★Z값 모르고 넓이는 알때 ppf★Z값 알고 넓이구할때cdf★sympy.solve★기초통계학-[Chapter06 - 연습문제-12]

13. Z ~ N(0,1)에 대하여 P(Z>=z_a) = a이고 X ~ N(뮤 , 분산)

 

1> P(X<= 뮤 + 표준편차*z_a) 

 

P(Z<= (뮤+표준편차*z_a - 뮤) /표준편차) = a

 

2>P(뮤 - 표준편차* z_(a/2) <= X <= 뮤 + 표준편차*z_(a/2) )

 

P(-z_(a/2) <= Z <= Z_(a/2) ) =  (P(Z<=z_(a/2) - P(Z<=0)) * 2 = a 

 

14. X ~ N(60,16)

==> 평균(뮤) = 60 , 분산 = 16

 

1> P(|X-뮤|<= 0.1* 뮤) 

a = scipy.stats.norm.cdf(1.5)*2 - 1
a

=  (P(X-뮤 <= 0.1*뮤) - P(Z<=0) ) * 2 = (P(Z<=  (뮤+0.1뮤 - 뮤) / 4  - 0.5) * 2 =  ((P(Z<= (0.1뮤)/4) - 0.5 ) * 2 = 2* P(Z<= (1.5) - 1 = 0.8663

 

2> P(|X-뮤|<= 2.5* 표준편차)  =  (P(Z<= (2.5 ) -0.5) *2 = 2* P(Z<= 2.5) - 1 = 0.987

 

a = scipy.stats.norm.cdf(2.5)*2 - 1
a

 

15. X ~ N(뮤,분산) , P(뮤 - k*표준편차 < X < 뮤+ k*표준편차) = 0.754

 

P(-k < Z < k) = (P(Z<k) - P(Z<0)) * 2 = 0.754

 

P(Z<k) =  (0.754 + 1) /2 = 0.877

print(1.754 / 2)

b = scipy.stats.norm.ppf(0.877)
b

==> k= 1.16

 

 

16. Z에 대해 만족하는 z_0

 

1> P(Z<=z_0) = 0.9986

 

b = scipy.stats.norm.ppf(0.9986)
b

==> 2.988

 

2> P(Z<=z_0) = 0.0154

 

==> -2.1596

 

3> P(0<=Z<=z_0) = 0.3554

 

P(Z<=z_0) - 0.5 = 0.3554

P(Z<=z_0) = 0.8554

 

==> z_0 = 1.0598

 

4> P(-z_0<=Z<=z_0)

 

= P(Z<=z_0)*2 -1 = 0.9030 

 

P(Z<=z_0) = (1.9030 / 2)

 

==> z_0 = 1.6595

 

5> P(-z_0<=Z<=z_0) = 0.2052

 

P(Z<=z_0)*2 -1 = 0.2052

 

z_0 = 0.26

 

6> P(Z>=z_0)

 

= 1 - P(Z<=z_0) = 0.6915

P(Z<=z_0) = 1 - 0.6915

 

z_0 = -0.5001

 

 

17. X ~ N(10,9) 만족하는 x_0

 

1> P(X<=x_0) = 0.9986

b = scipy.stats.norm.ppf(0.9986)
b

P(Z<= (x_0 - 10) /3) = 0.9986

 

(x_0 - 10 / 3) = 2.98888

x = sympy.Symbol('x')
print(2.98*3)
equation = (x -10) - 2.98 *3
b = sympy.solve( equation)
print(b)

x_0 = 18.94

 

2> P(X<=x_0) = 0.0154

 

P(Z<= (x_0 - 10) /3 ) = 0.0154

b = scipy.stats.norm.ppf(0.0154)
b

x_0 = -2.159 *3 + 10

#%%
x = sympy.Symbol('x')
equation = (x -10) + 2.1596 *3
b = sympy.solve( equation)
print(b)

x_0 = 3.5212

 

3> P(10<=X<=x_0) = 0.3554

 

P( (10-10)/3 <= Z<= (x_0 - 10) /3) = 0.3554

 

P(0<= Z< = (x_0-10)/3) = P(Z<= (x_0 - 10) / 3) - 0.5 = 0.3554

 

P(Z<= (x_0 -10) /3) ) = 0.8554

b = scipy.stats.norm.ppf(0.8554)
b

x_0 = 3*(1.0598) +10

 

x = sympy.Symbol('x')

equation = (x -10) - 1.0598 *3
b = sympy.solve( equation)
print(b)

x_0 = 13.1794

 

4> P(-x_0<=X<=x_0) = 0.9030

 

(P(Z<= (x_0 - 10) /3) - P(Z<=0) ) * 2 = 0.9030

 

P(Z<=(x_0 - 10) /3)) = 0.9030 /2 + 0.5 = 

b = scipy.stats.norm.ppf(0.9030 / 2 + 0.5)
b

x_0 = 3*(1.6595) + 10

 

x = sympy.Symbol('x')

equation = (x -10) - 1.6595 *3
b = sympy.solve( equation)
print(b)

x_0 = 14.9785

 

5> P(-x_0 <= X <= x_0) = 0.2052

 

P(Z<=(x_0 - 10) /3)) = 0.2052 /2 + 0.5 

b = scipy.stats.norm.ppf(0.2052 /2 + 0.5)
b

x_0 = 3*(0.26) + 10

 

x = sympy.Symbol('x')

equation = (x -10) - 0.26 *3
b = sympy.solve( equation)
print(b)

x_0 = 10.78

 

6> P(X>= x_0) = 0.6915

 

1 - P(X<=x_0) = 0.6915

 

1 - P(Z<=(x_0 - 10) /3)) = 0.6915

P(Z<=(x_0 - 10) /3)) = 1 - 0.6915 

b = scipy.stats.norm.ppf(1-0.6915)
b

x_0 = 3*(-0.5) + 10

 

x = sympy.Symbol('x')

equation = (x -10) - (-0.5) *3
b = sympy.solve( equation)
print(b)

x_0 = 8.5

 

 

18. X ~ N(4,9) 만족

 

1> P(X<7) = P(Z< (7-4)/3) = P(Z<1) = 0.8413

b = scipy.stats.norm.cdf(1)
b

 

2> P(4<=X<=x_0) = 0.4750

 

P(X<=x_0) - p(X<=4) = P(Z<=  (x_0 -4 ) /3 ) - P(Z<=0) = 0.4750

 

P(Z<= (x_0-4)/3) = 0.975

b = scipy.stats.norm.ppf(0.975)
b

x_0 = 3* 1.9599 +4

 

x = sympy.Symbol('x')

equation = (x -4) - (1.9599) *3
b = sympy.solve( equation)
print(b)

x_0 = 9.8797

 

 

3> P(1<X<x_0) = 0.756

 

P( (1-4) /3 < Z < (x_0 - 4) /3) = 0.756

 

P(-1 < Z < (x_0 - 4) /3) = P(Z<(x_0 - 4) /3) - P(Z<-1) = 0.756

b = scipy.stats.norm.cdf(-1)
b

P(Z<(x_0 - 4) /3) = 0.1586 + 0.756

b = scipy.stats.norm.ppf(0.1586 + 0.756)
b

x_0 = 3* 1.3696 + 4 

x = sympy.Symbol('x')

equation = (x -4) - (1.3696) *3
b = sympy.solve( equation)
print(b)

x_0 =  8.1088

 

출처 :  [쉽게 배우는 생활속의 통계학]  [북스힐 , 이재원] 

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