# #$ # 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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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! % !