Week8-4up - 175 STA 2023 c D.Wackerly Lecture 13 STA 2023 c...

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Unformatted text preview: 175 STA 2023 c D.Wackerly - Lecture 13 STA 2023 c D.Wackerly - Lecture 13 176 Thought: Vital papers will demonstrate their vitality by moving from where you left them to where you can’t find them. Last Time : ¡ Sampling Distributions ¡ Central Limit Theorem : (p. 280) For large For Today: pages 260 – 264, 265 – 270, 279 – 285 ¢ Assignments : , then regardless of the shape of the population dist. the sampling dist. of Exercises 6.15, 21, 24, 28, 37, 38, 42 – 46, is : £ For Tuesday: – approximately normal 7.1, 3–5, 10 – 11, 15–20 ¢¦ ©  §¥ © ¤¦ ¨§¥ ¤ – with mean Wednesday : P. 283 – 285, 299 – 302 – standard error Thursday: . Exercises 6.8, 6.27, 6.33, 6.41 (finish Chapter 6), 7.13, 7.37, 7.42, 7.44–7.46, QUIZ 3 177 STA 2023 c D.Wackerly - Lecture 13 STA 2023 c D.Wackerly - Lecture 13 178 Ex. : Time spent at the cashier at a movie theater : . time until all 36 have tickets min )' % # 0(&£ $" score for 30 : min min sec ) ¦ ¥¤ ! ! Want ¦ §¥ © ¢ Standard Error : before the movie starts? %£ $¦ #" % #" £ $¦ distribution. the probability that all 36 ticket buyers get tickets ) has an approximately % #"¦ £ $8) £ is “large” There are 18 minutes until the movie starts. What is ) tickets for all 36 buyers is less than 30 seconds? C DAB% 9 #5" ¦ @! 9 What is the probability that the average time to buy ¡  ¦ ¤ buyers. seconds. Thirty six ticket ¦ © seconds, 31 42! ¦ ¦ ) # 876% 1 5" !  ¡ Ex. : (#6.41, p. 275) Study relating IQ and juvenile STA 2023 c D.Wackerly - Lecture 13 b. ASSUME : Non-delinquent juveniles have same mean IQ with same std. dev. Look at non-delinquents: A © ¦ £¡ ¤¢ Want 'D(A A 3 2! 1 ) D(B©£ $" 'AA ¨ # '¦ DA ¤ juveniles given IQ test. score for § £ has an approximately . : £ distribution. . Do you think true popn. mean IQ is A¦ ¡ ¢ ! ¦ ¢© ¦ ¥ © 'A ¦ D7A @§ £ std. dev. (like for ALL juveniles)? , !  STA 2023 c D.Wackerly - Lecture 13 ¡ 181 STA 2023 c D.Wackerly - Lecture 13 182 Confidence Coefficient (p. 282) : the proportion of Chapter 7 : Estimation based on a Single the time that an interval estimator actually encloses Sample a parameter when we compute a large number of intervals (based on diff. samples). Ex., but Unknown mean, etc. Useful new idea! % 5'   ¦ 1$# 8) " &%1 $" £&  Standard Error of Est.  ! Notation: for any A B% , let , etc. be such that A ¢ © Estimator £ ¤ ¤ ¦ £! £ Parameter §  SEE FORMULA SHEET : 1 is a point estimator for " # ¡ parameter value. ' ( expressed as percentage. Ex.,  ¡ NUMBER that is intended to be “close” to the 0 ¤ Interval estimator (p. 282) : formula that tells us zA how to compute TWO NUMBERS that are intended .5-.025=.475 to ENCLOSE the value of the parameter .025 ¦ 1 1) ' 20(  BETWEEN THEM , Confidence Level (p. 256) : confidence coefficient ¡     ¤ ¡ Point estimator (p. 261) : formula for a SINGLE  7 ' Have : Large Sample from Population with fixed ¡ £   £ ¡ distribution of IQ scores for juveniles? non-delinquents, got 'A ¦ 0(C ¦B¨ 1 $" ! ) £A # Does your answer depend on the shape of the C ¦ ¥¤ ! c. Actually took sample of ¥ ¦ ¢ Standard Error : C ¦ ¢ ¥ ¦ a. What shape for the sampling distribution of ? C¦ ¢ delinquency. For ALL juveniles, IQ : is “large” 180 ¥ ¦ 179 STA 2023 c D.Wackerly - Lecture 13 0 z.025 ¡ 183 STA 2023 c D.Wackerly - Lecture 13 STA 2023 c D.Wackerly - Lecture 13 184 Confidence .5-.05=.45 Coefficient ¡ .05 ) ¡ 3 A# .99 1 1 0( ' ¦ z 0 (alpha) .05 .98 .95 Note: .90  8) 1 ) ( ' 1 % 1 % ¦ £¡  £ # 1 3 5" 1) 20( ' .025 α/2 1−α .025 1 α/2 .95 1 !  and the resulting interval . ¡ What proportion of the values of ¦ ¥© ¡ “middle” (shaded) region? with . the £ £A  ¥ ©  ¦£ are in the !  ¡ If we obtain a value of of the intervals will enclose the true value of . in the “tails” (unshaded) ! Plug that value into the formula § and the resulting interval ¤ the value of % A ¥ ©  £ ¨£ region. .95 £ . ! What fraction of the time will this occur?  © .025 Thus, in repeated sampling, approximately ¤ ¡ © .025 Plug that value into the formula value of , use it!! If not, estimate in the “middle” (shaded) region. ¤ ¡ If we obtain a value of 186 £ A3 13 £ ¦ 1) 20( ' ' ©¤¦ ) ¨¢ § STA 2023 c D.Wackerly - Lecture 13 A ¥ ©  £ ¨£ A 1 If you know  ) ¢ 1# £ 1 ) (' Recall £ 1 13 ) ¥£ ¤¢ ' 185 A 95% Confidence Interval for 1 STA 2023 c D.Wackerly - Lecture 13 not ) ¥¢ ¤ NOTE : £ £A  ¥ ©  ¦¤   ¤ £ ¥ ©  ¦A 3 ¤   ¡ STA 2023 c D.Wackerly - Lecture 13 ¥   ¨ ¦ ¥ ©§¢¤ Ex. : (#7.71, p. 314) Metalic Glass, 4 times stronger than steel, but brittle at high temps. Want to estimate 188 Confidence Interval for (large sample) p. 283 1 £  § is the z value that cuts off an area of  !¡ ) ¥¢ ¤ and ¥© 36 specimens randomly selected, measured temp.  the mean temp. at which metalic glass becomes brittle. ) ¥¢ ¤ 'C¥ ¦ (0&@§ £ 1¡ ¦¤ ¡ A7A ¦  ) ¢ # ¡ at which each became brittle :  187 STA 2023 c D.Wackerly - Lecture 13 in the upper tail of a standard normal distribution true mean temp. metalic glass gets brittle is unknown, ) ¥¢ ¤ ' 13 NOTE : the basic form of this interval is & $ %# " ' 7 ¦ a unicorn. ¢ ¡£ ¥ ¦ A ¨§ £ ¦  ¥ © ¦ 1 1 ( ' ¦ § £¦ 3A ¡ ¡ ' ( ¦ ¡ ¡ ¦ 7'   (7&¦  £ 'C¥ ¦ ¢   ¥ ¦ § £ £ ¡ £A For tomorrow: ¥ #¤ ' ( ¡   – the region of “believable” values for confidence level. 483.02 Exercises 6.8, 6.27, 6.33, 6.41 (finish Chapter 6), 7.13, 7.37, 7.42, 7.44–7.46, QUIZ 3 ¤ ) £C C 77' (0¥ ( £ : )' 0( £ # interval!! 476.98 Assignments : Today : P. 283 – 285, 299 – 302 Note : Lower confidence coefficient confidence interval for 190 Thought: A truly wise person never plays leap-frog with © ¡ ¤ CI and STA 2023 c D.Wackerly - Lecture 14 confidence   . formula sheet interval for 1 & %# " $  ) ¥¢ ¤ Ex. Back in Exercise 7.71, find a #" 189 table $ %# formula sheet interval. HOW? standard errors ) estimator £& ¡ confidence STA 2023 c D.Wackerly - Lecture 13 1 Problem actually asks for a 1−α ) ¨¢ ¤ ' ( ¡ ¦ ¡   (7&¦  £ 'C¥ ¦ ¢    ¦ § £ £ £A ¢ §£ £A ¡ ©   £¦A ¦  ¥ ©  ¦ § £ ¦  A (A ¦  (7&¦ § £ ¢ ¡ 'C¥ ¦ CI α/2 1 α/2 at the . For Monday: Read pages P. 306 – 308, MINITAB, COMPUTER DEMO ' 7   ¡ STA 2023 c D.Wackerly - Lecture 14 STA 2023 c D.Wackerly - Lecture 14 191 Ex. : (#7.71, p. 314) Metalic Glass, 4 times stronger A LARGE SAMPLE 95% Confidence than steel, but brittle at high temps. Want to estimate  Interval for the mean temp. at which metalic glass becomes brittle. £A  ¥ ©  ¦£ ¦¤ ¡ A7A ¦  ) ¢ # ¡ at which each became brittle :  ¢ £ and true mean temp. metalic glass gets brittle is unknown, with . © ' ( ¡ ) £ C¥ 0¡ (72¦A ¥ £ ¥ # ¦ ¡   (7&¦  £ 'C¥  ¡¢ # 'C¥ ¢ £A A7A  )  £ A  77&¦    ¦ § £ £ ¢ §£ £A ¡ ©   £¦A ¦  ¥ ©  ¦ § £ ¦  A (A ¦  (7&¦ § £ ¢ ¡ 'C¥ ¦ , use it!! If not, estimate  ¢ © ¦ ¥ © ¡ £ © ¡ If you know 36 specimens randomly selected, measured temp. 'C¥ ¦ (0&@§ £ Recall 192 CI .025 .95 1 £ 1 ) (' Problem actually asks for a confidence A interval. HOW? STA 2023 c D.Wackerly - Lecture 14 194 A ¨§ £ ¦  ¥ © § £¦  ¦ 7'   (7&¦  £ 'C¥ ¦ ¢   ¥ ¦ § £ £ ¡ £A : ¥ #¤ 1 ) ¨¢ ¤ ' ( ¡ confidence interval for ¤ ) £C C 77' (0¥ ( £ ¡ interval!!  !¡ ¥©   ' 7   1 ) ¥¢ ¤ 1 £  § ) ¥¢ ¤ ' ) standard errors £& $ %# formula sheet ¡£ ¥ ¦ ' ( ¦ ¡ ¡ #" 1 & %# " $  ) ¥¢ ¤ table 1 ) ¥¢ ¤ 1¡ 13 & $ %# " formula sheet Note : Lower confidence coefficient – the region of “believable” values for NOTE : the basic form of this interval is estimator 1 ( ' α/2 1−α ¦ 3A ¡ ¡ ¥   ¨ ¦ ¥ ©§¢¤ the upper tail of a standard normal distribution α/2 ¦ CI and ¡ in . ¦  is the z value that cuts off an area of ¤ (large sample) p. 283 ' 7 interval for   Confidence Interval for confidence )' 0( £ # Ex. Back in Exercise 7.71, find a ¢ A3 13 £ ¦ 1) ' 20(  ' 193 1 STA 2023 c D.Wackerly - Lecture 14 © .025 confidence level. 476.98 483.02 at the . ¡ 484.27 is ¥  A (A  'C¥ (7&¦ ¢     £ ¤ ¡ ¤ ¥ ¦ ¡ § ¦ £ £ £ ¡ 3A ¡ C ( ¦ ¡ ¡ 'C¥ (7&¦  ¥ C 7   ¡ ¤ )  £ ¦0¥  ¡ ¥ # ¤ ¥C  £ C ( ¡   – the region of “believable” values for . 475.73 472 at the C 7   – 484.27 this claim at the because C ( ¦ : .   ¤ believable values for Claim : confidence level in the region of ¤  because C ( this claim at the   ¡ – interval!! confidence level. ¥ & 475.73 472 Note : Higher confidence coefficient (.98) confidence interval for ¦ ¤ Claim : and . 196  confidence interval for CI STA 2023 c D.Wackerly - Lecture 14 195 £ ¤ Find a STA 2023 c D.Wackerly - Lecture 14 confidence level is in the interval. 484.27 STA 2023 c D.Wackerly - Lecture 14 C 0¥ 475.73 484.27 478 confidence level: ¤ A 7 C ¥ ¦£ ¦ ¥  £  ¥ £ 2 ¤¡ ¥ C (   in the interval. confidence level: ( so are ¥ £ 484.27 487 etc.). C C (   of C ( – At the C¥ ¦ 0&¤ this claim at the is ¤  – ¡ confidence level ¡ because ALL believable values for 475.73 Claim : C &¤ ¥¦ 484.27 487 this claim at the Claim : 198 £  ¤ 475.73 – STA 2023 c D.Wackerly - Lecture 14 £ Claim : 197 475.73 is C 0¥ ¡ sheet) ¢¡ If  A ' (   stay between 4 and 7 nights on a typical trip. is “large”, : (p. 300,formula is approximately normally £¢  7¦ ¢ for the true proportion of post-retirement travelers who ¦ ¨© £ ¢ ¤¦ ¡ Standard error of confidence interval ¢ post-retirees. said that they stayed away from home between 4 and 7 nights on trips. Find a 200 © Ex.: #7.79 p. 316 Survey of Estimation of a Proportion, STA 2023 c D.Wackerly - Lecture 14  ¢ 199 STA 2023 c D.Wackerly - Lecture 14 distributed. (p. 300) LOOKS FAMILIAR!!! with the Interval for a Proportion, $# ¡ " ¢¦ £ @£ ¢ ¢ £¢ ¡ £¢ ¡ ¢ ¤¦ §¦ ¥¤ ¢  7¦ ¢  A ' (   for the true proportion of post-retirement travelers who stay between 4 and 7 nights on a typical trip. ¦ ¡ ¦ ¡ ¡ ¡ ¡ ¦ £¢ ¦ £ ¢ ¡ ¦ Confidence interval : ©  or equivalently . ' ¡¥ £ ¦ & ¡  7¡ £ ¡ Thus, the believable values, at the the true proportion of ¦ @¢ post-retirement travelers who stay between 4 and 7 nights on a typical trip are those between . ¢ ¡ confidence interval  ¢ and 7 nights on trips. Find a 1 said that they stayed away from home between 4 ) ¥¢ ¤ post-retirees.  Ex.: #7.79 p. 316 Survey of © 201 ¢ STA 2023 c D.Wackerly - Lecture 14  . (p. 300) formula sheet ¢¦ £ @ ¢ table standard errors and ¡  ¢ 3 A ¦  & is a random variable. and 1 the attribute. confidence level for &$#"  ) ¨¢ ¤ formula sheet #" estimator , the sample proportion with is an unbiased estimator for , Confidence $# ¢ Estimate for : £ ¡ attribute. and get )& ¢ Large Sample Take: a random sample of size attribute (“ ”) is .   ¨ ¦ ¥ ¥ ©§¢¤ Have: a popn. where the proportion with a particular . ¡ ...
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