Week7-4up_001 - 147 STA 2023 c D.Wackerly - Lecture 11 STA...

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Unformatted text preview: 147 STA 2023 c D.Wackerly - Lecture 11 STA 2023 c D.Wackerly - Lecture 11 148 For a continuous random variable, Probabilities of intervals are areas under curves: graph of f(x) Thought: “I personally think we developed language because of our deep need to complain” - Lily Tomlin P(a<X<b) a Today : pages 215–226 b The area under a curve above a single point Line has area 0 For tomorrow: Exercises 5.15, 5.16, 5.19, 5.20, is 0. ¡ Assignments : For Wednesday: Read more on pages 198–208 a For Thursday: ¢ For any single value , UNLIKE for discrete random variables, if continuous random variable, is a ¦ ¢ 96, 98, 6.1, 6.3, 6.4  ¨¡ ¨ ¤ ©§¦¥£ ¡ Exercises 5.24, 5.25, 5.31, 5.36–37, 5.39–40, 95,  #!"%'¦&%0§¡2$1#'&0§¡¥)¨ ¤ £ ¨ ! % ¦  ¤ £ !  ¦ % ¤ £ ¨ ! ¦  ¤ ("'&¡¥$#" ¡¥£ 149 STA 2023 c D.Wackerly - Lecture 11 STA 2023 c D.Wackerly - Lecture 11 150 The Normal Distribution The graph of the normal density is a symmetric, 5.6, p. 216) 3 if its probability density is (p. 216) µ =8 σ =1 ¢ ¢ CDB@ 894 A ¨6§¦¤ 5 @VUS IPQGH(¤ E TR F 4 7 4 XX g X@ #(X g @ (7 fed$¨ C XX ca7` XW ¨ ##X Vb#7 YDA ¢ No need to know this formula. and std dev 4 3  4p srq§3¤ h i¦ ” MEANS “ has a normal distribution  7 da tp a ¤ X@ and std dev X 7 [email protected] ¢ ¢ h i¦ ¢ with mean 8 10 12 14 16 µ =10 σ =3 As a shorthand for “ has a normal distribution with mean N 6 µ =10 σ =1 ¦ N µ =12 σ =1 Two Normal Curves with Diff Means, Same Std Dev Formula traces out the “bell shaped curve”. Will write 3 Recall: , measures the spread of the distribution. (See Figure A random variable has a normal dist. with mean and std. dev. . The std dev, 4 bell-shaped curve, centered at One very important continuous distribution. ” 0 5 10 15 Two Normal Curves with Same Mean and Differnent Std Devs 20 151 STA 2023 c D.Wackerly - Lecture 11 STA 2023 c D.Wackerly - Lecture 11 152 The Standard Normal Distribution – p. 217 For any normal random variable, areas under normal Xt 7 Y¤ ¡ h p and std dev , given in Table IV, p. 809 . and , for X34¦ ¨ E Recall : -score :  i.e., curves between z-scores of 7 is normal with mean  distribution if is said to have a standard normal  ¤ £ The r. v. are Tabled area The -score gives the number of standard is from 3 ¦ deviations . ¦ ¢ ¦¨  7 ¦ ¨ 3 4 ¦ ¨ ¢¢ E E ¦ 3¨©¦ ¦¨ ¢¢ ¥ 3 ¦ ¦ % ¢¢ ¥ ¦ ¦ ¤¢¢ ¥ £ 3 is larger than 0 . ¢ is smaller than z . Given in table . is a standard normal random variable: 1/2 -z Find 0 . 0 Find z Find is already a -score. .4826 .4864 . . . .4982 .4987 .4830 .4868 . . . .4982 .4987 ... ... .. . ... ... .3830 .4015 .4177 .4319 .4441 . . . .4767 .4817 .4854 .4887 . . . .4986 .4990 .4857 .4890 . . . .4986 .4990 Find ¨ .4821 .4861 . . . .4981 .4987 ... ... .3810 .3997 .4162 .4306 .4429 . . . .4761 .4812 ¨7 ¨¤  c X ¤ ¥£ 2.1 2.2 . . . 2.9 3.0 ... ... ... ... ... .. . ¢ .3686 .3888 .4066 .4222 4357 . . . .4726 .4783 Find X t 7 c X © ¥£ ¨¤ .3665 .3869 .4049 .4207 .4345 . . . .4719 .4778 Find .09 .0359 .0753 .1141 . . . .3389 .3621 ¨ .3643 .3849 .4032 .4192 .4332 . . . .4713 .4772 ... ... .08 .0319 .0714 .1103 . . . .3365 .3599 ¨  g XW 7 % ¥£ ¤ 1.1 1.2 1.3 1.4 1.5 . . . 1.9 2.0 ... ... ... ... .. . ¢ § .02 .0080 .0478 .0871 . . . .3212 .3461 is a standard normal r.v., then ¢ 0.0 0.1 0.2 . . . 0.9 1.0 .01 .0040 .0438 .0832 . . . .3186 .3438 154 X t g XW 7 % ¥£ ¤ ¨  g XW 7 % % 2£ ¤ Ex. If (Part of the) Standard Normal Table .00 .0000 .0398 .0793 . . . .3159 .3414 STA 2023 c D.Wackerly - Lecture 11 X t g XW 7 % % ¥£ ¤ 153 STA 2023 c D.Wackerly - Lecture 11 That is, the value of 1/2 If STA 2023 c D.Wackerly - Lecture 11 is a STANDARD NORMAL random variable ¢ @ ¤ ¢ ` X 7 %  7 [email protected]@ E ¥£  7 ¨ 4 p ¨ §3¤  Find , find: 156  @"%  @ E ¥£ ¤ If 155 STA 2023 c D.Wackerly - Lecture 11 Find ¨@  "%  @ E 2£ ¤ ¨  @ ` X 7 %  7 @ d@ E 2£ X¤ ¨ ¨  c [email protected] E  ¥£ ¤ ¨  c X@ E  ¥£ ¤ ¨  @ c d"% % g X@ 7 ¥£ X@ ¤ ¨ ¨ STA 2023 c D.Wackerly - Lecture 11 have a standard normal dist. . score on dexterity test Normal distribution, Employer requires dexterity score of at least 80. Want ¨   g £ §¦¤¥£ ¢ E ¢¢ W V X   ¥£ ¤ `¨ ¡ . ¨ such that ¨   g £ §¦2£ ¤ ¢ ¨ ¨ ¤ ` a g c X   2£ score : Note : Used table in reverse. Find a number Ex.: #5.31, p. 228 ¢ such that ¤ ` a Hc X   ¥£ g¨ Find a number   g £ §¦2£ ¤ Ex. Let 157 Xe 4e 3 a X )¨ rp a )¨ ©¦ ¨ ¢ STA 2023 c D.Wackerly - Lecture 11 a Hc X g ¢ Find  @ c d"% % g [email protected] 7 ¥£ X@ ¤ Find Recall the Empirical Rule!!! . 158 ¨ W V X 1 % ¥£ ¤ `¨ ¡ ¢ 159 STA 2023 c D.Wackerly - Lecture 11 STA 2023 c D.Wackerly - Lecture 11 160 Ex. Not in book Normally distributed, so that Limit for interference: What fraction use opener beyond the limit? ¤ ¦ ¦ ¨ ¤ score : ¤ ¨ score . Xt @ g X 7 © 2£ £¤ E X t  a £ §¦2£ ¤ score is ¢ , its E – Whatever the value of X £ a ¡ 7 4 ¡W H7 ¨ rp¢H¨ 3 ¨ ©¦ – Find distance at which garage door opener is used. ¡ ¢ Want - 98th percentile of dexterity scores. ¨   a £ §¦2£ ¤ Five users randomly selected. What is the probability that all 5 used the openers beyond 50’? ¨ ¢ ¨¦ X a e W Y c ¨ a e XW b7 a 161 E e a )¨ STA 2023 c D.Wackerly - Lecture 11 STA 2023 c D.Wackerly - Lecture 12 162 Thought: If at first you don’t succeed, skydiving is not for you. Assignments : If you randomly select 5 owners of garage door openers and all 5 use the opener beyond 50’, what Today : P. 215 – 226, 253 – 259 For tomorrow: do you think about the claim that ¡ 4¡ W ¨ 7 H7 ¨ rp¥ '3 – ? Exercises 5.24, 25, 31, 36 – 37, 39–40, 95, 96, 98, 6.1, 6.3, 6.4 For Monday: Read pages 260 – 264, 265 – 270, 279 – 285 For Thursday: Exercises 6.15, 21, 24, 28, 37, 38, 42 – 46, 7.1, 3–5, 10 – 11, 15–20 ¢ 163 STA 2023 c D.Wackerly - Lecture 12 STA 2023 c D.Wackerly - Lecture 12 164 Last Time : Continuous Random Variables Whatever the value of ¢ ¨ ¢  © £ in Table IV, p. 809 .  ¢ Key to finding correct areas (probabilities) : draw and , for , the score for 5 is .48 .5 pictures Ex. : Like #5.40, p. 229 (Same scenario, different 0 ¢ From table, paint is normally distributed, ¢ X ` X '4 ¨ Thus, the ¨3 3 unusable. must be approx. . More than 5 ml dispensed – paint unusable. What should we set score for g b7 X ` ¨ X ¤ X ¨ X@ ` X # a  d@ ¤ E a '3 ¥ ¨ a  d$¨ 3 ` E a something we can set z0 E a ¨  a  d" ¥£ X@ ¤ numbers) Amount of dye, in ml., dispensed into can of to so that only 2% of cans are milliliters ¨ i¦ ¢ amount dispensed. so that ¢ STA 2023 c D.Wackerly - Lecture 12 165 3 Find 3 ¢ Areas under normal curves between z-scores of E Normal distribution is special case. STA 2023 c D.Wackerly - Lecture 12 166 Sampling Distributions In class demonstration of a Sampling Distribution. Different samples yield different values for . with a Population. ¢ – A Parameter is a meaningful number associated There is a probability distribution associated with This probability distribution is called the SAMPLING 4 R p p¦ (p. 254), , etc.. – used to make inferences about parameters. DISTRIBUTION of the statistic . (p. 255) ¦ 3 observations in a Sample are called Statistics the values of . ¦ R4 ( ), Popn. Range, Popn. Median, etc. Meaningful numbers computed from the tend to pile up in certain regions. ¢ ), Standard Deviation The values of ¢ – Mean ( ), Variance ( is a RANDOM VARIABLE. ¦ more parameters.(p. 254) ¦¢ ¢ Desired inference often phrased on terms of one or ¦ contained in a sample from the population. Results of Repeated Computation of the Statistic, ¢ inference about a population based on information ¡ ¢ Recall (p. 4 – 6), objective of statistics is to make an ¢ 167 (p. 266) F 3 ¦ 3p equal to the true population mean 168 , is ¢ ¡ parameter 8 ¨F4 4 F4 ¦ 8 ¤ . mean) . R¢ @7  F4 a , standard deviation (sample mean) is a point estimator for (popn. (sample variance) is a point estimator for ¢ ¤ 7 ¡ observations. ¦¢ , smaller (popn. variance). ¨ 48 ¨ F 4 ¨F3¢ ¢ Standard error : STA 2023 c D.Wackerly - Lecture 12 169 ¢ error of the mean . Ex. : Population with mean in a sample to compute a single number that we intend to be “close” to the value of the population (p. 266) often called the standard Note: Bigger a rule or formula telling how to use the use the data R 94 ¤ 3 ¨ F 3 A point estimator for a parameter (Defn. 6.4, p. 261) The standard deviation of the sampling distribution popn std dev of , , is equal to sample size Take STA 2023 c D.Wackerly - Lecture 12 3 ¢ The mean of the sampling distribution of , STA 2023 c D.Wackerly - Lecture 12 STA 2023 c D.Wackerly - Lecture 12 170 An unbiased estimator for a population parameter if (Defn. 6.5, p. 261) the mean of the sampling distribution Biased Estimator of the estimator equals the parameter. estimator for 3 ¦¢ 3 ¨ F 3 ¢ is an . underestimates µ µ overestimates Tends to overestimate too often. Unbiased Estimator If we have two unbiased estimators, prefer the one with the SMALLER standard error. µ 3£ when we computed the sample to get an unbiased estimator for the population variance , R4 variance, R  7 E ¤ Divided by µ . µ 3£ proportion of the time. £ ¤¡ Tends to over and underestimate the same close to close to ¢ ¥3 £ ¤¡ overestimates ¢ ¥3 µ underestimates ¢ 171 Summary: If we plan to take a random sample of size STA 2023 c D.Wackerly - Lecture 12 STA 2023 c D.Wackerly - Lecture 12 172 Know the mean and standard error of , HOW ABOUT THE DISTRIBUTION ? ¡ sampling distribution of is called its sampling distribution. the sampling dist. of – approximately normal – standard error ¢ Larger sample size, ¢ For most populations, ¢ Bigger ¢ Values of ¦ are close to probability. . with when n = 35. Effect of Increasing larger n Sample Size on ¦ Sampling Dist. of distribution of x Smaller standard error : original population distribution when n = 50. 174 : ¢ distribution of x is “large enough”. STA 2023 c D.Wackerly - Lecture 12 3 ¦ 173 better approximation. 8F 4¨4 Sample. (p. 254) . ¡ – Statistic – meaningful number assoc. with a 3 Population. (p. 254) ¦ ¡ – Parameter – meaningful number assoc. with a W ¨ F4 – with mean STA 2023 c D.Wackerly - Lecture 12 is : ¥  ¤ F 8 4 ¨ 4F 3 ¨ 3 8 T 4¨ F 4 ¢ 3 ¨ F3 ¢ is called the standard error of .(p. 266) , then regardless of the shape of the population dist. is more for larger sample sizes. New terms Central Limit Theorem : (p. 280) For large ¡ 3 ¨ 16¦ ¤ concentrated around ¢ ¦ (p. 266), so dist. of . ¦ ¢ ¢ is an unbiased estimator . So . (p. 266, 261) of is normal, with mean and standard deviation (standard error) F 8 4 ¨¨ F 43 3 ¦ ¦¢ (p. 255) Distribution of If the population has a normal distribution, the 4 is a random variable. ¦ , and standard ¢ deviation 3 from a population with mean smaller n µ ...
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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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