Week8-2up - ¡ Key : DRAW PICTURES Tables to find...

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Unformatted text preview: ¡ Key : DRAW PICTURES Tables to find probabilities Working with the Normal distribution Last Time : For Thursday: Exercises 6.1, 6.3, 6.4. 6.8 bin If can be 157 distribution with mean and standard deviation approximated with probabilities based on a Normal is “large” the probabilities involving (p. 185, Chpt. 4) Probability of a “success” : Sample size : has a binomial distribution Suppose p. 236 – 240 Distribution Normal Approximation to the Binomial For Wednesday: P. 254 – 264, COMPUTER DEMO 5.64 , 5.68 ¡ For tomorrow: Exercises 5.55 , 5.59 , 5.60 , 5.63 , Today : P. 236 – 240 Assignments : them. ¡ moving from where you left them to where you can’t find ¡ Thought: Vital papers will demonstrate their vitality by ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 13  ¢ ¢  ¦ ¦ ¦ ¡ £ ¨©  ¢ ¤ ¨©   156  ¢ ¦ §¥ ¦    © STA 2023 c B.Presnell & D.Wackerly - Lecture 13 ¦ ¢ ¢ ¨ ¦ ¥  £ © © © ¢ ¢ £ 0 1 2 Binomial Histogram 3 5 6 P(3 < x < 5) n = 10, p = .5 4 See Figure 5.18, p. 237 7 8 Normal Density µ = 5, σ = 1.581 9 10 158 STA 2023 c B.Presnell & D.Wackerly - Lecture 13 Smallest values of interest gets a little smaller. .5) to get to edge of box in the binomial histogram. The largest value of interest gets a little larger (by The .5’s are called corrections for continuity. ¢ table. Probabilities involving ’s obtained from ¢ ¤  ¥  ¢ ¤ § STA 2023 c B.Presnell & D.Wackerly - Lecture 13  ¥ £ ¥ £ ¢ ¢ ¢ ¢ ¦ ¡ © £ £  ¡ §  ¦ ¡ ¢  ¤¥£ ¤ ¦ ¡ ¡ © ¡ ¢  ¤¥£ ©  ¦ ¡ §¦ © §¦ © ¨ © © ¨ ¨ 159 How “large” should ¦ © ¦   1 2 µ−3σ 3 to ..... . larger of smaller of and µ+3σ ..... n-1 and n completely contained in This turns out to be the same as 0 © © © © ©     ©  ¦ © ¡  ¤     ¥   ¢  the interval from ¡ ¢  Ex.  (Figure 5.19, p. 238, p. 240) approximation is “good enough” to use? be so that the normal STA 2023 c B.Presnell & D.Wackerly - Lecture 13 © © © ¢ ¦ ¦ £ ¦ ¦ ¦ ¦ ¦ © ©© ¢ ¦ ¦§£ © ¨¨ ©© ¢ ¦ ¦§£  ¨ 160 161 What is the probability that more than 190 show up? all who show up? What is the probability that there will be a room for Reservations : 215 Capacity : 200 rooms reservations DO NOT show up. Ex. : (Not in Book) Motel records : 10% of those with STA 2023 c B.Presnell & D.Wackerly - Lecture 13 ¢ ¢ ¢ ¢ ¢ ¢ show up is binomial ! ! ©    £ £ ! ©    £ ¦ ¡  ¦     £ © ¨        ¦ ¥ "     ¢ ¦ ¢ ! ! Want all who show up? ! Thus, P( x < 200 ) ............................ 200 201 What is the probability that there will be a room for ¢ ! ¦ ! © ¥ £ ! ¤ ¥   ¥ £      ¢¡  ! ¤ ¥ ¡ ¥ £  ¢¡  ¤  ¥ ¢ ! ! ! ! ! ¡    ¤ ! ¤ ¥ Thus, ! ¤ ¥  ¡¡  £ ¥ £ © © © §  ¥ £ ¥ Want ¢ 163 What is the probability that more than 190 show up?  big enough to use normal approximation?  STA 2023 c B.Presnell & D.Wackerly - Lecture 13 ¤ ¥  £    ©  §   £ ¢ ¢ £ £ ¡¢  ¢  ¡ ¤   ¢ © ¤     ¥ ¡¢ ¤   ¤   ¢ 162  STA 2023 c B.Presnell & D.Wackerly - Lecture 13 ¢ ¦  and . and/or , with no or . Then make use the normal tables to approximate the probability. 3. Subtract off the mean, divide by the std. dev. and appropriate continuity correction(s). only © random variable, rewriting if necessary so that have 2. Set up needed probability involving a binomial 1. Compute © Binomial Probabilities  £ ¤ # of heads, getting 4,5 or 6 heads. , ¥ Systematic approach to approximating   ¥ , . even for (binomial table gives .656). Approx. is very good here, Note: The exact prob. in this example is 0.65625  ¦ 165 Ex. Coin tossing. In 10 tosses, approximate the prob. of ¦ © STA 2023 c B.Presnell & D.Wackerly - Lecture 13  . ¡ £   £ ¢   ¦  ¦ © ¦   ¥ §  ¤  ©  ¥  ¢ £  ¡  ¢ ¤  ¡ © ¡  ¡  ¡ ¥    ¥    ¡  ¥ ¤ 164 ¡  ¢   ¡ ¡ ¤ ¡   ¡  ¢  STA 2023 c B.Presnell & D.Wackerly - Lecture 13 ¨ large enough? ¥ “=” sign.  smaller of larger of Use “Continuity Correction.” £ 166 and and Write probabilities for the binomial variable with the ¢ ¦ Normal Approximation to the Binomial Distribution ¢ © £ £ ¡  ¢ Last Time : For Monday : SPRING BREAK!!! For Thursday: Exercises 6.1, 6.3, 6.4. 6.8 Today : P. 254 – 264 Assignments : a unicorn.  167 with a Population. – A Parameter is a meaningful number associated more parameters.(p. 254) Desired inference often phrased on terms of one or contained in a sample from the population. inference about a population based on information Recall (p. 4 – 6), objective of statistics is to make an Sampling Distributions STA 2023 c B.Presnell & D.Wackerly - Lecture 14 ¢ ¢ ¢ Thought: A truly wise person never plays leap-frog with  – Mean ( ), Variance ( ), Standard Deviation , etc.. – used to make inferences about parameters. (p. 254), observations in a Sample are called Statistics Meaningful numbers computed from the ( ), Popn. Range, Popn. Median, etc.  STA 2023 c B.Presnell & D.Wackerly - Lecture 14 © © ¢¥ ¢ ¢ ¦  £  ¨£   ¨ © ¨  ¤  ©  ¢ tend to pile up in certain regions. the values of . There is a probability distribution associated with The values of is a RANDOM VARIABLE. Different samples yield different values for . DISTRIBUTION of the statistic . (p. 255) This probability distribution is called the SAMPLING Sample numbers. Prob. Sample Spin the spinner twice, record – – For the spinner, with probability 169 or , each Prob average of 2 Ex. Consider a spinner that can land on STA 2023 c B.Presnell & D.Wackerly - Lecture 14 ¢ ¢ Results of Repeated Computation of the Statistic, £ ¢ ¢ ¢ 168  ¥ ¥ ¢  ¨ ¥ ¨  ¨    ©    ¥ ¥ £ ©  ¡  "¥ ©  £ ¨ ¨  £  ¥ ¥  ¨ £      STA 2023 c B.Presnell & D.Wackerly - Lecture 14 £ ¤  £ ¨ ¨ § ¡   ¥ ¨ ¥   £ ¥ ¡   £  ¨ ¥ ¨   ¡  , is The standard deviation of the sampling distribution popn std dev , is equal to of , sample size (p. 266)  ¢  £ £ £ ¥ ¡ ¥ ¡    ¥ £ ¥ © £ ¡ equal to the true population mean Note: Bigger , smaller Ex. : Population with mean Take 171 observations. Standard error : , standard deviation . (p. 266) often called the standard error of the mean . ¡   £ £ The mean of the sampling distribution of ,  What is the mean of ?   ! ! ¡   ¡ © ©  ¡ !  £      ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 14 ¦  The sampling distribution of . 170  "¥  £  ¦ ¢   £   ¦ ¢ ¨ STA 2023 c B.Presnell & D.Wackerly - Lecture 14 ¡  ¡¡ ¡   ¢ ¡ . ¥ ¢ ¢ ¢ mean) (sample mean) is a point estimator for parameter (popn. intend to be “close” to the value of the population in a sample to compute a single number that we a rule or formula telling how to use the use the data (popn. variance). (sample variance) is a point estimator for 173 ¡  £ underestimates Unbiased Estimator µ . overestimates estimator for Divided by to get an unbiased estimator for the population variance , variance, µ when we computed the sample proportion of the time. Tends to over and underestimate the same is an  of the estimator equals the parameter. (Defn. 6.5, p. 261) the mean of the sampling distribution An unbiased estimator for a population parameter if ¢ ¢ ¢ A point estimator for a parameter (Defn. 6.4, p. 261) £   © © STA 2023 c B.Presnell & D.Wackerly - Lecture 14 ¦ §¥ 172 § ©  STA 2023 c B.Presnell & D.Wackerly - Lecture 14 . ©    µ overestimates µ µ µ ¢  ¢   close to close to  the SMALLER standard error. £ £ 174 If we have two unbiased estimators, prefer the one with Tends to overestimate too often. underestimates Biased Estimator STA 2023 c B.Presnell & D.Wackerly - Lecture 14   ...
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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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