Week7_001 - STA 2023 c D.Wackerly - Lecture 11 Thought:...

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