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Unformatted text preview: STA 2023 c B.Presnell & D.Wackerly  Lecture 11 Thought: “I personally think we developed language
because of our deep need to complain”  Lily Tomlin Assignments :
Today : pages 21011, 215–226
For tomorrow: EXAMS RETURNED
Exercises 5.15, 5.16, 5.19, 5.20,
For Wednesday:
Read more on pages 198–208
For Thursday:
Exercises 5.24, 5.25, 5.31, 5.36–37, 5.39–40 133 STA 2023 c B.Presnell & D.Wackerly  Lecture 11 134 Last Time : The Binomial Probability Distribution
¨
& 2¡
¨ &¨
652¡ 43
7 ©
¤ 1
£ 2 3 ¢ §
98 ¢ ¢ © 1 & CB'''@(2
¡!AAA!#! 8 for ¢ .....
0 : (p. 1183) ¢ ¡ §©
¤ ¡ ¨ Tables : Contain © ¢ Variance (p. 185) : for ¢ £ Mean (p. 185) : § ¦¤
¥ number of trials, ¡!&&&!#!
0)'('%$" If ¨ Binomial Experiment p. 179 .....
k k+1 n1 n STA 2023 c B.Presnell & D.Wackerly  Lecture 11 135 Ex. Claimed Psychic. Five cards shufﬂed, one chosen.
Psychic tries to identify card (without looking). Do this
number of correct
¢ © 20 times and record identiﬁcations. Assume 20 trials are indep.
correct ? ¢ § ¤ £
© # correct choices in 20 trys
¢ Let £ correct ¤ – § If the psychic is guessing, what is – What is a “success” in this case?
¥
¢ is a binomial random variable with ¨
2¡ ¢ 43 ¢ and ¨ and .
¢ © – . ¡ ¨
2¡ 1 ¢
1 ¢ ¢ 3 STA 2023 c B.Presnell & D.Wackerly  Lecture 11 136 What is prob. psychic is correct “6 or more
§
¡ ©
¤ £ times”? .....
1 2 3 5 6 19 ¢ §
¢ ¡ ©
¤ ¡ § 0 ..... ©
¤ £
£ A
A (Table II(h), p. 772)
Discrete Random Variables Countable number of distinct values Look for “jumps” between the possible values. Assign a “chunk” of probability to each distinct
value. 20 STA 2023 c B.Presnell & D.Wackerly  Lecture 11 137 Continuous Random Variables Take on values associated with points on one or
more line intervals No gaps between values : Time, Height, Pressure,
etc. Too many values go assign a “chunk” of probability
to each
Distribute our 1 unit of probability over appropriate
interval like we would spread a handful of sand.
– It will pile up in certain places.
© – The depth or density depends on the value of .
© – The “density” can be written a function of ,
(p. 210). §©
¤ A mathematical model for the population
histogram STA 2023 c B.Presnell & D.Wackerly  Lecture 11 138 graph of f(x) P(a<X<b)
b §©
¤ The total area under graph of is
# a (p. 210). graph of f(x) Total Area = 1 Over a given interval, the area under the graph of
§ ¤
© a value in that interval (p. 210).
– How do I ﬁnd these areas?
Calculus (not in this class) Use a table. © corresponds to the probability that will take STA 2023 c B.Presnell & D.Wackerly  Lecture 11 for all , § ¢ ©
¤ ¢ £ Note: 139 Line has
area 0 so, for contin. r.v.’s, “endpoints”
don’t make any difference in
probs. of intervals, unlike discrete a (p. 210).
is a continuous r.v.,
§
' 7
¡ §
(
¡
¢ ©
¢ ¤ £ ©
¢ §
( ¤ ¢ ¡ £
¢ §
( 7
¡ © ¢
7 ¤ £ ©
¢ 7 ¤ £ © If STA 2023 c B.Presnell & D.Wackerly  Lecture 11 140 The Normal Distribution A random variable has a normal dist. with mean if its probability density is (p. 216) ¨
©4 § ¥
¦§
¤ ¤ £ ¢ ¡ #
3 ¢ 3 and std. dev. 1 One very important continuous distribution. §©
¤ Recall: #
& ¢ # &
9# " &&
'(& £ ¢ ¡ Formula traces out the “bell shaped curve”. Will write &&
''& No need to know this formula. © As a shorthand for “ has a !
© and std dev ” MEANS “ has a normal distribution
#& and std dev § #& ! ¤ !
© with mean © N 1 3! 1 ¤ normal distribution with mean 3 § N STA 2023 c B.Presnell & D.Wackerly  Lecture 11 141 The graph of the normal density is a symmetric,
. The std dev,
3 1 bellshaped curve, centered at , measures the spread of the distribution. (See Figure
5.6, p. 216)
µ =8
σ =1 4 µ =12
σ =1 6 8 10 12 14 16 Two Normal Curves with Diff Means, Same Std Dev µ =10
σ =1 0 µ =10
σ =3 5 10 15 Two Normal Curves with Same Mean and Differnent Std Devs 20 STA 2023 c B.Presnell & D.Wackerly  Lecture 11 142 The Standard Normal Distribution – p. 217 is said to have a standard normal
and std dev , & © ¢ 3 #!
$" !
¡ The score gives the number of standard . © is larger than
© is smaller than .
. 1
1 is from
1 © deviations .
1 ¢ ¢ £ ¢ £ © ¢
£ is a standard normal random variable:
¢ . is already a score. © # ©
¢ 1 ¤
3 © ¢ That is, the value of © ¤ © If 1 & ¤ Recall : score : ¤ § i.e., is normal with mean distribution if # The r. v. STA 2023 c B.Presnell & D.Wackerly  Lecture 11 143 For any normal random variable, areas under normal
are
¢ and , for curves between zscores of given in Table IV, p. 809 .
Tabled area 0 z Given
in table 1/2 1/2
z
0 Find 0
Find z
Find STA 2023 c B.Presnell & D.Wackerly  Lecture 11 144 (Part of the) Standard Normal Table 0.0
0.1
0.2
.
.
.
0.9
1.0 .00
.0000
.0398
.0793
.
.
.
.3159
.3414 .01
.0040
.0438
.0832
.
.
.
.3186
.3438 .02
.0080
.0478
.0871
.
.
.
.3212
.3461 ...
...
...
...
..
. 1.1
1.2
1.3
1.4
1.5
.
.
.
1.9
2.0 .3643
.3849
.4032
.4192
.4332
.
.
.
.4713
.4772 .3665
.3869
.4049
.4207
.4345
.
.
.
.4719
.4778 .3686
.3888
.4066
.4222
4357
.
.
.
.4726
.4783 ...
...
...
...
...
..
. 2.1
2.2
.
.
.
2.9
3.0 .4821
.4861
.
.
.
.4981
.4987 .4826
.4864
.
.
.
.4982
.4987 .4830
.4868
.
.
.
.4982
.4987 ...
...
..
. ...
... ...
... ...
... .08
.0319
.0714
.1103
.
.
.
.3365
.3599 .09
.0359
.0753
.1141
.
.
.
.3389
.3621 .3810
.3997
.4162
.4306
.4429
.
.
.
.4761
.4812 .3830
.4015
.4177
.4319
.4441
.
.
.
.4767
.4817 .4854
.4887
.
.
.
.4986
.4990 .4857
.4890
.
.
.
.4986
.4990 STA 2023 c B.Presnell & D.Wackerly  Lecture 11 is a standard normal r.v., then ¢ &
§ ¤ £ ¢ @#
&
¢ ¤
¢ %#
&
& § ¢ £ @#
& § ¢ ¤ @#
& £
¤ ¢ Find Find ¢ Find § Ex. If 145 £ ¢
& § ¢ ¢ § #
2 & # & ¤ ¤
£ £ STA 2023 c B.Presnell & D.Wackerly  Lecture 12 146 Thought: If at ﬁrst you don’t succeed, skydiving is not
for you. Assignments :
Today : P. 215–226
For tomorrow:
Exercises 5.24, 5.25, 5.31, 5.36–37,5.39–40, 5.95,
5.96, 5.98
For Monday:
Read pages 254–264, COMPUTER DEMO
For Tuesday:
Exercises 6.1, 6.3, 6.4, 6.8 STA 2023 c B.Presnell & D.Wackerly  Lecture 12 147 Last Time : Continuous Random Variables Probabilities are areas under “density function”
(p. 210).
graph of f(x) P(a<X<b)
a b If Normal distribution is special case. Areas under normal curves between zscores of is a continuous r.v., (p. 210)
§
( 7
¡ §
(
¡
¢ ©
¢ ¤ £ ©
¢ §
' ¤ ¢ ¡ £
¢ §
( 7
¡ © ¢
7 ¤ ©
£ 7 ¤ £ ¢ ¢ and , for in Table IV, p. 809 . Key to ﬁnding correct areas (probabilities) : draw
pictures STA 2023 c B.Presnell & D.Wackerly  Lecture 12 is a STANDARD NORMAL random variable
, ﬁnd: § ¢ § ¢ 7 @#
&
¢ 7 @#
& 32
! & ¤
# ¢
¤ £ # & ¤ ¢ ¤ 1 Find # Find § If 148 £ ¢
§
¢ ¢ § & ¤ 7 & ¤
7 ¤ £ ¤ £ ¤ STA 2023 c B.Presnell & D.Wackerly  Lecture 12 § § ¢ ¢
¢ ¤
7 ¤
7 ¤ ¤ £ Find 149 £ ¢ Recall the Empirical Rule!!!
§ &
§
¢
& ¢ ¢ ¢
& ¢ &
@# ¤ ¢ £
&
%# Find ¤ £ STA 2023 c B.Presnell & D.Wackerly  Lecture 12 have a standard normal dist. & 7 ¢ ¤ £ such that . & §
¢ 7 Find a number § Ex. Let 150 ¤ £ ¢ Note : Used table in reverse.
& ¢ § 7 ¤ £ such that ¡ § ¡
& ¢ ¢ Find a number . ¢ ¤ £ STA 2023 c B.Presnell & D.Wackerly  Lecture 12
&
¢ 3! ¢
¢ 1 Normal distibution, score on dexterity test
& © Ex.: #5.31, p. 228 151 Employer requires dexterity score of at least 80.
©
¤
¢ @ £ ¤ score : § Want . ¢
¢ ¢ § § ©
¤
¢
¢ ©
¤ £ £ STA 2023 c B.Presnell & D.Wackerly  Lecture 12 152 Want  98th percentile of dexterity scores.
so that
, its © score is ¢
¢ score
¢ ¢ 2
&
¢
& #
© – Whatever the value of ¤ – Find
¢ © ¤ STA 2023 c B.Presnell & D.Wackerly  Lecture 12 153 Ex. Not in book
distance at which garage door opener is used.
¢ ## ¢ 3! . & ¢ 1 Limit for interference: © Normally distributed, What fraction use opener beyond the limit?
§
& ©
¤
¢ % £ & §
¢ § ¢ %#
&
¢ ¤ score : ©
¤ ¤ £ £ Five users randomly selected. What is the
probability that all 5 used the openers beyond 50’? STA 2023 c B.Presnell & D.Wackerly  Lecture 12 154 If you randomly select 5 owners of garage door
openers and all 5 use the opener beyond 50’, what
do you think about the claim that ? ## ¢ 3! ¢ 1 –
Ex. : Like #5.40, p. 229 (Same scenario, different
numbers) Amount of dye, in ml., dispensed into can of
paint is normally distributed,
& &
¢ 3 something we can set
¢ 1 More than 5 ml dispensed – paint unusable.
1 What should we set to so that only 2% of cans are unusable. © Find amount dispensed.
so that
1 ¢ STA 2023 c B.Presnell & D.Wackerly  Lecture 12 , the ¤ 1 Whatever the value of 155 score for 5 is ¢
.5 .48 0 & ¤§
& ¤ ¤
#& 1
¢ ¢ & ¤ £ ¤ £ . must be approx. § ¢ score for ¢ Thus, the § ¢
& From table, z0 milliliters ¢ ¤
1 & ...
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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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