Week10 - STA 2023 c B.Presnell & D.Wackerly -...

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

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