Week8-4up_001 - 156 STA 2023 c B.Presnell &...

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Unformatted text preview: 156 STA 2023 c B.Presnell & D.Wackerly - Lecture 13 STA 2023 c B.Presnell & D.Wackerly - Lecture 13 157 Thought: Vital papers will demonstrate their vitality by moving from where you left them to where you can’t find Normal Approximation to the Binomial them. Distribution Assignments : p. 236 – 240 £¢ §¢ 45'§ 2310©('§& $%¢  ) ¢ Probability of a “success” : ¡ ¡ ¡ ¡ For Thursday: Exercises 6.1, 6.3, 6.4. 6.8 If Last Time : can be distribution with mean and standard deviation 4  2 )©  § 5'§ 310('& $ DRAW PICTURES STA 2023 c B.Presnell & D.Wackerly - Lecture 13 ¢ S 5'§ 2 4 'RQI7UQ7 EC A @8&1¨£ § P " P T  9 7 T ¦ S 5'§ 2 4 § P " F 6 CA  9 7 6 ¦ 'RQI7HG7 EDB@8&1¨£ See Figure 5.18, p. 237 45'§ 2 A @9   7 6 £ 6 ¦ 8H%UEXWV ¢ Normal Density µ = 5, σ = 1.581 4 5 6 7 8 9 The largest value of interest gets a little larger (by .5) to get to edge of box in the binomial histogram. 10 ¢ Smallest values of interest gets a little smaller. n = 10, p = .5 P(3 < x < 5) ¢ 3 ¢ 2 ¢ 1 The .5’s are called corrections for continuity. C Binomial Histogram 159 Probabilities involving ’s obtained from P `C ¢ 158 6C VY6 ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 13 S ¢ Tables to find probabilities 0 is “large” the probabilities involving approximated with probabilities based on a Normal Working with the Normal distribution Key : (p. 185, Chpt. 4) £ ¢ Sample size : "  ¦ #! ¡ For Wednesday: P. 254 – 264, COMPUTER DEMO has a binomial distribution § ¡ 5.64 , 5.68 ¤ ¥£ For tomorrow: Exercises 5.55 , 5.59 , 5.60 , 5.63 , bin © ¨§¦ Suppose Today : P. 236 – 240 table. be so that the normal reservations DO NOT show up. completely contained in ¢ n-1 What is the probability that there will be a room for What is the probability that more than 190 show up? µ+3σ all who show up? S4 show up is binomial £¢ 4 2 5'§ 3 ) 3 £¦ # $ 3 4 $# "  W T § $!¡"  S !  A £T § " ¢ S  A £T § ©¨"  ¢    3 T§ 5 3§ ¢ T EC  9  $C81¨£ 3 6 ¢  T ¦ ¨£@  C8B¨£ 3 ¦  ¡¢ A ¦  !" P VC 3 " T¦ all who show up?  7 9  6 ¨£ ¦ 3 Want Want T EC  63 What is the probability that there will be a room for 6¦ 1¨£ 3  9I7" 8VC 3 ¢6¦ 6 VC  ............................ P( x < 200 ) 9  ¡9¨@1¨£ ¡6¦ 3  9I7" 8VC 3 ¢6¦ Thus, 201 9  ¢C8B¨£ ¡A¦ 3  !" P VC 3 " T¦ 200 163 What is the probability that more than 190 show up? D" D  big enough to use normal approximation? Thus, 7 "W T§  §¦"  T§ STA 2023 c B.Presnell & D.Wackerly - Lecture 13 D" D I7" T§ 162 7 A ¥£T § ¤¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 13   Ex. and 1 02) ( ' & £ ¢ %  smaller of and 4 larger of 3 ¢ This turns out to be the same as ¨£ ¦ µ−3σ Reservations : 215 n ¢ 3 ¢ 2 .....  §¡ § ) $ 4 5'§ 2 W '&18W %¢ 1 Capacity : 200 rooms . ..... 0 ¢ to 161 Ex. : (Not in Book) Motel records : 10% of those with (Figure 5.19, p. 238, p. 240) 3 § approximation is “good enough” to use? the interval from STA 2023 c B.Presnell & D.Wackerly - Lecture 13 B§ How “large” should 160 STA 2023 c B.Presnell & D.Wackerly - Lecture 13 ¢ ¢ 164 STA 2023 c B.Presnell & D.Wackerly - Lecture 13 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.  ¡ 87I" ¦ 8 "7 ¦  C ¦ 4 '§ 2 1)  7¡ @ 8I7" ¦ ¡ C ¦ 'H $ § 4 I7"  C § ¡ ¥£ A . Then make T 6 appropriate continuity correction(s). or C , with no and/or  ¢ D !¢" only S random variable, rewriting if necessary so that have 2. Set up needed probability involving a binomial C . . § '& $ and 4 5'§ 2 %) 1. Compute ¢ ¦ D ©¢" P V@ A @9   £6 U6 D  £¦ Binomial Probabilities , " $9I7"  Systematic approach to approximating , I7" # of heads, 3. Subtract off the mean, divide by the std. dev. and use the normal tables to approximate the probability. Note: The exact prob. in this example is 0.65625 (binomial table gives .656). Approx. is very good here, ¡ C § even for 166 STA 2023 c B.Presnell & D.Wackerly - Lecture 14 STA 2023 c B.Presnell & D.Wackerly - Lecture 14 Thought: A truly wise person never plays leap-frog with a unicorn. . 167 Sampling Distributions Assignments : ¢ Recall (p. 4 – 6), objective of statistics is to make an Today : P. 254 – 264 For Thursday: Exercises 6.1, 6.3, 6.4. 6.8 inference about a population based on information contained in a sample from the population. ¢ For Monday : SPRING BREAK!!! Desired inference often phrased on terms of one or more parameters.(p. 254) Last Time : – A Parameter is a meaningful number associated with a Population. – Mean ( ), Variance ( A ¥£T § ¤¢ ¢ “=” sign. Meaningful numbers computed from the observations in a Sample are called Statistics (p. 254),  ¤© £© £ ¢¢   §¢ Write probabilities for the binomial variable with the "  D 81UE¢ V  D 6£6¦ 7 £ " V ¦ Use “Continuity Correction.” ), Standard Deviation ( ), Popn. Range, Popn. Median, etc. ¢ and S4 4 smaller of and ) larger of ) large enough? $ ¢ Normal Approximation to the Binomial Distribution , etc.. – used to make inferences about parameters. ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 14 169 Ex. Consider a spinner that can land on ¢ Prob  W  "7 ¨  W 0© W  ¨© ¨ C© ¨ W© ¨©  C©  ¨  I7" ¨  £ W© ¨ 171 $£ ¢ The mean of the sampling distribution of , equal to the true population mean $ $3 £ (p. 266) , is   !¡" W I7" ¨ !¡" ¨ ¢ The standard deviation of the sampling distribution popn std dev of , , is equal to sample size £ 2 (p. 266) often called the standard ¥ ) §2 )3 )£  £¦  £  C observations. , standard deviation £ §2 £ ) $ %¢ Standard error : ) Take ¡ 97 3 Ex. : Population with mean . ¨ § , smaller ) Note: Bigger £ £ £ error of the mean . ¦ The sampling distribution of . $© ¢ ¢ £ STA 2023 c B.Presnell & D.Wackerly - Lecture 14 £ £ ¢ 170 ¨© W ©W £¢ DISTRIBUTION of the statistic . (p. 255) Sample ¨ ¢ This probability distribution is called the SAMPLING the values of . Prob.  £ £ Sample average of 2 £ ¢ numbers. There is a probability distribution associated with ¢ ¢  #£ ¦  ¢ £ £ What is the mean of ?  $ P   ¨£¦ #  ) 4 Spin the spinner twice, record tend to pile up in certain regions. STA 2023 c B.Presnell & D.Wackerly - Lecture 14 £ " ¦ !" $ The values of ¡ – – is a RANDOM VARIABLE. or , each For the spinner, Results of Repeated Computation of the Statistic, Different samples yield different values for . " with probability W © ¨© 168 STA 2023 c B.Presnell & D.Wackerly - Lecture 14 . ¤  £ ¦ # $ 4 £ 172 STA 2023 c B.Presnell & D.Wackerly - Lecture 14 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. is an estimator for $ $ %¢ £¢ ¢ in a sample to compute a single number that we £ a rule or formula telling how to use the use the data $ A point estimator for a parameter (Defn. 6.4, p. 261) . Unbiased Estimator intend to be “close” to the value of the population parameter (popn. µ underestimates Divided by variance, when we computed the sample to get an unbiased estimator for the 174 Biased Estimator 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 B.Presnell & D.Wackerly - Lecture 14 µ µ proportion of the time. population variance , underestimates overestimates Tends to over and underestimate the same ¢ (popn. variance).  £P ¦§  ¢ ¢ (sample variance) is a point estimator for ) £¢ mean) $ (sample mean) is a point estimator for . ...
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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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