Week8 - STA 2023 c B.Presnell & D.Wackerly -...

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Unformatted text preview: STA 2023 c B.Presnell & D.Wackerly - Lecture 13 156 Thought: Vital papers will demonstrate their vitality by moving from where you left them to where you can’t find them. Assignments : Today : P. 236 – 240 For tomorrow: Exercises 5.55 , 5.59 , 5.60 , 5.63 , ¡ ¡ ¡ ¡ ¡ For Wednesday: P. 254 – 264, COMPUTER DEMO For Thursday: Exercises 6.1, 6.3, 6.4. 6.8 Last Time : Working with the Normal distribution ¢ Tables to find probabilities Key : DRAW PICTURES ¡ 5.64 , 5.68 ¢ ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 13 157 Normal Approximation to the Binomial Distribution p. 236 – 240 £ ¤¢ bin ¨¦ ©§¥ ¡ Suppose has a binomial distribution £ ¨  ¢  (p. 185, Chpt. 4)   ¥ £ ¦   £ is “large” the probabilities involving  £ ¦ If  ¢ Probability of a “success” : ¦ ¢ Sample size : can be approximated with probabilities based on a Normal distribution with mean and standard deviation  £ ¦   ¥ £ ¦   ¢ ¢ ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 13 158 See Figure 5.18, p. 237 Normal Density µ = 5, σ = 1.581 Binomial Histogram 0 1 2 3 4 5 6 n = 10, p = .5 P(3 < x < 5) 7 8 9 10 STA 2023 c B.Presnell & D.Wackerly - Lecture 13 159 ¢ ¦ £ ¡ ¥ ¤ ¡ ¦£ ¦ £ ¥ ¡ ¥ £ ¢  ¡ ¦ ¦£ ¨¡ £  ¢ ¢ ¨¡  ¢ ¦  ¢ ¨¡ £  £ ¦ § ¨¢   ¢ The .5’s are called corrections for continuity. ¢ The largest value of interest gets a little larger (by ¢ .5) to get to edge of box in the binomial histogram. Smallest values of interest gets a little smaller. ¥ £ Probabilities involving ’s obtained from table. £ ¢ ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 13 £ How “large” should 160 be so that the normal approximation is “good enough” to use? (Figure 5.19, p. 238, p. 240) §  £ ¦ to £ ¦ £ ¡ §   . ..... 0 1 2 ..... 3 n-1 µ−3σ µ+3σ This turns out to be the same as ¦ £ ¦ ¢ smaller of and and ¢ £ Ex.  ¦ larger of  ¢ the interval from completely contained in n  § £ ¦   ¤ ¥ £ ¦  § ¡ £ ¦ £ ¢   ¨   §¨  ¨ ¦ ¦ ¦ ¦ § ¦ © ¨   £ ¦ ¦   £ ¢ ¦   ¡   ¦ ¦ STA 2023 c B.Presnell & D.Wackerly - Lecture 13 161 Ex. : (Not in Book) Motel records : 10% of those with reservations DO NOT show up. Capacity : 200 rooms ¢ Reservations : 215 ¢ What is the probability that there will be a room for ¢ all who show up? What is the probability that more than 190 show up?  show up ¤ ¥ ¢ £¡  £    ¨ ¢  £ ¦ §   ¦   ¦ ¦ ¦ ¢ ¢ is binomial ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 13 162 £ big enough to use normal approximation? ¢ £ ¦ ¦ ¦ What is the probability that there will be a room for all who show up? ¨ ¢ ¢  ¨ ¢ £ ¦ Want ¦   ¨ ¡ ¢  ¢ £ ¦  ............................ 200 P( x < 200 ) ¢  ¨ ¦ ¨ ¡ ¡ ¢ ¡  ¢  ¢ ¦  ¦ Thus, 201 £ ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 13 163 What is the probability that more than 190 show up? ¨ ¢ ¡ ¨ ¢  ¨  ¦ ¡  ¡ ¢  ¦ ¡ ¢  ¨ ¡ ¢     ¨ ¦ ¦    ¨ ¦  ¢ ¦ £ ¡ ¢  ¢     ¥   ¢ ¦ ¥ ¢ ¦ £  ¦ £ ¦ ¢  ¦  ¦ Thus, ¨ ¢ Want £ ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 13 164 Systematic approach to approximating Binomial Probabilities   and  £ ¦ ¦ £   1. Compute . 2. Set up needed probability involving a binomial random variable, rewriting if necessary so that have , with no ¦ and/or or only . Then make appropriate continuity correction(s). 3. Subtract off the mean, divide by the std. dev. and use the normal tables to approximate the probability. STA 2023 c B.Presnell & D.Wackerly - Lecture 13 165 Ex. Coin tossing. In 10 tosses, approximate the prob. of getting 4,5 or 6 heads.  ¡ ,   .  ¡   ¡   ¦ ¨ ¡ ¢ ¨ ¡ ¢ ¨   , ¡ ¡ ¨ ¡ ¢ ¨  ¡ ¢  ¡ ¢ £    # of heads, ¦ £   £ ¦    ¡ ¨   ¡ ¨ ¢ ¢  ¢ £ ¢ ¡  £ ¢ ¢  ¥¢    Note: The exact prob. in this example is 0.65625 (binomial table gives .656). Approx. is very good here, ¡   £ even for . STA 2023 c B.Presnell & D.Wackerly - Lecture 14 166 Thought: A truly wise person never plays leap-frog with a unicorn. Assignments : Today : P. 254 – 264 For Thursday: Exercises 6.1, 6.3, 6.4. 6.8 For Monday : SPRING BREAK!!! Last Time : Normal Approximation to the Binomial Distribution large enough? ¦ and  ¦ £ £ ¢ ¢ £ ¦ ¢ smaller of and  larger of Write probabilities for the binomial variable with the ¢ “=” sign. ¡ ¨  ¢ ¢   ¨¡   ¢  Use “Continuity Correction.” ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 14 167 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  ¥ ¥ (p. 254), , etc.. – used to make inferences about parameters. ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 14 168 Results of Repeated Computation of the Statistic, Different samples yield different values for . is a RANDOM VARIABLE. ¢ ¢ The values of tend to pile up in certain regions. ¢ There is a probability distribution associated with ¢ the values of . This probability distribution is called the SAMPLING DISTRIBUTION of the statistic . (p. 255) ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 14 169 ¥ ¥ ¦  or , each   with probability § Ex. Consider a spinner that can land on For the spinner,       ¥¨ ¢  §    Spin the spinner twice, record  ¤  –  average of 2 numbers. Sample Prob ¦   ¦ §  ¥§  ¡  Prob.  ¦  Sample ¥  ¥§  ¥ ¦ §¥ § §¥   ¥    ¦   ¡   ¦ ¥ ¦ ¦  §¥ ¦ ¦ ¢ – ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 14 170 §  ¡ ¦ ¡  ¦ ¨ ¢   ¦  What is the mean of ? ¡ The sampling distribution of . ¨ ¢ ¦  ¨ ¢ §    ¡  STA 2023 c B.Presnell & D.Wackerly - Lecture 14 171  , is ¥ ¢ equal to the true population mean ¡ The mean of the sampling distribution of ,  (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 . ¡ , standard deviation ¡ ¡ ¦  observations.  £   ¡   Standard error :  ¡ Take . ¦ £ Ex. : Population with mean  , smaller ¡ Note: Bigger . ¢ ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 14 172 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. ¢  ¢ (popn. variance).  (sample variance) is a point estimator for  mean) STA 2023 c B.Presnell & D.Wackerly - Lecture 14 173 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 B.Presnell & D.Wackerly - Lecture 14 174 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 close to  µ ...
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