Week10-2up - 7.15–20 For Thursday: Exercises 7.1,...

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

Ask a homework question - tutors are online