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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 £¢ §¢
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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 ¤
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¨§¦ Suppose Today : P. 236 – 240 table. be so that the normal reservations DO NOT show up.
completely contained in ¢ n1 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
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" 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§
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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. ¡
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£¦ 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 leapfrog 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©
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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
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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.
 Spring '08
 Ripol
 Statistics, Binomial

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