# #$
# μ & ( ( ! # ! * !
+
# ,
# '
2
(x μ)
=
2
n
2
s &
s2 =
(x x )
2
x
=
n
n 1
μ 2 #'
2
2
s2 =
x
2
( x ) 2
n 1
n
* !
s +
) $ "! %
s
n
!! %
1 1
s12 s22
+ s +
& n1 n 2
n1 n 2 2
!"!%" " '!" #"#"! " !$#
#'"!!243+!""!"""#"!#"!$#2μ43
" "$'"!!23+!""!"""!"#1! " "1!!!"μ4
"-!""!"+!" (!" "% μ4
t=
x μ0
s
n
! 23>
"""-!" #" "$#! "-$#2 "'"""!%!"'
$μ4!"#"3#!"#""-!""!" ! ,
+μAμ4""-!""!"!'!"$* "!$
+μ@μ4""-!""!"!'"$*""-!" #"!!'" ! " "'!""!""!"% !"$
"+""-!""!"!"/% 0!)"-$#!5#!"$" "
+μ?μ4 "-$#!""-!""!"% !"$#"2"! "!!" " "3
"-$#!!!"" " $# !2!#!#'4,493) ""#
'"!!""" "$'"!!,
+
# & !!!! """"#!!#"&"',#"
!# "!#$!'# "%" ' %,# !#"! $%, &#""$ ! %" !4,491,!"!#$'# "%" &
""2!!#>4,493.
4+μ>4,49*+μA4,49
'
#
1
#""!" $"'# !+
5
4,495
x = 0.0508 6
4,4949
(x x ) 2 (0.051 0.0508) 2 + (0.0505 0.0508) 2 + etc...
2
s
=
=
= 9.15 107 7
4,48=
6
n
1
8
4,495:
s = s2 = 9.56 104
9
4,496
:
4,494<
;
4,494:
"-!""!"!+ t
=
x μ0 0.0508 0.05
=
= 2.17 " ! >;-5>:
s
9.56 104
n
7
"""-!" #" "$#!")6,5;%": ! !"%4,49>4,469,
!!"""-$#!!!"4,49)!'# "4#"""!#$'# "
%" &!"",
!"!%" "!"%#"! !"' " " $# # !! "'"$#"" #*+ " $#!" #
" "" $#""
#" """%$#! $#""!"$#!2" !3#!
μ4>4" -!"-"!"
"%#"! "
4+!""!"""!""%#"! #2μ5>μ63
+!""!"""!""%#"! ## ! " """ 2μ5?μ6)μ5Aμ6)μ5@μ63
!+ !! !)
""# ) t
=
x1 x 2
x1 x 2
"""# ) t =
2
2
1
1
s
s
1
2
s2 + + n
n
n
n
1
1
2
2
;.1+1/:.2+1/
!!!+ !"!!#" !+#"" !"!!+ !! ! * !! !!+ !! !. "!!! /!
"!+#"μ1<μ2*
! 18! *12$ !!!##700& . !
#!;21& /* !6" !! !##668& . !
#!;30& /* !!! ! ! . ";0*05/,
μ1;700'1;21'1;12(μ2;668'2;30'2;6
0)μ1;μ2
)μ1=μ2." $& !!! /
! "!!!# !!$"! "" !!! " !
!#!& *
!+ !! ! ) t
=
x1 x 2
s12 s22 + n1 n 2 =
700 668
212 30 2
+
12
6
= 2.342 ;.1+1/:.2+1/;.12+1/:.6+1/;16
!!+ !"!!'!+#" !$0*010*02' $!0* !!
! ! ! *
$!! #!!!! # !"!
0)! #!!! !# !"!
)! #!! !!!# !"!
(O E) 2
. ! ##" !%!#"/
+ " !! ! ) = E
2
;"! ! !"!-1
!!+#"!!2!#" " !"!2#" *!+#" !'!
#! !!!%! !"!*='!!&! !%! !"!
"" $"!!!!!!!"!&*
" !"!" * ""'"$!
"'" *#" !1!"* 2 $
$*! !"!& !!&. ";0*05/,
"
" "
" 143
60
55
18
! !! !&!9)3)3)1' )
0)! # !"!2 ! 9)3)3)1 !"!
)! # !"!2 !!9)3)3)1 !"!
%!#" )
"
" "
" 155*25
51*75
51*75
17*25
2 = (O E) 2 (143 155.25) 2 (60 51.75) 2 (55 51.75) 2 (18 17.25) 2
=
+
+
+
= 2.519 E
155.25
51.75
51.75
17.25
<5,2<4
!##3!#%$")#,%$"!#!#1+36)"#"!#!$!!#(
""!##($""+
$!"#$(###!"!"$"#!#!#
!!"#"!($"+#"$+#$""($!
"#!$#.""$<1+16/-
1*#!"$!"#!$#$"
*#!"#$!"#!$#$"
!!##36$")"#!"$!"#!$#)#!"$3+889
$"!" $!+ !#$"!%%$"!3)5)5)4)4)4)3)4)2)"#3"##"#"*
2 = (2 2.778) 2 (3 2.778) 2 (4 2.778) 2 (O E) 2 (1 2.778) 2
=
+ 2
+ 4
+ 2
= 2.72 E
2.778
2.778 2.778 2.778 <:,2<9
!##3!#%$")#,%$"!#!#1+36)"&#!#1+ $""(
$!"#!$#+
"&#!#&#!%!"!!#!#./
1*##&%!"!#
*##&%!"!##
"#(""$#"$#'#"#!$#
"!%%$".02)03)04)05/!$"$(!"#"#+!&"#!(%!2
$"#!(%!3+
!2
#"
#!( #!( !3
#!( 02
0 3
02;03
#!( 04
0 5
04;05
#"
02;04
03;05
02;03;04;05
!!##!(%!2"#$!%$"#!(%(###$!
+""$)#'#$!%$"##&##!(
#1 +# 3
%$" #1 +# 2 +# 3 +# 4 %!3"#!!##!($#(#$!%$"#!(
#1 +# 3
+ $")#'#%$"*
( #1 +# 2 )
#1 +# 2 +# 3 +# 4 E=
(#1 +# 3 )(#1 +# 2 ) (row total)(column total) =
grand total
#1 +# 2 +# 3 +# 4
!"!<.,2/.,2/ &!"#$!!&""#$!$"
2
," $!"##"#""# 2 = (O E) E
#,%$"!##3!#%$"+
%##&)"#!!#"#&#""%"#".""$<1+16/-
#"" %
&
$, &
$,
%!"
224
224
221
26:
5:6
!!"!"
22:
246
283
2:1
727
2#:!##"(
88
:2
97
76
42:
=21!##"(
292
263
235
84
641
5:1
5:2
5:3
598
2:71
,&! &! %" # '%# &
E=
(row total)(column total) (495)(490)
=
= 123.75 grand total
1960
!
-5 $
7-, $
2 = !
"
(" (
-./'31 -.0
-.0'.2
-10
-10'/-
-10'2/
35'31
35'5-
4,',4
-/.'1
-/.'33
-//',0
-..'55
-1/',2
35'.2
-/-'25
(O E) 2 (113 123.75) 2 (113 124) 2 (110 124.26) 2
=
+
+
+ etc... = 91.73 E
123.75
124
124.26
6*(-+*(-+6*0(-+*0(-+65
. ! %(! ,',,-%", "! '
$ $
$ $
$! P(A and B) = P(A) P(B) "! $!! P(A or B) = P(A) + P(B) P(A and B) $! ! ! P(A | B) =
P(A and B)
P(B)
P(A | B) =
P(B | A) P(A)
P(B)
&
$ $ $"" '" ' "($ "$ ($" !' $' $ ! $)
$
" "
0.5 0.5 0.5 = 0.125 = P(A and B) = P(2 children = bb and 1st child bb) $
" . % P(B) = P(1st child = bb) = 0.5 P(A and B) 0.125
$ P(2 children = bb | 1st child bb) = P(A | B) =
=
0.25 P(B)
0.5
"$ 1-2 # /-2 !' $ #& "$ " #(
)%+ 3 #*3 ,
) + 3 # 3 , & $%
P(A | B) =
P(B | A) P(A)
P(B)
P(S = Z |W = Zz) = 0.5 +0-2 ,
P(W = Zz) = 0.7 +,
P(S = Z) = (0.7 0.5) + (0.3 1) = 0.65 + #
0.5 0.7
P(W = Zz | S = Z) =
= 0.538 0.65
" m
P(X = m) =
,
$" "
(nm )
n! p (1 p)
m!(n m)!
3
" " "(
P(1 boy of 5 children) =
5!0.51 0.5 4
= 0.15625 1!(4)!
$ "&
&
P(X = m) =
enp (n p) m
m!
% !