Unformatted text preview: 42 STA 2023 c B.Presnell & D.Wackerly  Lecture 4 STA 2023 c B.Presnell & D.Wackerly  Lecture 4 43 StemandLeaf Displays – P. 28 Thought for the day: Friends may come and go, but
enemies accumulate. Given are widths (inches) of the dominant hands Assignments (domhwdt) of 37 honors STA 2023 students. Problems : 2.12, 2.14, 2.16, 2.20, 2.24, 2.89, 2.90,
2.92, 2.97–99, 2.102, 2.105, 2.116, 3.3, From data set you will soon use in the ﬁrst project. 3.5–3.8, 3.14, 3.16–3.18, 3.20, PROJECT 1 Character StemandLeaf Display For Wednesday : Read pages 111–124 Stemandleaf of domhwdt
Leaf Unit = 0.010 Last Time :
value ¤ ¤¢
¥£¡ score : mean standard deviation 1
1
2
4
5
11
12
(10)
15
13
12
2
2 (p. 65) ¡ Tchebysheff’s Theorem (always works) (p. 56) ¡ Empirical Rule (bellshaped distns) (p. 57) ¡ Percentiles (p. 64) 25
26
27
28
29
30
31
32
33
34
35
36
37 N = 37 0
5
01
0
000000
2
5555555555
07
9
0000000000
55 ¡ Quartiles (p. 69) Descriptive Statistics Rare Events and Inference  requires the assess Variable
domhwdt STA 2023 c B.Presnell & D.Wackerly  Lecture 4 Variable
domhwdt 44 Minimum
2.5000 Mean
3.2446 Median
3.2500 Maximum
3.7500 TrMean
3.2515
Q1
3.0000 StDev
0.2849 SE Mean
0.0468 Q3
3.5000 ¡ ment of how likely or probable an occurrence is. N
37 STA 2023 c B.Presnell & D.Wackerly  Lecture 4 45 ¡ Leaf unit = 0.010 (leafs are in last columns in ¡ Stem, middle columns (25, 26, etc.). In the 25, 5 is display) ¡ the “tenths” digit and 2 is the “units” digit. So, entry Median is the 19th value from the lowest. ¡ with stem 28 and leaf 1 is 2.81. 12 values are in the stems above that labeled (10), ¡ From the top of the ﬁrst row down to the entry in (), seven more in that stem would be the 19th (middle) entries in ﬁrst column are total number of data value. ¦ median is 3.25. ¡ 3.00. ¦ with stem 30 indicates that there are 11 values ¡ last entry in that row. EX. 11 in the row values Rotate picture 90 degrees, get something much like
a frequency histogram. ¡ stem containing the median. Number in () is the
number of leaves on that stem.
The rows below that beginning with the number in ()
are the number of values in that row, on down. Ex.
13 in front of stem 34 indicates that there are 13 § values 3.40. ¡ The stem beginning with the number in () is the Locate the point on the stem and leaf diagram
corresponding to handwidth 3.37”. ¡ 46 STA 2023 c B.Presnell & D.Wackerly  Lecture 4 ¡ STA 2023 c B.Presnell & D.Wackerly  Lecture 4 47 Interquartile Range: IQR = upper quartile  lower BoxPlots – Section 2.8 quartile (measure of variablity) Boxplots of proportions of undervotes in 2000 ¡ Top vertical line (whisker) extends up to the largest Presidential election in various counties in Florida. “O” value that is not more than (1.5)IQR above the indicates optical scanner, “P” indicates punch cards. upper quartile ¡ Bottom vertical line (whisker) extends down to the
smallest value that is not more than (1.5)IQR below
the lower quartile ¡ Values beyond the end of the whiskers are marked with a and are “unusually” large (or small) – Outliers. ¡
¡ 19 counties used punch cards, 31 used optical ¡ Top horizontal line (upper hinge) in each box is Why outliers? (mistake recording, measurement upper quartile (75th percentile) (approx. 0.02 in P) from diff. popn., rare event). ¡ Bottom horiz. line (lower hinge) is lower quartile ¡ Middle horiz. line is median (approx. 0.013 in P) scanners. Essentially the whole boxplot for P is
above that for O. Conclusion? (25th percentile) (approx. 0.01 in P) ¡
48 counties using optical scanners? STA 2023 c B.Presnell & D.Wackerly  Lecture 4 What would be the shape of the histogram for the STA 2023 c B.Presnell & D.Wackerly  Lecture 4 49 Chapter 3: Probability
Scatter Diagram: Section 2.9
The study of randomness Plot one variable on horiz axis, another on vert. axis Experiment : the process by which an observation is
Plotted the number of votes for Buchanan in the ‘00 made. (Def. 3.1, p. 100) election versus the number for Perot in the ‘96 election EXAMPLES : (both candidates from the same “minority party”) ¡ Weigh something ¡ Obtain someone’s I.Q. ¡ Toss two coins : a penny and a dime. Intuitively – the PROBABILITY of an outcome is (p. 102) ¡
point in the graph. ¡ Guess which county corresponds to the “unusual” the long run relative frequency of occurrence
how likely the outcome is. EXAMPLE : What is the probability that both coins turn
up “heads”? ¡ STA 2023 c B.Presnell & D.Wackerly  Lecture 4 %&$¤#! ¤
"
¤ 50 STA 2023 c B.Presnell & D.Wackerly  Lecture 4 51 A Sample Point is the most basic outcome of an To each sample point, , we assign a number, the experiment . An event that can occur only one way. probability of 1. the probability of each sample point is between (
3
12
( % "!
0¦ &$¤#)¦ $'" ( and . for each %
&¤ EXAMPLE : Toss two coins : one penny, one dime 2. the sum of the probabilities of all of the sample ¡
¢ (Head on penny, Tail on dime) points is . all simple events ,is the EXAMPLE : Coin toss. etc.) ¤ ¡ each single SAMPLE POINT often denoted (with G7( F 7% $¤#BE% D¤CB7% $¤CB7% ¥ $¤#!
"! "! "! " collection of ALL possible sample points. £ Def. (Def. 3.3, p. 101) The Sample Space, 9A ( 4 "
@876%5$¤#! ¡ One simple event : § ¤§¥
©¨¦¤ a subscript, so that: (shaded box, p. 104) ' (Def. 3.2, p. 101) , EXAMPLE : Toss two coins: Simple events: :
¤
¥
¤ §§§
£ HH HT TH TT or TT Def. (p. 105) An event,
points. (p. 106) The probability of an event the probabilities of the sample points in STA 2023 c B.Presnell & D.Wackerly  Lecture 4 52 , we must know which sample points %"
¦IH#! G 8ESR£C! ¡
( % "
(Q¦ %¦IH"CP¦ ' ¡
!
H
. . 53 EXAMPLE : (Similar to several HW problems)
Breakdown of the workers in a state according to : . Political Afﬁliation ¡ H for any event, ¡ ¡ are in is the sum of STA 2023 c B.Presnell & D.Wackerly  Lecture 4 NOTES :
To ﬁnd , is a collection of sample HT TH H : H : ¤ § ©§ ¤ § ¥ ¤
¤
¤
¤ HH H : Job Type EXAMPLE : Toss two coins Job Type Political Afﬁliation
Rep. Dem. Ind. White collar 12% 12% 6% Blue collar 23% 43% 4% get a match (both coins show the same face). G (` ( F Y( F E¦IH#!
X % "
G 5¡ § V QTH
W¡ U Select one worker at random from this group. To get at least one head describe this person, we need to specify: TH b¡
a : HT : TT Political Afﬁliation h¦g
¡ : £ HH ¡ G7ef ( F Y( F Y( F 7% a #!
F
XX"
¤
¤
¤
¥¤
a
W ¡ U
d¡ § c § V Q a
BH ¡ : Job Type
Republican, White Collar worker. STA 2023 c B.Presnell & D.Wackerly  Lecture 4 55 e ` GPX ` G(
X % h¦g "C!BB% a C!
"
W a g § h¦g UQ
U
Q a person is a Republican combinations: eG
¥
% a g #!
" Sample space consists of all political afﬁliation, job type W 54 STA 2023 c B.Presnell & D.Wackerly  Lecture 4 a h ¡ a
¡ h
Y ag
h
¦g
d£ Ge ¤' )X ` ( PX ` G(
G
¢G
% h ¡ C! X % Y #! X % g CBBIH#!
"
h"
h "!% "
W h ¡ § Y § ¦g Q
h hU
U
QTH
W
e
e "
"
"
F G' 7% a ¡ C! F G E% a C! ` G E% a g #!
G' E% h ¡ C! ` G( 7% Y #! ` G( E% ¦g #!
"
h"
h" ¢
£
Two “duds” Select a door “at random”. % h C! ¡
"
h¡ Win the good prize.
? 56 One good prize. ¡ STA 2023 c B.Presnell & D.Wackerly  Lecture 4 Three Doors ¡ person is a white collar worker LETS MAKE A DEAL!! STA 2023 c B.Presnell & D.Wackerly  Lecture 4 57 Simple Events – If you selected initially Monte will show you If you switch – you win!
initially Monte will show you ¥ If you switch – you win! – If you “switch” ¥ ¦
£ e
% "! % "! "
( E¨ #[email protected]¥ #BE% ¦ #! e
7% W § ¥ 6UCTE% h C!
"! " If you “STAY” with your initial choice, ` (He can always do this!!) ¨
©¨ ¡
¥ ¡ – If you selected ¡ ¦
§ Sample space : After you choose a door, Monte Hall shows you a “dud”. initially and switch – you lose! ¥ Dud #2. – If you selected ¡ Dud #1. If you “SWITCH” ¦ Good door "! "
¥ 7% ¦ #BE% h C! ¡ STA 2023 c B.Presnell & D.Wackerly  Lecture 5 a H their UNION, , , is all points in (Def. 3.5, p. 111). ¡ Exercises : 3.21–23, 3.27, 3.32, 3.33, a
H For Monday : Read pages 127–135 LAST TIME :
Entire shaded area is ¡ Experiment (p. 100), or both a H 3.35, 3.37–3.39, 3.41, 3.43, 3.46 or £ are two events in H For tomorrow: and a Def. If Assignments H most of the cars today could see their real owners. a Thought for the day: Drivein banks were established so 59 £ 58 STA 2023 c B.Presnell & D.Wackerly  Lecture 5 In the Job Type–Political afﬁliation example G e ¥ G e ` )X £' PX ` ( PX ` G( 7% a IH#!
G ¢G G
"
W a g § h ¡ § Y § g 0 a H ¡
h hU
G
b¡
a
W a g § ¦g Q
hU
BH ¡
W h ¡ § Y § ¦g Q
h hU
white collar worker
Republican , (p. 101), : ¡
¡ Find the probability of an event (p. 106) STA 2023 c B.Presnell & D.Wackerly  Lecture 5 60 STA 2023 c B.Presnell & D.Wackerly  Lecture 5 Def. (Def. 3.8, p. 118) If and simultaneously have no sample contains no points), then are mutually exclusive (or disjoint ) events. See ﬁgure 3.11, p. 118. a £
H and a H a ¡
a ¢H ¡ (Def. 3.6, p. 111). ¡
a ¢H sample points in both , is the set of all a their INTERSECTION, points in common, ( and 61 £ ¡ Events (p. 105), ¡
a £H
H ¡ Properties of Probabilities of Sample Points (p. 96), a £ Sample Space, . ¡ Sample Point (p. 101), H ¡ Probability (p. 102), H EXAMPLE : Toss two dice
In the Job Type–Political afﬁliation example and sum of up faces is 11
up face on ﬁrst die is 3 are mutually exclusive.
are mutually exclusive, then (p. 118) % a C! X ¦IHCB7% a IHC!
" % "!
" Note. If and H ' E% a ¥IH#!
¡" . Then a : a
` G( 7% ¦g #BEH % a
h " !¡
WhU
8¦g Q a ¢P
G W a g § ¦g Q
hU
W h ¡ § Y § ¦g Q
h hU
Republican ¤ . ¤
a white collar worker HIC!
"
aH¡
b¡
a
BH ¡ 62 STA 2023 c B.Presnell & D.Wackerly  Lecture 5 Ex. Odds are 2:1 that when Henry and Thomas play , racquetball, Henry will win. H and T play 3 matches, . and the winner of each match recorded. There are
eight sample points: ¤
¤
¤
¥
¦¤ H HHH HTH
HTT (why?), we have TTH
TTT (p. 115) ?
2/27
1/27 %"
a C!
%"
IH#!
Ua
UH
% ¤$¤C!
£" a. Find . b. Let H wins at least two matches . . ©
` X ` X ¨` X
©
© ©
` § ¢ ` Last one much easier using idea ofcomplement :
” H wins no matches TTT ¡ H"
©¨` ¨ 7% a ¢I#!
§
W UQ
U
Q a £H
¡ H wins at least 2 and H wins an odd #
HHH (NOT mut. excl.) W ( "
"
©` § ¢ ` (© ` ¤ 87% ¤ C! ¤ ( % ¤ #!
U
Q W
U
0 ¤
¤
“not W W ©
©
©
©
G ©` § ' ` ` X ` X ¨` X `
FFF
% ¡ ¤$#! X % $¤"#! X % ¤DC! X % ¥ D¤"CBB¦IHC!
"
"
!%"
UQ
U
QTH W W W HHH, HTT, THT, TTH F H wins an odd number of matches THH, THT, TTH ` ©§
` ` ` ¨£` ¤ 8B% £ $#!
© § ¥ ( ¤"
c. HHH, HHT, HTH, HTT, ©¨` X ©¨` X ©` X ©` B% ¤ #!
"
(
8 HHH, HHT, HTH, THH H wins at least 1 match F b. Let
H wins at least 2 matches UQ
U
Q ¤ , 65 ` %" ( "
¦IHC! ¤ 87% IHC! a. Since the sum of the probs. of all sample points STA 2023 c B.Presnell & D.Wackerly  Lecture 5 F % IHC! ¤ ( E$HC!
"
% "
64 and H wins an odd # of matches . Find c. Find
STA 2023 c B.Presnell & D.Wackerly  Lecture 5 W so and 4/27 ` (8E% IH#! X %¦IH#!
"
"
( E7R£#!
% "
¡
%7R£#B7% HI#! X ¦IH"#!
"! " %
¡ H¡
H¡ THT Since 2/27 W £T H H . 4/27 ¦
¤¤
¥¢¤
£
¤¤
¡
¢¤ THH Thus, 4/27 W H H are mutually exclusive 8/27 W £ Sample Point HHT
and 63 6%$¤#!
4" , is the set of all sample points not in H denoted H Def. (Def. 3.7, p.115) The complement of an event STA 2023 c B.Presnell & D.Wackerly  Lecture 5 U
Q
U
Q a "
©
` § F ( ¨`© ` X `© ` X `© ` X © ` B% a #! 66 STA 2023 c B.Presnell & D.Wackerly  Lecture 5 STA 2023 c B.Presnell & D.Wackerly  Lecture 5 67 Additive Rule of Prob. (p. 117)
EXAMPLE : Five equally qualiﬁed job applicants, ¡
a £H
H
£a
% a ¢IH#! ¤ % a C! X ¦IH#B7% a IHC!
¡"
" % "!
" W
§¥
¡ h ¥¡ h ¡
aH¡
G % a "C©§ % a "#©§ %¦IH"C!
!
!
¡
U b¡
a
W
U H ¡
hire at least 1 woman hire two of the same sex Find If and and . STA 2023 c B.Presnell & D.Wackerly  Lecture 5 and are mutually exclusive since OLD STUFF!! 68 Men : mutually exclusive? % a C! X %¦IH"C!B7% a IH"#!
"
H Women : and a Are Entire shaded area is G ' 7% a ¡ IH#!
" Randomly select 2 to hire. a H 2 Women and 3 Men. STA 2023 c B.Presnell & D.Wackerly  Lecture 5 69 Conditional Probability
EXAMPLE : Political Afﬁliation/Job Type
be two events, where . The conditional probability of a a
is also in H H
a (What proportion of ?) £ % C!
"
% a ¢IHC! 7% a ¥IHC!
¡a " ¤ " ' £% a #!
¢" G% e G e )X £' PX ( )X G( E%
G ¢G G
"
a IHCD!"
¥
G e ` ¥ G ` G( ¤ ` ¥ e PX ` ' e G 7% a IHC! ¡
G
"
G ` G( 7% a ¡ IHC! § ¥ e G S% a #©§ ' e G E¦IHC! ¡
"
"!
% "
U b¡
a
W
U
)BH
e W "
e "¡
"
F G' 7% a ¡ C! F G E% a C! ` G E% a g #!
G' E% h ¡ C! ` G( 7% Y #! ` G( E% ¦g #!
"
h"
h" ¢
£ Republican H H white collar worker is and a Def. (p. 122) Let given 70 STA 2023 c B.Presnell & D.Wackerly  Lecture 5 STA 2023 c B.Presnell & D.Wackerly  Lecture 5 71 EXAMPLE : Pol. party – job type. person is a Republican . If I know that a person is a Republican, what is the
probability that (s)he is a white collar worker? RW DW IW H RB DB IB a "
a g #!
h"
¦g C!
$HC!
" ¡ . Thus, eG
% #!
"
Ge e
F G ¥ G( % a ¢IHC! E% a ¤
¡a "
e` "
e
"
F G' 7% a ¡ #! F G E% a C! ` G E%
G' E% h ¡ #! ` G( E% Y C! ` G( E%
"
h"
get a number greater than 3 If a person is NOT a Democrat, what is the probability that (s)he is a blue collar worker? ¢
¤ e
£
"
G % ¡a C! " 7% ¥IHC! ¡
¤"
` £ % a £IHC! a
G £ 7% a ¢$HC!
¡"
U
¡
W ¢ § F 0 a ¢H ¡
G £ 7% a C! ¡
"
G W ¢ § ¥ § F UQ W
U b¡
a
G £ ¥£ X ¥£ X ¥£ E¦IHC! ¡
% "
G W ¢ § F § ` UQ W
U
)BH ¡
4"
¡
¥£ 76%$¤#!
G W ¢ § ¥ § F § e § ` § ( Qd£ ¡
U
get an even number £ ¡ . W U b¡
a
U
)BH ¡ Balanced W EXAMPLE : Toss a balanced die. person is a white collar worker . ...
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 Spring '08
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 Statistics, Quartile, Whitecollar worker, Bluecollar worker

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