¡ ¢ ¢ ¢  ¡ #$ #$ STA 2023 c D.Wackerly - Lecture 16 £ 219 ¢ STA 2023 c D.Wackerly - Lecture 16 ¢ 8.61, 8.67–69 Thursday : Exercises 8.29, 8.33, 8.34, 8.38–41, 8.59, Wednesday : P. 334 – 338, 347–351, , . (p. 322) 222 HYPOTHESIS, – (p. 322) , light bulb ex. The “other” hypothesis is called the NULL manner What we are “trying to prove” in an objective, fair light bulb ex. ALTERNATIVE or RESEARCH hypothesis, – © ©  ¡ ¢ £ Tuesday : EXAM 2 – During your discussion section . The hypothesis of MAIN INTEREST is the ¡ Monday : OPTIONAL review . (p. 309)  if you have one, Parts of a statistical test. (p. 322) otherwise use  ¢ For Tomorrow : Exercises 8.18, 8.21–23, 8.25, 8.27 Use “ballpark” value for and solve for ¤ #$ Today : P. 328 – 332 confidence.   Assignments $ to within “B” units with ¡ and decide that nothing can be done. '( & ©  Estimate  £ individually, can do nothing, but collectively can meet, ¢ Last Time: & ¦ ¦© ¨   #$ Thought: A committee is a group of people who, STA 2023 c D.Wackerly - Lecture 17 £ ¤ ¡ 221 ¦    STA 2023 c D.Wackerly - Lecture 17 ¨ ¢  ¥ #¤ ¦  ¢  © Type I error £ © © ¢ accepting and/or Type II error Correct Type II error (p. 325) when true ¡ ¡ for the ’s that we pick. The test that we will discuss have the SMALLEST saying when ’s saying what we “want” to say when we should not (p. 323), SIGNIFICANCE Type I error £  In our lightbulb example, ¡ LEVEL of the test. ¢ Reject Ho £ Correct #$  . forms the basis for our decision. Make decision Compute value of TS Get data Do experiment depends on the choice of gives values of TS for which 4. Rejection Region : (RR) Then . 224 is REJECTED computed from the sample data using a formula 3. Test Statistic : (TS) 2. Alternative Hypothesis : 1. Null Hypothesis : Parts of a Statistical Test (p. 326) STA 2023 c D.Wackerly - Lecture 17 ¡ Accept Ho ¥   ¡ Ha true ¡ © £  Ho true ¢ © ¤ ¡ Decision  © ¥ Reality ¢ © © © ©    ¤ ¤  ¢ ¢ ¢  ¢ #$ ¤ Errors: p. 325 ¢ 223 ) STA 2023 c D.Wackerly - Lecture 17 £ © © © © ¡ © If the value of the TS is NOT in the REJECTION REGION, we and If the value of the TS is in the REJECTION Decision : ¡ © . 225 because we usually do , what kind of error could we judgement Don’t want to accept usually that is really true , so we reserve depends on the value of the parameter in – What is the probability of a TYPE II error? make? – If we accept do so. ¡ not know the probability of making an error if we – We do not ACCEPT REGION, we DO NOT REJECT. ¡ ¡ : innocent Courtroom Analogy STA 2023 c D.Wackerly - Lecture 17 Proof “beyond a reasonable doubt” : small. Put burden of proof on Prosecutor : Experimenter Experimenter : Prosecutor : guilty © © © ¤ ¢ ¤ ¡© ¢ ¤© ¢ STA 2023 c D.Wackerly - Lecture 17  226 ¡ ¡ ¡ Test Statistic, TS Need ¡ ¢ £ ¤ ¢ £ Rejection Region, RR . has an APPROXIMATE Recall : Large Sample is true If has a is true, is a TEST STATISTIC if is a That is  ¡ ¤ ¢ £ £ In this case, ¡ Lightbulb Example : ¢ ¡¡ % a fixed particular value of   £ ¡¡ #$ £ ¡ ¡ ¢© ©  ¡ #$ ¢ ¡¢ £ £ about a Population Mean, ¡ STA 2023 c D.Wackerly - Lecture 17 ¢ ¡¡ ¢ ¢ $ ¡ Large Sample Hypothesis Testing 227 ¢0 ¢0 STA 2023 c D.Wackerly - Lecture 17 © © ¢ ¡¡ $ ¢ ¢0 ¢0 © © ¥ £ ¥¤ ¡ ¢£   ¡ ¢£  ¡ ¢£ ¡©  ¡¡   ¢ ¡¡  ¤ ¢0 ¢0 £ ¢ ¢ ¡ ¥¤ £ ¤  228 distribution distribution ¤ £ ¡¡ ¥¤ ¡¡ ¥¤ £ ¢© © © ¤¢ £ $ Rejection region : If we are interested in © Want © 0 zα α type I error Rejection region : ¡ ¡ “something” in favor of than . by a “lot” of standard errors. true mean lifelength of ALL BULBS Ex. : Lightbulb Example “Upper Tail test”, “One Tail Test” (p. 329) Data : level test, RR :  ¢ Should ¡ " is probably ¢ The true value of than ¢ ¡ ¡ £ ¤ £ $ £ ¡ ¡ ¡¡ ¡ ¡ £ ¢ is © © ¡© © $ $ is POSITIVE and LARGE  ¢ ¢ ¢ If ¢ true value is. ¤ , whatever that  " ¡ £ ¡ ¤ ¢  ) ¥ ¡ is close to the true value of  £ £ ¢ ¢ £  ¡ 1 ¢ ¡     #$ #$ " ¢ STA 2023 c D.Wackerly - Lecture 17  229 ¢ FACT: ¡¡ ¢0 2  # )  ¢ ¡¢ £ ¢ ¢0 ¡ ¡ ¤¢ £ £$ £$ ¡¡ ¢ ¡ " ¥¤ ¡¡ $£  ¤ £ &  ¤ 2 ¡ " ¥¤ #  STA 2023 c D.Wackerly - Lecture 17 ¢ 230 LEVEL!! in favor of ¡ ¢   ¢ $ ©  © If we wanted © at the level level of confidence ). at the ? confidence ). at the is true!! – NOTE: This DOES NOT mean that with " £ , is AT THE level” ( or claim that the mean lifelength of all bulbs is larger than “ – In terms of this problem – – Conclusion : Is – RR : ¢ significance” ( or with $  #$ “ Claim that the mean lifelength of all bulbs,  232 and α − Zα If we are interested in : ? one-second runs? Use 0 . “refute the claim” based on data for on average, at least 10 boards per second”. Evidence to printed circuit boards claims that “product can inspect, Ex. : #8.24, p. 333 Manufacturer inspection equip. for STA 2023 c D.Wackerly - Lecture 17 ¥ ) ¡  $   $ ¤ ) ) ¡ ¤ ¡© ¢ £ ¢ In terms of this problem:  ¡ ¤ Conclusion :  # £  )  231 ¢ ) $ #$ ¢   ¢  ¢0 STA 2023 c D.Wackerly - Lecture 17 $ ) #$ ¡ ¢ £ “Lower Tail test”, (p. 329) ¤¢ £ ¡   )  ¢ ¢   ¢ 1  ) ¡¡ ¡¡ ¢0 ¡ ¡ £$ ¡ " ¤ £  ¤ £ ¢ ¡ ¢ £ ¥¤ ¥  ¤ 2 ¡ " ¥¤ 10 8 RR : at the 9 10 6 7 9 12 9 8 7 11 10 12 9 9 10 11 6 13 9 12 12 9 9 ¥ 10 8 9 10 10 enough evidence at the boards inspected per second is less than 10 .” level to indicate that the mean number of circuit application: “There level of significance. In terms of this level test 9 10 0 7 9 Minitab? Must have actual data (not just and ). 1-second runs. Data : 48 actual numbers Ex. # 8.24 Number of solder joints inspected in 48 Median 9.000 Maximum 13.000 Mean 9.292 Minimum 0.000 N 48 Q1 9.000 TrMean 9.432 Q3 10.000 StDev 2.103 SE Mean 0.304 234 Variable JntsInsp N 48 Mean 9.292 StDev 2.103 Test of mu = 10.000 vs mu < 10.000 The assumed sigma = 2.10 Z-Test SE Mean 0.304 Z -2.33 P 0.0099 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 Variable JntsInsp Variable JntsInsp Descriptive Statistics Minitab File Open Worksheet; find M8-24.mtw in MiniData (double click). Stat Basic Statistics Display Descriptive Statistics, select (double click) variable, Click OK  8 © Hypothesis test for a mean.  12  © 11 $ STA 2023 c D.Wackerly - Lecture 17 ¢ 9 233  9 11 10  ¢ 7 ¢ ¡ 11 ¢ ¡ 10  10 # ¢ ¢ ¢ 9 # ) ¥ © © # ¢ ¢ ¢ ¢ ¢ ) $  ¡ 10 ) ¢ ¢ ¢ Data:    Back to #8.24:  )  STA 2023 c D.Wackerly - Lecture 17   ) ) 2 ¤   $ © $ )  #    )  ) 2 ...
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