Week11-2up - Tuesday: EXAM 2 Monday: OPTIONAL review day...

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Unformatted text preview: Tuesday: EXAM 2 Monday: OPTIONAL review day 8.7, 8.9, 8.10, 8.13 For Thursday: Exercises 7.51, 7.53, 7.59, 7.64, 7.67, Wednesday: P. 306 – 308, 322 – 326 7.77, 7.79, 7.82, 7.91 such that Confidence Level (p. 282) Confidence Coefficient Interval Estimator (p. 282) . ¦ ¤ For Tuesday: Exercises 7.37, 7.42, 7.44–46, 7.72, ¡ ¡ ¡ Point Estimator (p. 261) ¢ Today: P. 299 – 302, COMPUTER DEMONSTRATION ¥ Last Time: © Assignments : § and decide that nothing can be done.  ¡ ¡ ¢ £ individually, can do nothing, but collectively can meet, ¦ Thought: A committee is a group of people who, STA 2023 c B.Presnell & D.Wackerly - Lecture 17 § ¨ § ¨ 194  STA 2023 c B.Presnell & D.Wackerly - Lecture 17  (p. 282) 195 § (large sample) p. 283 α/2 1−α formula sheet table standard errors NOTE : the basic form of this interval is estimator α/2 formula sheet in § 197 and unknown, true mean temp. metalic glass gets brittle is at which each became brittle : 36 specimens randomly selected, measured temp. the mean temp. at which metalic glass becomes brittle. than steel, but brittle at high temps. Want to estimate Ex. : (#7.71, p. 314) Metalic Glass, 4 times stronger STA 2023 c B.Presnell & D.Wackerly - Lecture 17 the upper tail of a standard normal distribution is the z value that cuts off an area of 196 ¨ ¡ ¢ § § ¢ 0 4 £¤   ¡ ¡ ¤ ¦ ¡¦ § £ ¥ %   © ( ¡ 5 ¦ "  &' § § 1 £ 9     $ %    ¤ ¥ #  £ 9 CI and ¥  78 ¥ ¡ 6 ¡ ¡ ! ¡ £ Confidence Interval for &' ¢ (% &' confidence interval for : ¦ ! ¦ () ¥ $ $ H F  ) 9  ) © $ © © 32 $A F §   £  32 )8 $ £  @ BC DE £ $  A2 )8  ) )8 A2  $   6 $! )7 4  ¢ A2 4 £ ) ¢ G   © 2 8Q 5 £ $ R S93 2 8Q  © )8  £  32  A2 ¢ £  32  $  32 ) ¦ G $! £ STA 2023 c B.Presnell & D.Wackerly - Lecture 17 ¦ 6 )8 )7 R 93 ¢ 0 $! £ 78 G 1 ¦ A2 32 I )7 P  ¤ £ CI and .  : – the region of “believable” values for confidence interval for interval!! Note : Higher confidence coefficient (.98) 93 475.73 confidence level. 484.27 @ at the . 198 £ ¡ – interval. because 484.27 is this claim at the 475.73 472 . 199 confidence level is in the confidence level 484.27 in the region of this claim at the believable values for because Claim : – 475.73 472 Claim : STA 2023 c B.Presnell & D.Wackerly - Lecture 17 F 93 ¡ ! confidence interval for  Find a ¥ 93 ¥ 93 ¥ F STA 2023 c B.Presnell & D.Wackerly - Lecture 17  5 ¡  32 ©   $ @ ) !2 F 5  4 6 Q H F ¡  5 5 £ 78 G  2 !Q 32 ¢ ¡ ¡ 2 !Q ) ) 2 !Q 93 93 $ 2 AQ ) RQ ¦ Q 7 32 ) !2 5 5 F F 484.27 487 confidence level because ALL believable values for this claim at the is 484.27 487 in the interval. this claim at the 475.73 confidence level: ¡ ¡ 475.73 ( so are – At the Claim : 478 of confidence level: 484.27 STA 2023 c B.Presnell & D.Wackerly - Lecture 17 5 Claim : – 475.73  – 32 )Q 32 )Q  Claim : 5 5 32 Q ¡ 93 93 93 F 2 3Q ) F 200 £ 5 F 5 2 AQ )R 9 2 8Q )R 7 32 )R 2 STA 2023 c B.Presnell & D.Wackerly - Lecture 17 etc.). 2 3Q is 201 ¢ ¡ ¢ the attribute. Estimate for : attribute.  ¢£ and get with the is an unbiased estimator for , . (p. 300) , the sample proportion with is a random variable. 6  Take: a random sample of size attribute (“ ”) is . 6 Have: a popn. where the proportion with a particular stay between 4 and 7 nights on a typical trip. ¢ If LOOKS FAMILIAR!!! Confidence estimator formula sheet table standard errors Interval for a Proportion, Large Sample (p. 300,formula is approximately normally : distributed. (p. 300) is “large”, sheet) Standard error of for the true proportion of post-retirement travelers who confidence interval F $ 9 and 7 nights on trips. Find a 5 £ 6 7!  7 said that they stayed away from home between 4 ¢¨ post-retirees. ¥ ¦¤ ¡ ¡ 6 % Ex.: #7.79 p. 316 Survey of ¢§ ¨ Estimation of a Proportion, ¨ (  "  ¡ ¡ ¢£ ¡ § ¢ ¢£  ¥¤ ¢ (% I and formula sheet ¢¨ £¤ ¢ %   &'   R 6   ¡¦  &' § 6 ¦ 6   STA 2023 c B.Presnell & D.Wackerly - Lecture 17 £ !Q ¡ ¡ ¢£ ¢£ 202  STA 2023 c B.Presnell & D.Wackerly - Lecture 17 ¤ ¢§ § &'  ¦ ( ¨ § 203 . P ¢¨ post-retirees. confidence interval  ¡ ¡ ¡ )A ¥ . the true proportion of 204 and . nights on a typical trip are those between post-retirement travelers who stay between 4 and 7 confidence level for Thus, the believable values, at the 7! ¡ ¥ ¢£   A   $  or equivalently ¢ ¡ ! Confidence interval :  ¡  6 @ Q A2 )  stay between 4 and 7 nights on a typical trip. for the true proportion of post-retirement travelers who and 7 nights on trips. Find a 9 said that they stayed away from home between 4 $ !Q Ex.: #7.79 p. 316 Survey of STA 2023 c B.Presnell & D.Wackerly - Lecture 17  F £ 6 7! 7 205 We cannot give out grades over the phone. Tuesday, 4/3/01: EXAMS RETURNED Wednesday: P. 328 – 332 Tuesday: EXAM 2 Monday: OPTIONAL review day 8.7, 8.9, 8.10, 8.13 For Thursday: Exercises 7.51, 7.53, 7.59, 7.64, 7.67, Today: P. 306 – 308, 322 – 326 Assignments : makes a pretty small package – (John Ruskin) Thought: When a man is wrapped up in himself, he STA 2023 c B.Presnell & D.Wackerly - Lecture 18 $ £ &' ¤ § ¥ ¦ %  ¤ 1−α table sample: standard errors formula sheet ¦ α/2 (P. 283), LARGE sample: (P. 300), LARGE § ¢ £ – Population Proportion,  © § – Population mean, #  α/2 ¢¨ ¢ ¡ F formula sheet  Estimate to be within sample mean, . confidence. units of the true value of That is : .025 is within of .95 .025 , how large should “ ”, the sample size be? Question : If the population standard deviation is with , by using the Choosing the Sample Size Estimate the population mean, Want: ¡ ¡ $ estimator 6  §  G ( 5 %   &' §  PARAMETER ¢§ ¨ £ Confidence Interval for a 6  5 (% $ ¢ ©   ¨ £ § 6 H "  £ () STA 2023 c B.Presnell & D.Wackerly - Lecture 18 5 &' §  ) ¤  £ Last Time: 206 ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 18 5 #  5 9A F 98 ) 4    G 7 98 ) 98 207 of the time ) ! 5   6 5  © £ ¡ ¡ ¡  95% of time. 95% of the time. STA 2023 c B.Presnell & D.Wackerly - Lecture 18 “” in prev. example) (p. 307) to within “ ” units with or , and use from past study or pilot study if available. – guess the range, – use Need : at least a “ballpark value for and solve for confidence level If I want to estimate  £ bound I want ( 7 Want : units of Want : to be within 98 £ ) 98 ) 98 £ 89 6 ! ! ) 3Q ¢ ¡ ) 7  6 G ) of ¢ is within ¢ 5  G 6 ¢ £ ¡ ¡ 5 ¡  6 ) A7 G! 7 ) !A7 ) 3Q 2 2 ¦  7  7  7 6 4  6 Know: 208 ¤ 6    ¤ 2 H 6 ! £ £ STA 2023 c B.Presnell & D.Wackerly - Lecture 18 ¤ G 8  £ )  5 §   ¥  G 6 ¤  G 8  Q 2  H8  209  5  98 £   ¤ £ have to be? ¥ $4800 and $13,000. How large does 2 6 bulbs on test, true mean Decision is to be made using sample data. UNKNOWN ) lifelength of the manufacturer’s bulbs. ( FIXED, but We do not know the true value of (ALTERNATIVE hypothesis) (NULL hypothesis) Choose between two HYPOTHESES. Can the claim be made? (VALIDLY) Data : record time to burnout for each. Put a random sample of hours. 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 year to within $500 with prob. .95. Know – cost between £ !   H  ¡ average cost of freshman £ § B )  Ex. Want to estimate $ $A STA 2023 c B.Presnell & D.Wackerly - Lecture 18  ED 98 4 210  !  6 !  STA 2023 c B.Presnell & D.Wackerly - Lecture 18 ! A ¢ 3 ) ) ¥ G $ ¡ ¡  ¡¢ £5 ¤¢ £5 6 $A $ $ $ ) 78 6 3  $ 78  ¦    6 7 £ £   7 H G 6 9 7! 7! @ G $ A A ¡ 2  9 )7 5 ¡ ¡ ¡ ¡ 6   211 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,  and © © Type II Error  ” Type II Error – Leading to an – Thus, “accept must decide to “reject – If we want to REDUCE How about both very small? and ” in 213 OFTEN. Type I Error we . OFTEN. to be is a fixed sample size. Ideally, want BOTH Suppose (p. 325) (p. 323) are PROBABILITIES ( numbers between 0 and 1 ) Note: beta ¢ Using sample data to make decision – always the ¡ ¡ ¡ ¡ % % &' Type I Error alpha ¥ ¡ ¥ &' ( ¢ ( ¢ about the POPULATION.  ¡ ¥ Sample DOES NOT contain ALL of the information ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 18 © ¢ 212  6 ¡ ) ¡ ) ¡ © STA 2023 c B.Presnell & D.Wackerly - Lecture 18 ¡ ¥ ¡ ¡ ¡ ¡ ¢ small enough to be “CONVINCING” directions. . SMALL we our confidence in our conclusion to ACCECT – By making is . 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 ¡ ¥ ¥ ¥ ¢ ¢ 6 R ¡ R ¢ ¢ ¡ move in ¥ and ¥ 215 . (p. 322) . (p. 322) hypothesis. (NULL hypothesis) (2) (ALTERNATIVE hypothesis) (1) specifying the ALTERNATIVE or “RESEARCH” PARAMETER EQUALS the value used in The NULL hypothesis is always that a In out set-ups – Denote by HYPOTHESIS The “other” hypothesis is called the NULL – Denote by manner – What we are “trying to prove” in an objective, fair Hypothesis. ALTERNATIVE HYPOTHESIS, or the Research ¤ £5 ¡ £5 . ¤ The hypothesis of MAIN INTEREST is the  ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 18 £ £ For fixed 214  STA 2023 c B.Presnell & D.Wackerly - Lecture 18 7! 7! ¡ ¡ ¤ ¡ A A ¥ ¥ © accepting when true ¡ ¡ © for the ¥ ’s that we pick. The test that we will discuss have the SMALLEST the test. ’s SIGNIFICANCE LEVEL of the test, LEVEL of saying when saying what we would like to say when we should not © ¡ ¡ ¡ ¡ ¡ 216 ¡ ¥ ¥ ¥      Type I error ¢ © ¤ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 18 ¡ ...
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