week12-2up_001 - Monday, 6:00 pm – 10:00 pm, TUR L005...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Monday, 6:00 pm – 10:00 pm, TUR L005 Wednesday 2/13/02 : Pages 334 – 338, 347–351 scheduled discussion section. Tuesday 3/26/02 : EXAM 2–during your regularly Monday, periods 3–8, FLO 104 Help for Exam 2 bring it to class with you. etc. Suggestion - print out Sample Exam 2 and £ © number of trials, – Tables : Contain – Variance (p. 185) : – Mean (p. 185) :  ¦ ( about homework, course material, sample exam, – If ¨ 0 ¤ ! Monday 3/25/02 : OPTIONAL REVIEW, ask questions ¤ – Binomial Experiment p. 179 – NOT ALL 1 2 3 for ..... k k+1 for ..... n-1 n : (p. 183) criteria – Some discrete random variables are binomial – The Binomial Probability Distribution ¤ " ) rise above your principles.   ) Thought: To succeed in politics, it is often necessary to variable.  £ ) ¡ ¡ Can COUNT the number of distinct values of the © poker with. Chapter 4 : Discrete Random Variables   $ #  ¤ #£ ' & ¥  ¦ £ © ¨ ¨ % §¦ ¥  ( ¤ ¢  ¤ £ # © © ¤ ! #" ¡ ¡ STA 2023 c D.Wackerly - Review for Exam II # Thought: Never do card tricks for the people you play 1 # STA 2023 c D.Wackerly - Review for Exam II £ 2 ¡ a P(a<X<b) Normal distribution is special case. ¤ ¥ '  ' ¥ ¦ ¨ is a continuous r.v., graph of f(x) b Probabilities are areas under “density function”. If ¡ ¡ ¡ ¥  and , for , Dist. of (p. 255) of and standard for larger sample sizes. is called the standard error of .(p. 266) concentrated around is more is an unbiased estimator (p. 266), so dist. of . (p. 266, 261) . So is called its sampling distribution. is a random variable. deviation from a population with mean Chapter 6: If we plan to take a random sample of size pictures Key to finding correct areas (probabilities) : draw in Table IV, p. 809. Areas under normal curves between z-scores of more line intervals. ¦ ¢ ¡ ¦ ¤ ' ¨ ¡ ¥ $ Possible values are all those associated with one or ' ¥ ¡ £ ¡  ¡ ¡ ¡ ¦ § Chapter 5 : Continuous Random Variables ¢ ¤ ¨ ¡ ¦ £ % % i.e., N . True for any . If the population has a normal dist., then so does ,  ¡ ¦ ©  % ¦ $   ¦ $ ¡ STA 2023 c D.Wackerly - Review for Exam II © ¢ £ ¨ ¥  ¤ ¤ ¥   3 % ¨ £ ¢  $ ¥ ¨ $  ¨ ¤ ¤ £ ¥  $ ¥  ¥ % STA 2023 c D.Wackerly - Review for Exam II £ ¡  ¦   ¥  4 ¢  ¦ $ £  ¡ , , , etc. (p. 254) Parameter All statistics have sampling distributions Confidence Coefficient (p. 282) Confidence Level (p. 282) ¡ ¡ Interval Estimator (p. 282) Point Estimator (p. 261) Chapter 7: a Sample. A Statistic is a meaningful number associated with with a Population. $ A Parameter is a meaningful number associated % ¡ Chapter 6: distribution. regardless of the shape of the population % ¡ & ¡ ¡ ¡ ¡ ¥  © £   ¥ ¨ £ $ Estimator Error of Est. Standard Error of Est. Standard Estimated is “large”, both estimators are approximately : – For valid CI for : – For valid CI for How large is “large”? NORMALLY distributed. If Both estimators are UNBIASED © N © ¥ approximately normal, i.e., . smaller of larger of is ¡ ), then the sampling distribution of £ $  ( ¦ is large £  ¡ STA 2023 c D.Wackerly - Review for Exam II ¤ © ¦ Central Limit Theorem (CLT): (p. 267) If 5 ¤ ¡ ¡ ¤  ¤ ©  £ %  £ © £ £  £ ¤ © STA 2023 c D.Wackerly - Review for Exam II ¤  ¤ © ¨   ¨ £ ¤ £ 6  ¡ ¨ ¤ α/2 £ ¨ ¦ – Population mean, ©  ¥ 1−α table formula sheet (P. 283) – Population Proportion,  £ ¡ " ¢ ¥¦ & formula sheet ¨ ¤ standard errors (P. 300) α/2 7 £ "  § estimator  PARAMETER % $ & ©   ¥¦ £ £ ¤¨ ©   © ¦ & £ £ §¤ ¨ # § Finding the sample size to estimate . STA 2023 c D.Wackerly - Review for Exam II ¡ ¡ Confidence Interval for a &  & ©  £ ¥  © ¨ ¤ © confidence. Want : Correct to within “ ” units with standard error (p. 307) and SOLVE if you have one. Maybe and solve for – Use ballpark value for Range use . Finding the sample size to estimate . confidence. Want : Correct to within “ ” units with £ ¥¦ £ ¤ £  £ ©  ©  ©  £# & ¦ £ " ¢ & standard error to get sample size that will work for any value of . use (p. 333) and SOLVE if you have one, if not and solve for – Use “ballpark” value for ©  ¨ ¡ " © & %   ¦ © £ ¤ ¡ ¨ ¦ £ ©  £  ¤ #¤ ¡ " & ©  " ¡ ¢ £ &  ¦ %  ¤    ¨ ¨ STA 2023 c D.Wackerly - Review for Exam II © £ %  ¤ £  ¢ £ ¡  ¤  £  © ¡   8 , ¡ ¤ ¡ ¢ . (p. 322) Correct Type I error Decision Accept Ho Reject Ho Correct Type II error Ha true Reality (p. 322) , light bulb ex. Ho true ¡ ¡ ¦ ¥ Errors: p. 325 ¤ $ £ ¤" HYPOTHESIS, – The “other” hypothesis is called the NULL manner What we are “trying to prove” in an objective, fair light bulb ex. ¤" ¦ ¨ ¡ $ ¡ £ ¥ ¢ ¢ ¡ ¨ ¡ ¡ (p. 325) (p. 323), SIGNIFICANCE when true 10 In our lightbulb example, saying when saying what we “want” to say when we should not accepting and/or Type II error LEVEL of the test. ¦ ¡ ¡ ¡ ¤ ¤ ¡ ¤ ¡ ¦ ¥ ¦ ¥ © ¥ ¡ Type I error $ ALTERNATIVE or RESEARCH hypothesis, – ¤ The hypothesis of MAIN INTEREST is the ¡ ¤" Parts of a statistical test. (p. 322) ¢ ¢ STA 2023 c D.Wackerly - Review for Exam II $ Chapter 8 – Large Sample Hyp. Testing 9 ¡ STA 2023 c D.Wackerly - Review for Exam II ¤" ¤  ¢ ¡ ¨ ¦ © ¥  ¨ ¡ ¦ § ¥ ¢ ¨ ¨ ¡ § ¤ ¤ ¥ © ¡  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 UNKNOWN population mean Test Statistic £ makes a pretty small package – (John Ruskin) Last Time: Large Sample Hypothesis Testing about STA 2023 c D.Wackerly - Lecture 18 $ ¡ OR estimator RR standard error Hypothesized Value from NULL hypothesis Sheet 236 hypothesized value Estimator and Standard Error from Formula   Thought: When a man is wrapped up in himself, he 235 ¤ ¤  % ¡ ¡ ¤ ¤ ¥ £ $ ¡ ¥$ © ¤ ¢ STA 2023 c D.Wackerly - Lecture 18 ¤ ¥¤£¡¢ ¡ ¥¦¥¡¡ ¥$  £ $ $ ¤ ¥$ ¥$ £ £ ¢ § §§ £ © £   ¥©£¡¡ ¨ ¥¥¡¡ © $  water specimens ¡ α/2 -z α/2 How? £ and 0 ¡ § £ ¡ ¥ ? pH is NOT that of neutral water at the z α/2 α/2 level? “Two Tailed test” (p. 331) or  from a recreational lake. Can we claim that the mean ¡ ¥ £ ¡ £ £ £ £ ¤ $ $ £ £ ¤ ¥$ ¥$ ¤ £ ¢ ¥  ¢  ¥ %  £ $ © £ Back to pH example: STA 2023 c D.Wackerly - Lecture 18 at the not .” the ¡ 238 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: £ indicates acidity. Randomly select ¡ Ex. : pH of 7 is neutral, over 7 is alkaline, under 7 £ ¡  ¥ © ¤  ©  ¢ ¡ ¡ ¡ ¡ #¤ $  £ £ £ ¡ # ¤ ¤ #¤  "   ¤ £ & £ ¥ ¢ " " & ©  ¤ 237  STA 2023 c D.Wackerly - Lecture 18 " ¤ # ¥  # ¤ # ¤ # ¤ ¤  ¡ ¤ ¢ ¢ $ ¥ ¥ # ¥ ¡ " ? 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 to reject is chosen BEFORE the test is performed Hypothesis Testing STA 2023 c D.Wackerly - Lecture 18 Agrees with two-tailed test!! 239 confidence interval the 99% ¡ — the value “ ” is ? . ¤ #¤ ¥ Do you think that ¡ confidence interval for ¤ Construct a ¡ £ If the mean pH is NOT , what is it?  £ ¡ ¡ STA 2023 c D.Wackerly - Lecture 18 £ ¡ ¥ £ £ ¡ ¥ ¡ ¨ ¨ # $ ¡ ¡ ¡ ¤ £ ¤ ¡ ¢  £ ¥ $ $ ¡ ¡ ¢ ¥ ¢ ¤ ¤ " ¤ # £ ¤" ¤" ¥ ¤ ¡ ¢ ¢ ¢ ¤ ¥ 240 ¥ ¡ the one observed p-value = indicative that Probability of a z-value Larger z-values are true. " rejection region ¡ $ $ p-value ¡ is 241 REJECT . . . . . . p-value allows him/her to assess the “rareness” of the observed event. 242 on a person who might be interested in your conclusions, the Instead of “imposing” YOUR CHOICE of – – – – – . CANNOT reject REJECT In our case, p-value = .0256 p-value p-value STA 2023 c D.Wackerly - Lecture 18 ¡ STA 2023 c D.Wackerly - Lecture 18 ¥ ¦ £ ¡ ¤ £ ¡ ¤" ¤ ¤ # # # # ¤" # ¡  ¢ ¢ ¡ ¢    ¡ " ¤ # ¤ ¤ £ £ £ £ £ ¤  ¢  ¡ ¤ ¤ # ¤ ¤ # ¢   ¡ ¢ ¡ ¡ ¡ ¡ ¡   £ ¤ ¤ ¡ #¤ #¤ #¤ #¤ #¤  ¨ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¥ ¢ ¥ ¢ ¥ ¢ ¥ " ¢  ¤ " ¡ ¡ ¡ ¡ ¥ ¥ ¥ ¡ $ $ for any ¡ 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 for any that is ¡ ¡ ¡ ¡ See page 234 of notes: p-value Z -2.33 that is  ¡ ¥ £ ¡ ¤ £ " " ¤ ¢   ¤ ¤ £ ¡ . P 0.0099  Smaller z-values are more indicative that ¥ . 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 © # ¤ ¥ ¥ £ ¡ £ the one observed ¥ # ¡ ¥ ¢ # $ Probability of a z-value  ¡ ¡ ¡ ¡  TWO - Tailed Test  ¤ ¤ p-value ¥ ¡ #¤ ¢ ¤ ¤ # $ $ £ ¤ " " " ¤ ¢ Ex. : #8.24, P. 333 ¥ STA 2023 c D.Wackerly - Lecture 18 ¤ ¤ " 243  STA 2023 c D.Wackerly - Lecture 18 £ ¡ #¤ . ¢ 244 (1) How??? would select Diet Pepsi in a blind taste test. true proportion of Diet Coke drinkers who 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. indicated that they preferred Coke drinkers were given unmarked cups of both Diet Ex. : #8.68, p. 352 In a “Pepsi Challenge”, 100 Diet ¡ ¡ © (2) Estimate for # of trials ; number of trials in the sample size trials based on a “large” in the sample # of GOAL : Test hypotheses about guarantee expires. the proportion of batteries that fail before select Diet Pepsi in a blind taste test. the proportion of Diet Coke drinkers who Recall the BINOMIAL EXPERIMENT. . UNKNOWN but FIXED PROPORTION of items with a particular attribute 246 (Section 8.5) Interested in a POPULATION that contains an Large Sample Tests About   ¤  © © £ ¡ ¤ ¤ ¤ ¢ £ ¥¢ £ STA 2023 c D.Wackerly - Lecture 18 ¤ © ¡ ¨  ¤ ¡ ¡ ¦ ©  ¤ ¡ § £ ¤ 245 ¡ §  © ¡ STA 2023 c D.Wackerly - Lecture 18 ¢ § £ # ¡  ¤ ¥ ¢ £ Consider testing distribution. a fixed particular value of ¥¤¥¡£¡¥¡¥¡¥¡¥¡¥¡¢ ¡ ¥¦£¡¥¡¥¡¥¡¥¡¥¡¥¡¡ versus ¤ has an approximate OR OR © ¤ £  © That is © distribution has an ¤  $ ¡ ¢ % ¡ is “large” £ © © © ¤ ¢ © ¡ © £ ¤ ¤ ¥© % ¥© ¥© ¥© ¤ ¤ © © § STATISTIC if hypothesized value standard error is true Rejection Regions (RR): , 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 D.Wackerly - Lecture 18 ¡ If 247 ¡ ¡ § §§ §§ ¤ ¥¤¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥¡£¡¥¡¢ ¡ ¤ © © £¦¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥¡£¡¥¡¡ ¢ ¤ © ¤ ¥© ¥© ¥© ¡ ¤ ¤ ©  £ ¥ ¢ ¥© ¥ © © ¦ £ ¥ ¤ ¥© ¨ STA 2023 c D.Wackerly - Lecture 18 £ ¡  ¥© ¥ £ £  £ § § £ £ £ §§ §§ ¤ ¢ £ ¤ ©  £ ©   ¢ £ © £ © & or & £  © ¤  248 RR ¥©¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥¡£¡¥¡¡ ¨ £¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥¡£¡¥¡¡ ¢ indicated that they preferred true proportion of all voters who think health level test, RR : £ Data : ¤ ¤   " £ SAMPLE of all Diet Coke drinkers. Note: the the Pepsi Challenge are a ¤ © ¤ ¡ ¡ (4) (3) 249 is “large” Assumptions : the 100 individuals participating in  ¡ care reform is the leading priority © 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. # ¡ © ¢ ¢  ¡ ¤ #¤ ¤ ¡ ¢ © £ ¥¢ £ ¤ # #¤   ¢ reject AT THE level of significance” ( or with Coke drinkers will select Diet Pepsi in a blind taste value? value = test. 250 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 D.Wackerly - Lecture 18 ¡ ¡ Coke drinkers were given unmarked cups of both Diet ¤ © Ex. : #8.68, p. 352 In a “Pepsi Challenge”, 100 Diet £  STA 2023 c D.Wackerly - Lecture 18 ¤ ¢ © ¡ ¥ ¢ #¤ ¡ ¡ ¡ ¥ £ ¢ ¡ Basic Statistics 1 Proportion Sample 1 X 56 N 100 Sample p 0.560000 Test of p = 0.5 vs p > 0.5 90% CI (0.462710, 0.657290) Test and Confidence Interval for One Proportion Z-Value 1.20 P-Value 0.115 Click Box “Use test and interval based on normal distribution”, OK, OK Click Options, Select Alternative, Type in Null Value 251 ¡ Number of trials, Number of Successes Click radio button “Summarized Data”, type in Stat Minitab? STA 2023 c D.Wackerly - Lecture 18 ¡ ¡ ...
View Full Document

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.

Ask a homework question - tutors are online