Week10-4up - 175 STA 2023 c B.Presnell & D.Wackerly...

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Unformatted text preview: 175 STA 2023 c B.Presnell & D.Wackerly - Lecture 15 STA 2023 c B.Presnell & D.Wackerly - Lecture 15 176 Recall from last time: If we plan to take a random from a population with mean £ ¡ standard deviation ¥§ ¦¤ (p. 255) ¡ #!£ £ ¤ "  ¨ ¢ ¢  ¥ § ©¢ ¤     ¨ Today: P. 265 – 271 . So . (p. 266, 261) of For Tuesday: Exercises 6.15, 6.21, 6.24, 6.28, is called its sampling distribution. is an unbiased estimator (p. 266), so dist. of ¢ concentrated around for larger sample sizes. ¥§ 6.33, 6.37, 6.38, 6.41–46 is more is called the standard error of .(p. 266) ¤ For Wednesday: P. 280 – 285 For Thursday: Exercises 7.1, 7.3–5, 7.10, 7.11, New terms ¨ $£ ¤ Distribution of Assignments : and is a random variable. ¥§ hair! , ¥§ run the country are busy driving taxicabs and cutting ¥§ Thought: Too bad that all the people who know how to ¢ sample of size – Parameter – meaningful number assoc. with a Population. (p. 254) 7.15–20 – Statistic – meaningful number assoc. with a Sample. (p. 254) 177 STA 2023 c B.Presnell & D.Wackerly - Lecture 15 STA 2023 c B.Presnell & D.Wackerly - Lecture 15 178 Know the mean and standard error of , HOW ABOUT THE DISTRIBUTION ? % ¤ If the population has a normal distribution, the § sampling distribution of is normal, with mean ¡"  £ ¢  £¢ and standard deviation (standard error) distribution of x when n = 50. . ¡ ¤ Central Limit Theorem : (p. 267) For large , regardless of the shape of the population dist. § the sampling dist. of is : & approximately normal & standard error 3) 4120¡ (© ¡  ¡" £ ' £ ¢ ' ¢  & with mean ¤ Larger sample size, For most populations, . better approximation. is “large enough”. original population distribution distribution of x when n = 35. ¤ 179 STA 2023 c B.Presnell & D.Wackerly - Lecture 15 STA 2023 c B.Presnell & D.Wackerly - Lecture 15 180 Ex. : Time spent at the cashier at a movie theater : 1£ ¡ ¢ seconds, with is “large” has an approximately distribution.   ¢ & Standard Error : ©§ ¨ score for 30 : ¤& & ¥§ Sampling Dist. of   £ Want Sample Size on ¤ larger n  3 1 ¦§  ¥ ¤ Effect of Increasing & ¥§ probability. ¢ are close to tickets for all 36 buyers is less than 30 seconds? £ Values of . ¡ ¤ Smaller standard error : What is the probability that the average time to buy smaller n .  £ : ¡" £  £ ¡ Bigger §( ¤ ¢ buyers. seconds. Thirty six ticket µ 181 ¥§ STA 2023 c B.Presnell & D.Wackerly - Lecture 15 STA 2023 c B.Presnell & D.Wackerly - Lecture 15 182 Ex. : (#6.41, p. 275) Study relating IQ and juvenile delinquency. For ALL juveniles, IQ : 3 ¢   a. What shape for the sampling distribution of ? £ has an approximately distribution.  ¡ £ " ' £  &  ¢ & Standard Error : Does your answer depend on the shape of the distribution of IQ scores for juveniles? £ ¤   sec  min ¡ min is “large” §( time until all 36 have tickets  ¥ ¦§  ¤  ¥¦§  ¤  § ¤   ¥ ¦ min juveniles given IQ test. ¥§ ¡ # ¢ before the movie starts? £ the probability that all 36 ticket buyers get tickets ! "  There are 18 minutes until the movie starts. What is  ¥    &  ¤ ¤ 183 STA 2023 c B.Presnell & D.Wackerly - Lecture 15 STA 2023 c B.Presnell & D.Wackerly - Lecture 16 184 Thought: An honest politician is one who, when bought, stays bought.—(Simon Cameron) b. ASSUME : Non-delinquent juveniles have same  3   ) § score for Assignments : . 3 ¤ Want ¡ non-delinquents: # ¢ mean IQ with same std. dev. Look at Today: P. 280 – 285 : For Thursday: Exercises 7.1, 7.3–5, 7.10, 7.11, 7.15–20 & ©¨ § For Monday: P. 299–304, COMPUTER DEMONSTRATION For Tuesday: Exercises 7.37, 7.42, 7.44–46 non-delinquents, got Last Time: SEE FORMULA SHEET , Estimator ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 16 the time that an interval estimator actually encloses a parameter when we compute a large number of intervals (based on diff. samples). Ex., Sample etc. ¤ , Confidence Level (p. 256) : confidence coefficient 3 ¢ ¤ ¤ Point estimator (p. 261) : formula for a SINGLE £¡ expressed as percentage. Ex., ¡ Have : Large Sample from Population with fixed £¡ ¤ Confidence Coefficient (p. 282) : the proportion of Chapter 7 : Estimation based on a Single but Unknown mean, 186 3 & 185 approximately NORMALLY ¡  ¡! ¢ ¡!  STA 2023 c B.Presnell & D.Wackerly - Lecture 16 ¥§ §  3    ¥§ For “large” sample size, distributed. Standard Error of Est. ¡" Parameter £ (like for ALL juveniles)? 3 . Do you think true popn. mean IQ is   std. dev. # ¢   1  ¢ ) §  ¤ & ! c. Actually took sample of , etc. Useful new idea! NUMBER that is intended to be “close” to the ¤¥ 3 ¥¤ ¡ !¤  ¦ § ©§ ¨  ¤ ¥§  ¢ ¢ :  ¥ is a point estimator for , let ¦ §§ Notation: for any parameter value. be such that A §& Interval estimator (p. 282) : formula that tells us how to compute TWO NUMBERS that are intended 0 to ENCLOSE the value of the parameter zA BETWEEN THEM .5-.025=.475 .025 0 z.025     § ¤ 187 STA 2023 c B.Presnell & D.Wackerly - Lecture 16 STA 2023 c B.Presnell & D.Wackerly - Lecture 16 Confidence .5-.05=.45 (alpha) Coefficient   ¢ © z ¢ .05  0 188 .99 .05    § .98 .95 Note: .90 !  ¡! α/2    § ¥ § ¥   § ©   ¤ .025 α/2 1−α .025 .95 and the resulting interval the .  “middle” (shaded) region? are in the & ¤ Thus, in repeated sampling, approximately ¤ If we obtain a value of in the “tails” (unshaded) & ¢ the value of § and the resulting interval   Plug that value into the formula ¡¡ !  § region. . . & What fraction of the time will this occur? £  £ £ .025 of the intervals will enclose the true value of % ¢  £ ¡¡ ¢ §  ! ¤ What proportion of the values of § ¤ & Plug that value into the formula value of with . ¥§ £   £ ¤ .95 in the “middle” (shaded) region. ¢ .025 If we obtain a value of 190  £ ¡¡ !  §  ! ¡¡ ¢ © ¥§  ©£ ¨§ STA 2023 c B.Presnell & D.Wackerly - Lecture 16 §    § © 3 , use it!! If not, estimate ¡¡ !  If you know  £ §    § Recall not ! §  ¦¤§ © ¥£ 3 189 A 95% Confidence Interval for  ¦£ § ¥ STA 2023 c B.Presnell & D.Wackerly - Lecture 16 § NOTE : ¨ !  £ ¡¡ ¢ ¢ ¢ ¨ ! £ ¡¡ ¢ © ¢ ¤ STA 2023 c B.Presnell & D.Wackerly - Lecture 16 ¥   ¨ ¦ ¥ ©§¢¤ than steel, but brittle at high temps. Want to estimate (large sample) p. 283 36 specimens randomly selected, measured temp. ¨ ¥  £  ¦£ § ¥ § 3 #  ¥ §  and is the z value that cuts off an area of $¢ the mean temp. at which metalic glass becomes brittle. ¤  ¦£ § ¥ 1¡¤ ¢¤     ¢  ¡ at which each became brittle : Confidence Interval for Ex. : (#7.71, p. 314) Metalic Glass, 4 times stronger 192  191 STA 2023 c B.Presnell & D.Wackerly - Lecture 16 in the upper tail of a standard normal distribution true mean temp. metalic glass gets brittle is unknown,  ¦£ § © ¥ " #! £¡ 3 ¤ £¡ 3  ( ! 3 ¡ !    §    3! 1  3  #   ¡  " # ¢  ¥ § £ ! ¥§   ¢  3 ! 1 # ¡  ¡ !  #  ¢  476.98 ¡"  ¨ ¥ § £   : ¤ ¢ ¢ ©  ¤ £¡ 3 3¤ £¡ – the region of “believable” values for confidence level. ! # ¢  ( confidence interval for 483.02 £ ¢ ¢ interval!! at the $ Note : Lower confidence coefficient . $ #! " £¡ CI confidence  ¦£ §  ¥  ¡ ! 1  3  #   ¡  " ¡¡ ¢  ¥ § £ !  ¨ ! ¥ § '£ ¡¡ ¢  ¥ §  . and 193 table  formula sheet standard errors " #! ¤ confidence $ ¡"  interval for 3 ! ¡¡ ¢ estimator STA 2023 c B.Presnell & D.Wackerly - Lecture 16 Ex. Back in Exercise 7.71, find a NOTE : the basic form of this interval is formula sheet ! £ interval. HOW? 1−α  ¥©£ § 3  #  ¥§  Problem actually asks for a α/2 §   CI α/2 £¡ 3 ¤ ...
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