Week8_001 - STA 2023 c D.Wackerly - Lecture 13 175 Thought:...

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Unformatted text preview: STA 2023 c D.Wackerly - Lecture 13 175 Thought: Vital papers will demonstrate their vitality by moving from where you left them to where you can’t find them. Assignments : For Today: pages 260 – 264, 265 – 270, 279 – 285 For Tuesday: Exercises 6.15, 21, 24, 28, 37, 38, 42 – 46, 7.1, 3–5, 10 – 11, 15–20 Wednesday : P. 283 – 285, 299 – 302 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 STA 2023 c D.Wackerly - Lecture 13 176 Last Time : Sampling Distributions ¡ Central Limit Theorem : (p. 280) For large , then regardless of the shape of the population dist. is : ¢ the sampling dist. of – approximately normal £ ¥ ¤ £ – with mean ¦ . ¥ ¦ ¡ ¤ – standard error STA 2023 c D.Wackerly - Lecture 13 177 Ex. : Time spent at the cashier at a movie theater : seconds. Thirty six ticket £ ¥ ¡ ¢ seconds, ¦ ¥ 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. ¥ . ¦ ¨§   ¦ score for 30 : ¦ ¦ £ Want ¤ ¤ ¤ ¡ Standard Error : ¥ £ buyers.  £ ¥ ©  ¢ © ¨ § ¦ ¤ STA 2023 c D.Wackerly - Lecture 13 178 There are 18 minutes until the movie starts. What is the probability that all 36 ticket buyers get tickets before the movie starts? min ¨§ © © ¢ min sec   ©  ¥ ¨§ ¢ ¢ ¨§ ¥ ¥ ¥ min  time until all 36 have tickets ¢ £¡ © ¥ ¨§ ¦ STA 2023 c D.Wackerly - Lecture 13 179 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 D.Wackerly - Lecture 13 180 b. ASSUME : Non-delinquent juveniles have same ¥ ¡ non-delinquents: ¥ ¦¢ mean IQ with same std. dev. Look at   ¡ ¡ . : ¥ . Do you think true popn. mean IQ is (like for ALL juveniles)? ¡ ¨ § ¦ ¡ ¢¡ ¥ ¡¡ § ¥ ¢ std. dev. non-delinquents, got  £ ¢£ ¡ c. Actually took sample of ¥ ¦¢   ¦ score for ¡¡ ¨§ ¢ Want , ¦ STA 2023 c D.Wackerly - Lecture 13 181 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. £ £ Estimator ¥ ¢ ¦ Parameter ¢ SEE FORMULA SHEET : § is a point estimator for Standard Error of Est. ¦ ¢ £ ¡ 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 D.Wackerly - Lecture 13 182 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: z.05 0 ¥ .05  ¨¦ ©§ .5-.05=.45 STA 2023 c D.Wackerly - Lecture 13 183 STA 2023 c D.Wackerly - Lecture 13 184 (alpha) Confidence Coefficient   ¡¨ .99 .98 .95 .90 α/2 1−α  ¤¡  £  ¤¢  £¡   ¨ ¡ not £ NOTE :  α/2 ¨ £ ¡¥ © ¤§¦ STA 2023 c D.Wackerly - Lecture 13 185 A 95% Confidence Interval for ¤ ¦ ¡¡ £¡ ¡ ¢ Recall ¥ with . ¦ , use it!! If not, estimate ¢ ¤ ¦ .025 .025 .95 § ¢ £ ¤ ¦ ¡¡ £¡ £ ¤ £ £ ¤ ¦ If you know ¦ ¢¡ £¡  £ STA 2023 c D.Wackerly - Lecture 13 in the “middle” (shaded) ¢ If we obtain a value of 186 region. ¦ £¡ ¡ ¢ ¦ the . What proportion of the values of ¢ £ value of ¤ and the resulting interval ¡¡ Plug that value into the formula are in the “middle” (shaded) region? ¦ Thus, in repeated sampling, approximately % ¢ If we obtain a value of . £ in the “tails” (unshaded) region. £¡ ¡ ¢ ¦ £ the value of . ¦ What fraction of the time will this occur? ¦ and the resulting interval ¡¡ Plug that value into the formula ¤ of the intervals will enclose the true value of ¤ STA 2023 c D.Wackerly - Lecture 13 187 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 ¡ ¢  ¡¡ true mean temp. metalic glass gets brittle is ¥ ¥ £ ¦ ¡ ¡¡ £¡ § ¡  ¢ ¥ ¢ ¢ ¥ ¤ £ ¦ ¡¡ ¡¡ ¢ ¥ ¡ ¥ ¡  ¡ ¡ ¡£ £ § £¡ ¥ § ¡ £¡ ¡ ¡ ¢ ¢ § ¥ ¡ ¥ ¡ ¡ ¡ £ ¢ ¢¥ £ ¥ Problem actually asks for a interval. HOW? ¥ CI ¡¡ ¨ ¢ unknown, confidence STA 2023 c D.Wackerly - Lecture 13 188 §§ ©¨¡ 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  £ ¡ £ ¡   NOTE : the basic form of this interval is     table   ¨   ¤¡  £ ¡     formula sheet standard errors formula sheet £  estimator STA 2023 c D.Wackerly - Lecture 13 189 ¡  . ¥ ¨ ©¦  ¥ ¡¥ £¡ ¡ § ¡ ¢ ¥ £ ¤ ¡ ¢ ¡¥ £¡ ¥ ¡ ¥ £ £ ¢ ¥ ¢ £ ¡ ¢¥  £ ¢ ¥¨ £ confidence level. 483.02 £ £ £ § ¥ ¡ ¥ ¡  – the region of “believable” values for at the  ¡ : ¡ ¢ confidence interval for ¥   £¨ § ¦ ¡ interval!! ¡ ¥ Note : Lower confidence coefficient 476.98 ¦ ¡ ¥ CI and confidence ¥ £ interval for ¡ Ex. Back in Exercise 7.71, find a . ¡  ¡ ¡ STA 2023 c D.Wackerly - Lecture 14 190 Thought: A truly wise person never plays leap-frog with a unicorn. Assignments : Today : P. 283 – 285, 299 – 302 For tomorrow: Exercises 6.8, 6.27, 6.33, 6.41 (finish Chapter 6), 7.13, 7.37, 7.42, 7.44–7.46, QUIZ 3 For Monday: Read pages P. 306 – 308, MINITAB, COMPUTER DEMO STA 2023 c D.Wackerly - Lecture 14 191 A LARGE SAMPLE 95% Confidence Interval for ¤ ¦ ¡¡ £¡ ¡ ¢ Recall ¢ £ ¥ with . .025 ¦ ¡ .95  £¡ ¢¡  ©§  ¨¦  ¦ ¤ .025 ¦ ¦ ¡ , use it!! If not, estimate ¢ £ If you know £¡  ¡¡  ©§   ¨¦ STA 2023 c D.Wackerly - Lecture 14 192 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 ¡ ¢  ¡¡ true mean temp. metalic glass gets brittle is ¥ ¥ £ ¦ ¡¡ ¡¡ £   ¡ ¡¡ § ¡ ¢ ¥ ¤ £ ¦ £ ¡¨ ¡ ¡ £ ¢ ¥ ¡ ¥ £ £  ¢ ¥ ¥ ¡ ¡ ¡¡ ¡¡ ¢ confidence ¡ £¡  ¢ ¥ ¢  ¥¨ ¡ ¥ ¡ ¡£ £ § £¡ ¥ § ¡ £¡ ¡ ¡ ¢ ¢ § ¥ ¡ ¥ ¡ ¡ ¡ £ ¢ ¢¥ £ ¥ Problem actually asks for a interval. HOW? ¥ CI ¡¡ ¨ ¢ unknown, STA 2023 c D.Wackerly - Lecture 14 193 §§ ©¨¡ 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  £ ¡ £ ¡   NOTE : the basic form of this interval is     table   ¨   ¤¡  £ ¡     formula sheet standard errors formula sheet £  estimator STA 2023 c D.Wackerly - Lecture 14 194 ¡  . ¥ ¨ ©¦  ¥ ¡¥ £¡ ¡ § ¡ ¢ ¥ £ ¤ ¡ ¢ ¡¥ £¡ ¥ ¡ ¥ £ £ ¢ ¥ ¢ £ ¡ ¢¥  £ ¢ ¥¨ £ confidence level. 483.02 £ £ £ § ¥ ¡ ¥ ¡  – the region of “believable” values for at the  ¡ : ¡ ¢ confidence interval for ¥   £¨ § ¦ ¡ interval!! ¡ ¥ Note : Lower confidence coefficient 476.98 ¦ ¡ ¥ CI and confidence ¥ £ interval for ¡ Ex. Back in Exercise 7.71, find a . ¡  ¡ ¡ STA 2023 c D.Wackerly - Lecture 14 confidence interval for . £ ¡ ¥ ¥ ¢ ¥ ¢  ¢ ¥ ¡ £ ¥ ¡ ¡ ¥ ¢¥ ¡ ¢ ¥ £ ¥¦¢ ¥ £ £ ¡ ¥¨ £ ¡ ¢ – the region of “believable” values for confidence level. 475.73 484.27 £ : confidence interval for at the  ¡ interval!! £¥ ¢ ¡ Note : Higher confidence coefficient (.98) ¥ § £ CI . ¡ ¥ and ¡¡ ¡ Find a 195 ¡ ¢ ¡ ¡ STA 2023 c D.Wackerly - Lecture 14 £ £ this claim at the is ¥ believable values for . £ ¥ £ Claim : in the region of £ because confidence level ¢ 475.73 472 because interval. ¢ this claim at the ¡ – 484.27 ¡ – 484.27 ¡ ¥ 475.73 472 ¡ ¥ Claim : 196 confidence level is in the STA 2023 c D.Wackerly - Lecture 14 £ ¢¥ confidence level because ALL believable values for 484.27 487 ¢ £ is ¡ ¥ this claim at the ¡ ¢¥ – £ 475.73 Claim : £ this claim at the ¢ – 484.27 487 ¡ £ 475.73 ¡ Claim : 197 in the interval. confidence level: ¢¥ STA 2023 c D.Wackerly - Lecture 14 £¢ ¥ ¥ £ Claim : 198 475.73 484.27 478 ¡ confidence level: ¥ £ ¡£ ¥ ¢ ¥ ££ ¡ ¥ £ ¡ ¥ ( so are etc.). ¢ of ¢ ¡ – At the is STA 2023 c D.Wackerly - Lecture 14 199 £ £ ¥ ¡ Ex.: #7.79 p. 316 Survey of Estimation of a Proportion, post-retirees. said that they stayed away from home between 4 ¡  ¡ ¡ and 7 nights on trips. Find a confidence interval for the true proportion of post-retirement travelers who stay between 4 and 7 nights on a typical trip. Have: a popn. where the proportion with a particular attribute (“ ”) is . ¢ and get ¢ ¡ ¡ Take: a random sample of size with the attribute. ¢ , the sample proportion with ¥ £ ¢ Estimate for : ¢ ¡ the attribute. is a random variable. ¥ ¢ ¥ ¤ £ is an unbiased estimator for , . (p. 300) ¢ £ ¢ £ ¢ STA 2023 c D.Wackerly - Lecture 14 200 ¢ (p. 300,formula ¥ ¤ ¦ : ¥ £ ¢ Standard error of ¡ sheet) ¢ is “large”, is approximately normally £ ¡ If distributed. (p. 300) LOOKS FAMILIAR!!! § ©§ ¡ ¥ ¤ £ ¢ ¡ Large Sample Confidence Interval for a Proportion, ¨  £ ¡  standard errors    ¡     formula sheet table     formula sheet estimator ¢ ¡ ¢ ¥ ¢ and . ¡ ¥ ¢ ¡  ¤¡  £ ¡ ¢ STA 2023 c D.Wackerly - Lecture 14 201 £ £ ¥ ¡ Ex.: #7.79 p. 316 Survey of post-retirees. said that they stayed away from home between 4 ¡  ¡ ¡ and 7 nights on trips. Find a confidence interval for the true proportion of post-retirement travelers who stay between 4 and 7 nights on a typical trip. ¥ ¥ ¥ ¢ ¥ ¥ ¥ £ ¢ ¡ Confidence interval : ¡ or equivalently . ¡ ¥ £ ¡ ¡ £ ¢¡ £ Thus, the believable values, at the ¥ ¢ confidence level for the true proportion of post-retirement travelers who stay between 4 and 7 nights on a typical trip are those between and . ...
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