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Unformatted text preview: 1 Thought: Never do card tricks for the people you play STA 2023 c D.Wackerly - Review for Exam II STA 2023 c D.Wackerly - Review for Exam II 2 Chapter 4 : Discrete Random Variables poker with. ¡ Can COUNT the number of distinct values of the ¡ The Binomial Probability Distribution variable. Thought: To succeed in politics, it is often necessary to rise above your principles. – Some discrete random variables are binomial – NOT ALL 9 £5YYY565 3 ¤ `B@A@8704¨V V U § XWET¦ 9%£ ¤ P S1GFRQI 9£ ¤ HGFED £CB9@A9@8670342 10(&$ "£  5955 ¤ #)'% # ! ¤  § ¤©¨¦ § ¤ ¥£ ¢ – Binomial Experiment p. 179 – number of trials, – If Monday 3/25/02 : OPTIONAL REVIEW, ask questions about homework, course material, sample exam, criteria : (p. 183) for etc. Suggestion - print out Sample Exam 2 and – Mean (p. 185) : bring it to class with you. – Variance (p. 185) : Help for Exam 2 – Tables : Contain ¡ Monday, periods 3–8, FLO 104 ¡ Monday, 6:00 pm – 10:00 pm, TUR L005 Tuesday 3/26/02 : EXAM 2–during your regularly for ..... scheduled discussion section. 0 1 2 3 ..... k k+1 n-1 n Wednesday 2/13/02 : Pages 334 – 338, 347–351 STA 2023 c D.Wackerly - Review for Exam II 4 ¡ Areas under normal curves between z-scores of ¡ Key to finding correct areas (probabilities) : draw r ¡ Possible values are all those associated with one or ¡ Probabilities are areas under “density function”. 3s e¨r and , for Chapter 5 : Continuous Random Variables in Table IV, p. 809. 3 3 STA 2023 c D.Wackerly - Review for Exam II pictures more line intervals. Chapter 6: If we plan to take a random sample of size is called its sampling distribution.  @fFhEihqaTg@WEiqa¨p¤ § ¦ ¤ f h  U § ¦ f U  h § ¦ ¤ f U  U § AFEicba¨g@Fedcba¨¦ (p. 266), so dist. of for larger sample sizes. t D concentrated around is more is called the standard error of .(p. 266) t . (p. 266, 261) is an unbiased estimator v# I £ €Q4¤ I ¡ y x I v# D D ¤  § w v# 4t  q4¤ D ¡ of . So t ¡ (p. 255) b is a continuous r.v., If the population has a normal dist., then so does ,  y x I5  £ €Q„ƒD§ ‚t  i.e., N . True for any £ ¡ ¡ Normal distribution is special case. is a random variable. t t u¡ If , Dist. of P(a<X<b) a I deviation and standard t £ from a population with mean D graph of f(x) . ¡ £ is large £ €Q„ƒD§ ¤t  y x I 5 £ t 3 ¢ £ ¡ ), then the sampling distribution of ¡ approximately normal, i.e., STA 2023 c D.Wackerly - Review for Exam II 5 Central Limit Theorem (CLT): (p. 267) If ( STA 2023 c D.Wackerly - Review for Exam II 6 Estimated is N , Standard regardless of the shape of the population Parameter Error of Est. Estimator Standard Error of Est. distribution.  £y £ % 1 § £ ¡ ¡ ¡ , etc. (p. 254) £  ¤ ¦  ¡ Both estimators are UNBIASED £¡ P QI D a Sample. ¨¦%¦  § D All statistics have sampling distributions , £y A Statistic is a meaningful number associated with with a Population. I A Parameter is a meaningful number associated ¥ Chapter 6: If is “large”, both estimators are approximately NORMALLY distributed. Chapter 7: ¡ How large is “large”? © ¢ £ ¡ Want : Correct to within “ ” units with and solve for (p. 307) if you have one. Maybe 7 – Use ballpark value for Range use . I 6 0I Finding the sample size to estimate . 4 ¡ Want : Correct to within “ ” units with confidence. and solve for  – Use “ballpark” value for (p. 333) if you have one, if not to get sample size that will work  9¢ ¤ 4¤  % use and SOLVE £ 45¤ % £ P ' % 1 § (&r ¡ §  P (&r ¡ '%  3   (3 A6 ƒ¨C6§ standard error 4¤ 5 $ ! §  P# (&r  # "  ' "%! ! " r P )% r ' 3 ¥ D P )% R 'r $ 9 £ P' 2¦%¦  § )% r ¦   $ £ y P' $ 0 )% r t  1£ I y P )% r t  ' (P. 300) and SOLVE £ 9 # – Population Proportion, standard error α/2 (P. 283) confidence. 4¤ 5 ¡  3   3 A6 ƒ¨C6§ ¡ – Population mean, 45¤ £ I y P (% r ¡ ' §  P (&r ¡ '%  3   (3 A6 ƒ¨C6§ standard errors 1−α 8 4 ¡ α/2 smaller of Finding the sample size to estimate . PARAMETER formula sheet larger of STA 2023 c D.Wackerly - Review for Exam II 7 Confidence Interval for a table : 3 ¡ STA 2023 c D.Wackerly - Review for Exam II . !  H¦ § ¦% 5  H¦ § ¦% 5 ¡ Confidence Level (p. 282) formula sheet  – For valid CI for Confidence Coefficient (p. 282) estimator : – For valid CI for Interval Estimator (p. 282) 3 ¢ £ D ¡ Point Estimator (p. 261) for any value of . 9 STA 2023 c D.Wackerly - Review for Exam II STA 2023 c D.Wackerly - Review for Exam II 10 Chapter 8 – Large Sample Hyp. Testing Parts of a statistical test. (p. 322) ¡ The hypothesis of MAIN INTEREST is the Type II error accepting , light bulb ex. (p. 322) when true saying what we “want” to say when we should not  ¢ §¡ 6 ¤ D ©£ § ¦ ¤¨ Errors: p. 325 ¡ In our lightbulb example, Accept Ho Correct Type II error Reject Ho Type I error saying when Correct STA 2023 c D.Wackerly - Lecture 18 235 STA 2023 c D.Wackerly - Lecture 18 236 Last Time: Large Sample Hypothesis Testing about UNKNOWN population mean Assignments r D ( % R h¨r #$# ¨ ih D " ¤ £ ¡ s % re¨r # ¨ Dis D  ! &' % D ¤ ¤¨ ¨ WED ©£ ¡ makes a pretty small package – (John Ruskin) ¤ ED ¡ Thought: When a man is wrapped up in himself, he OR P. 341 – 345 (Sec. 8.4) estimator hypothesized value standard error ) Wednesday: P. 288 – 294 (Sec. 7.2), RR Test Statistic 1 x ¤ £ y HI ¤ r ¨D   8.61, 8.67–69 ) Tuesday : Exercises 8.29, 8.33, 8.34, 8.38–41, 8.59, ¡ Today: P. 334–338, 347–351 Estimator and Standard Error from Formula 8.55 – 8.57 Sheet 1 21 Thursday: Exer. 7.27, 7.30, 7.33, 8.49, 8.50, 8.53, Hypothesized Value from NULL hypothesis ) 0) Ha true D Ho true Decision ¦ ¥g¤  Reality ¢ §¡ 6 s D ¦ ¨ ¢ HYPOTHESIS, – and/or ¦ ¤ ¢ §¡ 6 ED ¡ The “other” hypothesis is called the NULL (p. 325) ¡ ¢ ¨  5  manner  What we are “trying to prove” in an objective, fair  LEVEL of the test. . (p. 322)  ¥¦ ¤¡ ¦ ¥g¤  ¡ 5  ¨ ¡ §¦  Tg¤ ¡  ¢ §¡ 6 s D ¥£ § ¢ ¦ ¤¡ ¡ , (p. 323), SIGNIFICANCE  §¦ ¨g¤  ¡ light bulb ex. Type I error  ALTERNATIVE or RESEARCH hypothesis, – 237 STA 2023 c D.Wackerly - Lecture 18 STA 2023 c D.Wackerly - Lecture 18 238 Ex. : pH of 7 is neutral, over 7 is alkaline, under 7 3¡ water specimens Back to pH example: ? 6 G 93 ¤  level? ¤¨ ¤ ¡£ ¨ ¡ ¡ £ and 6G 93 ¤  ¡ 3¡ ¤ £ ¡ pH is NOT that of neutral water at the 9¦ ¤ ¡ S§2t  from a recreational lake. Can we claim that the mean 9 ¦ 0g¤ ¥ 93 indicates acidity. Randomly select level test: RR : at the 6 G 93 ¨ ¡ level of significance. In terms of this application: “There α/2 the 6 G 93 α/2 ¤r 'P r (% es¨r P )% R ¨r ¤ ¥¤ 'r h ¤ y £ €x yx ¥ 0 ¢££ D S HI  ¤ r ¤ © ¡ ¢£D  t  Dt ¡ ¨ W ¤ D ¤ £ D ¨ W¤ D ¤ ¨ Reject or ¤ How? enough evidence at level to indicate that the mean pH reading is not .” ¦ α/2 0 z α/2 “Two Tailed test” (p. 331) STA 2023 c D.Wackerly - Lecture 18 239 -z STA 2023 c D.Wackerly - Lecture 18 240 Hypothesis Testing If the mean pH is NOT , what is it? to reject . CONFIDENCE in our SO). in favor of ¡ the 99% ? ( IF we DO for which could The p-value or observed significance level (P. 335) Recall the lightbulb example ¢ © 86 ¤ r 9 ¢ §¡ 6 s D ¤ £ ¦ ¡ ¢ §¡ 6 ED ¤ ¨ ¦¤ ¤ ¨5 © © 9 ¤   6  ¤ 6 © 93 be rejected in favor of ¡ ¥ ¤ ¦ 5 £y What is the SMALLEST value of  ¤ ©¢ R¨¢ £ r 5 decision to reject ¨ – Provides ¨ ¢ ©© $ ¦ ¤  D$ 9 ¡ S¦ ¦ Agrees with two-tailed test!! – ¡ D ¡ confidence interval Smaller ¨ ¢ . ? — the value “ ” is ¡ Do you think that ¡ confidence interval for  ¦ Construct a is chosen BEFORE the test is performed ¡ 241 STA 2023 c D.Wackerly - Lecture 18 STA 2023 c D.Wackerly - Lecture 18  rejection region ¡ . . . . . ¤ ¤ ¤ ¦ 93 6 G 93 ¤ ¢ §¡ 6 s D ¥£ ¦ ¤¡ ¢ §¡ 6 ED ©£ ¦ ¤ ¤¨ ¤ ¤ Instead of “imposing” YOUR CHOICE of is  REJECT . ¨£ ¨ £ ¡6 ¢G 93 ¡ ¦ 93 ¡ ¡ 93 ¡£¢ 93 ¡ ¢ 96 – on a person who might be interested in your conclusions, the ¡ ¡ ¡ p-value allows him/her to assess the “rareness” of the observed event. s§ ¨br¨¦ STA 2023 c D.Wackerly - Lecture 18 Ex. : #8.24, P. 333 ¤ ¡ ¡9 ¦  ¤ r 3A6 h D ¥£ ¤¡ 3 A6 ¤ D ¤ ¨ TWO - Tailed Test ¡ Probability of a z-value the one observed ¡ is true. ¤ StDev 2.103 – SE Mean 0.304 Z -2.33 P 0.0099 ? claim that 6 ¤ D ¨ £ ¨ Mean 9.292 . ¦  ¡ ¡ ¡ Test of mu = 10.000 vs mu < 10.000 The assumed sigma = 2.10 value = with ¢ € 93 ¤  ¡ Z-Test . ¢ 93 ¤  ¡  ¡ that is . 9 © © 3 3 ph  for any © © 3 93 that is ¢9 8¦ 86 ¤ r for any See page 234 of notes: N 48 and DOUBLE IT. EX. : Have done a two-tailed test: Smaller z-values are more indicative that p-value Find the area in whichever “tail” the -value is in ¦6 ¤ D ¤¡ ¦ 6 ¤ED ¤¨ ©£ p-value 244 r ¤ 243  STA 2023 c D.Wackerly - Lecture 18 Variable JntsInsp CANNOT reject ¨ ¨£ ¨ ¢ 93 p-value = ¡ ¡ 93 Probability of a z-value true. . ¨ ¢ ¢6 93 – indicative that ¡ 3 ¡ ¡9 ¦ s r 3 8¢ 93 ¦sr sr sr ¢ 96 s r h¡ ¡ – – Larger z-values are p-value REJECT In our case, p-value = .0256 – the one observed p-value ¨ 6 p-value 242 . STA 2023 c D.Wackerly - Lecture 18 246 Large Sample Tests About (Section 8.5) 7 245 STA 2023 c D.Wackerly - Lecture 18 Interested in a POPULATION that contains an Ex. : #8.68, p. 352 In a “Pepsi Challenge”, 100 Diet UNKNOWN but FIXED PROPORTION of items with a Coke drinkers were given unmarked cups of both Diet particular attribute ¡ indicated that they preferred ¡ © § Recall the BINOMIAL EXPERIMENT. ¤1 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. select Diet Pepsi in a blind taste test? ¤1 the proportion of batteries that fail before guarantee expires. ¡ GOAL : Test hypotheses about number of trials ¤ ¥£ 1 # of trials ; ¡ Estimate for # of based on a “large” in the in the sample ¤ £ 2¦  ¤ ¡ © £¢  sample size STA 2023 c D.Wackerly - Lecture 18  ¤¨ ¨ ¤  ©£ ¡ is “large” if ¤¦ I   ¦ ¤ r £ ¤¦ % 1 § ¤ §I 1 standard error Estimator and Standard Error from Formula Sheet r ¨ ¤ £ § ¨ ¡ ¨%¨  § ¤ r ¨ ¨   © 1 21 If is true , has a STANDARD NORMAL distribution Rejection Regions (RR): rs % e¨r P (% rR h¨r ' ## ¨  ¤  " # ¤¡ # # ¨  h  ! £ rh % R ¨r RR or ( ¨ s  P (% es r 'r OR ' #  OR &%  a fixed particular value of ¡  ¤¦ ¤ ¥D 1 1 Consider testing # ¨ s    ¤¨ ¨ ¤  ©£ #$# ¨  ¤  # " # # ¨  h  ! ¤ ¡ OR ¤r distribution. OR hypothesized value Hypothesized Value from NULL hypothesis has an approximate versus estimator ) £¡ ¦ 1 That is is the null hypothesis, TEST STATISTIC has an distribution 248 ) If 247 STA 2023 c D.Wackerly - Lecture 18 trials ) 0) ¡ How???  (2) ¤ 2 ¤¨ ¤¡ ¥£ (1) 9 would select Diet Pepsi in a blind taste test. ¡ © ¤ q ¡ true proportion of Diet Coke drinkers who £ ¢ the taste of Diet Pepsi. Is there sufficient evidence to  Coke and Diet Pepsi. . 249 STA 2023 c D.Wackerly - Lecture 18 STA 2023 c D.Wackerly - Lecture 18 Ex. : #8.68, p. 352 In a “Pepsi Challenge”, 100 Diet 250 Coke drinkers were given unmarked cups of both Diet indicated that they preferred indicate that a majority of the Diet Coke drinkers will Conclusion : reject ¡ care reform is the leading priority confidence ) to indicate that the majority of Diet  ¤ ¢ ¡ 38 9¢ ¤ 39 8¢ ps  ¡ ¤ ¨ £ ¤ ¦ ¤ ¤ ¢ 93 ¢ 93 ¤  ¡ 3 A6 ¤ £ 3 ¤ ¡ ¡ ¤r 251 1 Proportion ¡ ¡ ¡ Click Options, Select Alternative, Type in Null Value Number of trials, Number of Successes 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 p 0.560000 90% CI (0.462710, 0.657290) Z-Value 1.20 P-Value 0.115  ¡ is “large” Click radio button “Summarized Data”, type in N 100 level of significance” ( or with test. Minitab? X 56 claim that there is sufficient evidence at Coke drinkers will select Diet Pepsi in a blind taste STA 2023 c D.Wackerly - Lecture 18 Sample 1 “ (4) SAMPLE of all Diet Coke drinkers. Note: Basic Statistics In terms of this problem: the the the Pepsi Challenge are a Stat AT THE (3) Assumptions : the 100 individuals participating in Data : LEVEL!! in favor of ¢© ¤ q ¡ true proportion of all voters who think health level test, RR : ¢ 93 ¤  select Diet Pepsi in a blind taste test? ¨ £ ¢ the taste of Diet Pepsi. Is there sufficient evidence to ¡ ¡ Coke and Diet Pepsi. value? value = ...
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