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Unformatted text preview: 147 STA 2023 c D.Wackerly  Lecture 11 STA 2023 c D.Wackerly  Lecture 11 148 For a continuous random variable, Probabilities of
intervals are areas under curves:
graph of f(x) Thought: “I personally think we developed language
because of our deep need to complain”  Lily Tomlin
P(a<X<b)
a Today : pages 215–226 b The area under a curve above a single point
Line has
area 0 For tomorrow: Exercises 5.15, 5.16, 5.19, 5.20, is 0. ¡ Assignments : For Wednesday:
Read more on pages 198–208
a For Thursday: ¢ For any single value , UNLIKE for discrete random variables, if
continuous random variable, is a ¦ ¢ 96, 98, 6.1, 6.3, 6.4 ¨¡ ¨ ¤
©§¦¥£ ¡ Exercises 5.24, 5.25, 5.31, 5.36–37, 5.39–40, 95,
#!"%'¦&%0§¡2$1#'&0§¡¥)¨
¤ £ ¨ ! % ¦ ¤ £
! ¦ % ¤ £ ¨ ! ¦ ¤
("'&¡¥$#" ¡¥£ 149 STA 2023 c D.Wackerly  Lecture 11 STA 2023 c D.Wackerly  Lecture 11 150 The Normal Distribution
The graph of the normal density is a symmetric, 5.6, p. 216) 3 if its probability density is (p. 216) µ =8
σ =1 ¢
¢ CDB@ 894
A
¨6§¦¤ 5
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TR F
4
7 4 XX g X@
#(X g @ (7 fed$¨ C XX ca7` XW ¨
##X Vb#7 YDA
¢ No need to know this formula. and std dev 4 3 4p
srq§3¤ h
i¦ ” MEANS “ has a normal distribution 7 da tp a ¤
X@ and std dev X
7 [email protected] ¢
¢ h
i¦ ¢ with mean 8 10 12 14 16 µ =10
σ =3 As a shorthand for “ has a normal distribution with mean
N 6 µ =10
σ =1 ¦ N µ =12
σ =1 Two Normal Curves with Diff Means, Same Std Dev Formula traces out the “bell shaped curve”.
Will write 3 Recall: , measures the spread of the distribution. (See Figure A random variable has a normal dist. with mean
and std. dev. . The std dev, 4 bellshaped curve, centered at One very important continuous distribution. ” 0 5 10 15 Two Normal Curves with Same Mean and Differnent Std Devs 20 151 STA 2023 c D.Wackerly  Lecture 11 STA 2023 c D.Wackerly  Lecture 11 152 The Standard Normal Distribution – p. 217
For any normal random variable, areas under normal Xt 7 Y¤ ¡ h p and std dev , given in Table IV, p. 809 . and , for X34¦ ¨ E Recall : score : i.e., curves between zscores of 7 is normal with mean distribution if is said to have a standard normal ¤ £ The r. v. are Tabled area The score gives the number of standard
is from 3 ¦ deviations . ¦
¢
¦¨ 7 ¦ ¨ 3 4 ¦ ¨ ¢¢ E
E
¦
3¨©¦ ¦¨ ¢¢
¥ 3
¦ ¦ % ¢¢
¥ ¦ ¦ ¤¢¢
¥ £ 3 is larger than 0 . ¢ is smaller than z . Given
in table . is a standard normal random variable: 1/2 z
Find 0 . 0
Find z
Find is already a score. .4826
.4864
.
.
.
.4982
.4987 .4830
.4868
.
.
.
.4982
.4987 ...
...
..
.
...
... .3830
.4015
.4177
.4319
.4441
.
.
.
.4767
.4817 .4854
.4887
.
.
.
.4986
.4990 .4857
.4890
.
.
.
.4986
.4990 Find ¨ .4821
.4861
.
.
.
.4981
.4987 ...
... .3810
.3997
.4162
.4306
.4429
.
.
.
.4761
.4812 ¨7 ¨¤
c X ¤ ¥£ 2.1
2.2
.
.
.
2.9
3.0 ...
...
...
...
...
..
. ¢ .3686
.3888
.4066
.4222
4357
.
.
.
.4726
.4783 Find X
t 7 c X © ¥£
¨¤ .3665
.3869
.4049
.4207
.4345
.
.
.
.4719
.4778 Find .09
.0359
.0753
.1141
.
.
.
.3389
.3621 ¨ .3643
.3849
.4032
.4192
.4332
.
.
.
.4713
.4772 ...
... .08
.0319
.0714
.1103
.
.
.
.3365
.3599 ¨ g XW 7 % ¥£
¤ 1.1
1.2
1.3
1.4
1.5
.
.
.
1.9
2.0 ...
...
...
...
..
. ¢ § .02
.0080
.0478
.0871
.
.
.
.3212
.3461 is a standard normal r.v., then ¢ 0.0
0.1
0.2
.
.
.
0.9
1.0 .01
.0040
.0438
.0832
.
.
.
.3186
.3438 154 X
t g XW 7 % ¥£
¤
¨ g XW 7 % % 2£
¤ Ex. If (Part of the) Standard Normal Table
.00
.0000
.0398
.0793
.
.
.
.3159
.3414 STA 2023 c D.Wackerly  Lecture 11 X
t g XW 7 % % ¥£
¤ 153 STA 2023 c D.Wackerly  Lecture 11 That is, the value of 1/2 If STA 2023 c D.Wackerly  Lecture 11 is a STANDARD NORMAL random variable ¢ @
¤
¢
` X 7 % 7 [email protected]@ E ¥£
7 ¨ 4 p ¨ §3¤
Find , ﬁnd: 156 @"% @ E ¥£
¤ If 155 STA 2023 c D.Wackerly  Lecture 11 Find ¨@
"% @ E 2£
¤ ¨ @ ` X 7 % 7 @ d@ E 2£
X¤ ¨ ¨ c [email protected] E ¥£
¤
¨ c X@ E ¥£
¤ ¨ @ c d"% % g X@ 7 ¥£
X@
¤
¨ ¨ STA 2023 c D.Wackerly  Lecture 11 have a standard normal dist. . score on dexterity test Normal distribution, Employer requires dexterity score of at least 80.
Want ¨
g £ §¦¤¥£ ¢
E ¢¢ W V X ¥£
¤
`¨
¡ . ¨ such that ¨ g £ §¦2£
¤ ¢ ¨ ¨
¤
` a g c X 2£ score : Note : Used table in reverse. Find a number Ex.: #5.31, p. 228 ¢ such that ¤
` a Hc X ¥£
g¨ Find a number g £ §¦2£
¤ Ex. Let 157 Xe 4e 3
a X )¨ rp a )¨ ©¦
¨ ¢ STA 2023 c D.Wackerly  Lecture 11 a Hc X
g ¢ Find @ c d"% % g [email protected] 7 ¥£
X@
¤ Find Recall the Empirical Rule!!! . 158 ¨ W V X 1 % ¥£
¤
`¨
¡ ¢ 159 STA 2023 c D.Wackerly  Lecture 11 STA 2023 c D.Wackerly  Lecture 11 160 Ex. Not in book
Normally distributed, so that Limit for interference: What fraction use opener beyond the limit? ¤ ¦ ¦ ¨ ¤ score : ¤ ¨ score . Xt @ g X 7 © 2£
£¤
E X
t a £ §¦2£
¤ score is ¢ , its E – Whatever the value of X £ a
¡
7 4 ¡W
H7 ¨ rp¢H¨ 3 ¨
©¦ – Find distance at which garage door opener is used. ¡ ¢ Want  98th percentile of dexterity scores. ¨
a £ §¦2£
¤ Five users randomly selected. What is the
probability that all 5 used the openers beyond 50’? ¨
¢ ¨¦ X
a e W Y c ¨ a e XW b7
a
161 E e
a )¨ STA 2023 c D.Wackerly  Lecture 11 STA 2023 c D.Wackerly  Lecture 12 162 Thought: If at ﬁrst you don’t succeed, skydiving is not
for you. Assignments :
If you randomly select 5 owners of garage door
openers and all 5 use the opener beyond 50’, what Today : P. 215 – 226, 253 – 259
For tomorrow: do you think about the claim that ¡ 4¡ W ¨
7
H7 ¨ rp¥ '3
– ? 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 ¢ 163 STA 2023 c D.Wackerly  Lecture 12 STA 2023 c D.Wackerly  Lecture 12 164 Last Time : Continuous Random Variables Whatever the value of ¢ ¨ ¢ © £ in Table IV, p. 809 . ¢ Key to ﬁnding correct areas (probabilities) : draw and , for , the score for 5 is .48 .5 pictures Ex. : Like #5.40, p. 229 (Same scenario, different 0 ¢ From table, paint is normally distributed, ¢ X ` X '4
¨ Thus, the ¨3 3 unusable. must be approx. . More than 5 ml dispensed – paint unusable.
What should we set score for g
b7 X ` ¨
X
¤ X
¨
X@
` X # a d@ ¤ E a '3 ¥ ¨ a d$¨ 3 ` E a something we can set z0 E a
¨ a d" ¥£
X@ ¤ numbers) Amount of dye, in ml., dispensed into can of to so that only 2% of cans are milliliters ¨
i¦ ¢ amount dispensed.
so that ¢ STA 2023 c D.Wackerly  Lecture 12 165 3 Find 3 ¢ Areas under normal curves between zscores of E Normal distribution is special case. STA 2023 c D.Wackerly  Lecture 12 166 Sampling Distributions
In class demonstration of a Sampling Distribution. Different samples yield different values for . with a Population. ¢ – A Parameter is a meaningful number associated There is a probability distribution associated with This probability distribution is called the SAMPLING 4 R p p¦ (p. 254), , etc.. – used to make inferences about parameters. DISTRIBUTION of the statistic . (p. 255) ¦ 3 observations in a Sample are called Statistics the values of . ¦ R4 ( ), Popn. Range, Popn. Median, etc.
Meaningful numbers computed from the tend to pile up in certain regions. ¢ ), Standard Deviation The values of ¢ – Mean ( ), Variance ( is a RANDOM VARIABLE. ¦ more parameters.(p. 254) ¦¢ ¢ Desired inference often phrased on terms of one or ¦ contained in a sample from the population. Results of Repeated Computation of the
Statistic, ¢ inference about a population based on information ¡ ¢ Recall (p. 4 – 6), objective of statistics is to make an ¢ 167 (p. 266) F 3 ¦ 3p equal to the true population mean 168 , is ¢ ¡ parameter 8 ¨F4
4
F4 ¦ 8 ¤ . mean) . R¢ @7
F4 a , standard deviation (sample mean) is a point estimator for (popn. (sample variance) is a point estimator for ¢ ¤ 7
¡ observations. ¦¢ , smaller (popn. variance). ¨ 48 ¨ F 4 ¨F3¢
¢ Standard error : STA 2023 c D.Wackerly  Lecture 12 169 ¢ error of the mean . Ex. : Population with mean in a sample to compute a single number that we
intend to be “close” to the value of the population (p. 266) often called the standard Note: Bigger a rule or formula telling how to use the use the data R
94 ¤ 3
¨ F 3 A point estimator for a parameter (Defn. 6.4, p. 261) The standard deviation of the sampling distribution
popn std dev
of ,
, is equal to
sample size Take STA 2023 c D.Wackerly  Lecture 12 3 ¢ The mean of the sampling distribution of , STA 2023 c D.Wackerly  Lecture 12 STA 2023 c D.Wackerly  Lecture 12 170 An unbiased estimator for a population parameter if
(Defn. 6.5, p. 261) the mean of the sampling distribution Biased
Estimator of the estimator equals the parameter. estimator for 3 ¦¢
3
¨ F 3 ¢
is an . underestimates µ µ overestimates Tends to overestimate too often. Unbiased
Estimator If we have two unbiased estimators, prefer the one with
the SMALLER standard error. µ 3£ when we computed the sample to get an unbiased estimator for the population variance , R4 variance, R
7 E ¤ Divided by µ . µ 3£ proportion of the time. £
¤¡ Tends to over and underestimate the same close to close to ¢
¥3 £
¤¡ overestimates ¢
¥3 µ underestimates ¢ 171 Summary: If we plan to take a random sample of size STA 2023 c D.Wackerly  Lecture 12 STA 2023 c D.Wackerly  Lecture 12 172 Know the mean and standard error of ,
HOW ABOUT THE DISTRIBUTION ?
¡ sampling distribution of is called its sampling distribution. the sampling dist. of – approximately normal – standard error ¢ Larger sample size, ¢ For most populations, ¢ Bigger ¢ Values of ¦ are close to probability. . with when n = 35. Effect of Increasing larger n Sample Size on ¦ Sampling Dist. of distribution of x Smaller standard error : original population distribution when n = 50. 174 : ¢
distribution of x is “large enough”. STA 2023 c D.Wackerly  Lecture 12 3 ¦ 173 better approximation. 8F
4¨4 Sample. (p. 254) . ¡ – Statistic – meaningful number assoc. with a 3 Population. (p. 254) ¦ ¡ – Parameter – meaningful number assoc. with a W ¨ F4 – with mean STA 2023 c D.Wackerly  Lecture 12 is : ¥ ¤
F 8
4 ¨ 4F
3
¨ 3 8 T 4¨ F 4 ¢
3
¨ F3 ¢ is called the standard error of .(p. 266) , then regardless of the shape of the population dist. is more for larger sample sizes. New terms Central Limit Theorem : (p. 280) For large ¡ 3 ¨
16¦ ¤ concentrated around ¢ ¦ (p. 266), so dist. of . ¦ ¢ ¢ is an unbiased estimator . So . (p. 266, 261) of is normal, with mean and standard deviation (standard error) F 8
4 ¨¨ F 43
3 ¦ ¦¢ (p. 255) Distribution of If the population has a normal distribution, the 4 is a random variable. ¦ , and standard ¢ deviation 3 from a population with mean smaller n µ ...
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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

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