★모분산 비에 대한 소표본 추론★기초통계학-[소표본 추론-08]
2023. 1. 18. 15:10
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1. 모분산 비에 대한 소표본 추론
==> 어느 회사에서 생산된 측정 기구의 정밀도가 더 정확한지 비교하는 경우도 있다. 이러한 상황에 대한 통계적 추론은 두 집단 사이의 모분산을 비교함으로써 해결할 수 있다.
https://knowallworld.tistory.com/310
★F-분포★두 표본분산의 비에 대한 표본분포★기초통계학-[모집단 분포와 표본분포 -11]
1. 두 표본분산의 비에 대한 표본분포 ==> 서로 독립인 두 정규모집단의 모분산이 다를때, 모분산 중에서 어느 것이 더큰지 비교하는 경우 ==> 모분산은 양수이므로, 두 모분산의 비의 값을 이용하
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EX-01) 서로 독립인 두 정규모집단으로 각각 표본으로 선정하여 다음 결과를 얻었다. 모분산의 비 모분산_1/ 모분산_2 에 대한 90% 신뢰구간을 구하라.
A = pd.DataFrame({'표본 1 ' : [10 , '|x = 17.5' , 's_1 = 3.5'] , '표본 2 ' : [8 , '|y = 21.2' , 's_2 = 2.8']})
A
X = np.arange(0,10, .01)
fig = plt.figure(figsize=(20,12))
# A = '12.5 11.5 6.0 5.5 15.5 11.5 10.5 17.5 10.0 9.5 13.5 8.5 11.5 15.5'
# A= list(map(float , A.split(' ')))
# Vars = np.var(A , ddof=1)
Vars = 0.2**2
n = 10
m = 8
dof = [[n-1 , m-1]] #자유도
trust = 90
trust = round((1- trust/100)/2,3)
sample_x = 17.5
stand_x = 3.1
sample_y = 21.2
stand_y = 2.8
STDS = math.sqrt(Vars)
MO_std = 0.3
for i in dof:
ax = sns.lineplot(X , scipy.stats.f(i[0] , i[1]).pdf(X))
X_r = scipy.stats.f(dof[0][0], dof[0][1]).ppf(0.95)
X_l = scipy.stats.f(dof[0][0], dof[0][1]).ppf(0.05)
# t_r = round( (x_0 - (0)) / (math.sqrt(33.463) * math.sqrt(1/16 + 1/16)), 3)
print(X_r)
ax.fill_between(X, scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X) , 0 , where = (X<=X_r) & (X>=X_l) , facecolor = 'orange')
ax.fill_between(X, scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X) , 0 , where = (X>=X_r) | (X<=X_l) , facecolor = 'skyblue') # x값 , y값 , 0 , X조건 인곳 , 색깔
ax.vlines(x = X_r ,ymin=0 , ymax= scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_r) , colors = 'black')
ax.vlines(x = X_l ,ymin=0 , ymax= scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_l) , colors = 'black')
plt.annotate('' , xy=(X_r , scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_r)/2), xytext=(X_r+2 ,.1) , arrowprops = dict(facecolor = 'black'))
plt.annotate('' , xy=(X_l , scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_l)/2), xytext=(X_l + .5 , scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_l)/2) , arrowprops = dict(facecolor = 'black'))
area = round(1- scipy.stats.f(dof[0][0] , dof[0][1]).cdf(X_r) ,4)
ax.text(X_r+1.5 , scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_r) + .08 , r'$P(F\leqq f_{%.3f,%2d,%2d})$' % (trust,dof[0][0] , dof[0][1]) + f'= {area}' , fontsize = 14)
ax.text(X_l + .8 , scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_l)/2 , r'$P(F\leqq f_{%.3f,%2d,%2d})$' % (1-trust,dof[0][0] , dof[0][1]) + f'= {area}' , fontsize = 14)
ax.text(X_l + 0.2 , scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_l)/6 , r'$P(f_{%.3f,%2d,%2d})$' % (1-trust,dof[0][0] , dof[0][1]) + f'= {round(X_l , 2)}' , fontsize = 14)
ax.text(X_r - 1.5, scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_r)/6, r'$P(f_{%.3f,%2d,%2d})$' % (trust,dof[0][0] , dof[0][1]) + f'= {round(X_r , 2)}' , fontsize = 14)
ax.text(6 , 0.6 , r'$\overline{x} = {%.3f} , \overline{y} = {%.3f}$' % (sample_x, sample_y) + f'\n' + r'$s_1 = {%.3f} , s_2 = {%.3f}$' %(stand_x, stand_y), fontsize = 15)
ax.text(6 , 0.4 , f'{ (1- (trust*2))*100}%신뢰도 에서의 신뢰구간\n' + r'$\left(\dfrac{s^2_1}{s^2_2}*\dfrac{1}{f_{0.05 , n-1 , m-1}} , \dfrac{s^2_1}{s^2_2}*\dfrac{1}{f_{0.95 , n-1 , m-1}}\right)$' + f'\n =' + r'$\left( {%.3f} , {%.3f} \right)$' % (round(stand_x**2 / stand_y**2 / X_r,4) , round(stand_x**2 / stand_y**2 / X_l,4)) , fontsize = 16)
신뢰구간 : (0.333 , 4.036)
EX-02) 서로 독립인 두 정규모집단으로 각각 표본을 선정하여 다음 결과를 얻었다. 모분산의 비에 대한 95% 신뢰구간을 구하라.
A = pd.DataFrame({'표본 1 ' : [7 , '|x = 161' , 's_1 = 7.4'] , '표본 2 ' : [6 , '|y = 169' , 's_2 = 9.1']})
A
X = np.arange(0,10, .01)
fig = plt.figure(figsize=(20,12))
# A = '12.5 11.5 6.0 5.5 15.5 11.5 10.5 17.5 10.0 9.5 13.5 8.5 11.5 15.5'
# A= list(map(float , A.split(' ')))
# Vars = np.var(A , ddof=1)
Vars = 0.2**2
n = 10
m = 8
dof = [[n-1 , m-1]] #자유도
trust = 90
trust = round((1- trust/100)/2,3)
sample_x = 17.5
stand_x = 3.1
sample_y = 21.2
stand_y = 2.8
STDS = math.sqrt(Vars)
MO_std = 0.3
for i in dof:
ax = sns.lineplot(X , scipy.stats.f(i[0] , i[1]).pdf(X))
X_r = scipy.stats.f(dof[0][0], dof[0][1]).ppf(0.95)
X_l = scipy.stats.f(dof[0][0], dof[0][1]).ppf(0.05)
# t_r = round( (x_0 - (0)) / (math.sqrt(33.463) * math.sqrt(1/16 + 1/16)), 3)
print(X_r)
ax.fill_between(X, scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X) , 0 , where = (X<=X_r) & (X>=X_l) , facecolor = 'orange')
ax.fill_between(X, scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X) , 0 , where = (X>=X_r) | (X<=X_l) , facecolor = 'skyblue') # x값 , y값 , 0 , X조건 인곳 , 색깔
ax.vlines(x = X_r ,ymin=0 , ymax= scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_r) , colors = 'black')
ax.vlines(x = X_l ,ymin=0 , ymax= scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_l) , colors = 'black')
plt.annotate('' , xy=(X_r , scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_r)/2), xytext=(X_r+2 ,.1) , arrowprops = dict(facecolor = 'black'))
plt.annotate('' , xy=(X_l , scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_l)/2), xytext=(X_l + .5 , scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_l)/2) , arrowprops = dict(facecolor = 'black'))
area = round(1- scipy.stats.f(dof[0][0] , dof[0][1]).cdf(X_r) ,4)
ax.text(X_r+1.5 , scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_r) + .08 , r'$P(F\leqq f_{%.3f,%2d,%2d})$' % (trust,dof[0][0] , dof[0][1]) + f'= {area}' , fontsize = 14)
ax.text(X_l + .8 , scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_l)/2 , r'$P(F\leqq f_{%.3f,%2d,%2d})$' % (1-trust,dof[0][0] , dof[0][1]) + f'= {area}' , fontsize = 14)
ax.text(X_l + 0.2 , scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_l)/6 , r'$P(f_{%.3f,%2d,%2d})$' % (1-trust,dof[0][0] , dof[0][1]) + f'= {round(X_l , 2)}' , fontsize = 14)
ax.text(X_r - 1.5, scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_r)/6, r'$P(f_{%.3f,%2d,%2d})$' % (trust,dof[0][0] , dof[0][1]) + f'= {round(X_r , 2)}' , fontsize = 14)
ax.text(6 , 0.6 , r'$\overline{x} = {%.3f} , \overline{y} = {%.3f}$' % (sample_x, sample_y) + f'\n' + r'$s_1 = {%.3f} , s_2 = {%.3f}$' %(stand_x, stand_y), fontsize = 15)
ax.text(6 , 0.4 , f'{ (1- (trust*2))*100}%신뢰도 에서의 신뢰구간\n' + r'$\left(\dfrac{s^2_1}{s^2_2}*\dfrac{1}{f_{0.05 , n-1 , m-1}} , \dfrac{s^2_1}{s^2_2}*\dfrac{1}{f_{0.95 , n-1 , m-1}}\right)$' + f'\n =' + r'$\left( {%.3f} , {%.3f} \right)$' % (round(stand_x**2 / stand_y**2 / X_r,4) , round(stand_x**2 / stand_y**2 / X_l,4)) , fontsize = 16)
# #=================================가설검정=====================================
#
ax.set_title('양측 검정' , fontsize = 17)
#
# X_L_1 = (n-1) * Vars / (MO_std**2) #검정값
# print(f'X_L_1 : {X_L_1}' )
# X_L_1 = abs(round(X_L_1,4))
#
# X_R_1 = round(float(scipy.stats.chi2(dof_2).ppf(1- scipy.stats.chi2(dof_2).cdf(X_L_1))),4)
# print(X_R_1)
# ax.fill_between(X, scipy.stats.chi2(dof_2).pdf(X) , 0 , where = (X>=X_r) | (X<=X_l) , facecolor = 'red') # x값 , y값 , 0 , X조건 인곳 , 색깔
#
#
# area = round(float(scipy.stats.chi2(dof_2).cdf(X_L_1) + 1 - (scipy.stats.chi2(dof_2).cdf(X_R_1))),4)
#
#
# ax.vlines(x= X_L_1, ymin= 0 , ymax= stats.chi2(dof_2).pdf(X_L_1) , color = 'green' , linestyle ='solid' , label ='{}'.format(2))
# ax.vlines(x= X_R_1, ymin= 0 , ymax= stats.chi2(dof_2).pdf(X_R_1) , color = 'green' , linestyle ='solid' , label ='{}'.format(2))
# #
# annotate_len = stats.chi2(dof_2).pdf(X_R_1) /2
# plt.annotate('' , xy=(X_L_1, annotate_len), xytext=((X_R_1-X_L_1)/2 , annotate_len) , arrowprops = dict(facecolor = 'black'))
# plt.annotate('' , xy=(X_R_1, annotate_len), xytext=((X_R_1-X_L_1)/2 , annotate_len) , arrowprops = dict(facecolor = 'black'))
# ax.text( (X_R_1-X_L_1)/2 + 5, annotate_len+0.005 , f'P-value : \nP(X<={X_L_1}) + P(X>={X_R_1}) \n = {area}',fontsize=15)
X = np.arange(0,10, .01)
fig = plt.figure(figsize=(20,12))
# A = '12.5 11.5 6.0 5.5 15.5 11.5 10.5 17.5 10.0 9.5 13.5 8.5 11.5 15.5'
# A= list(map(float , A.split(' ')))
# Vars = np.var(A , ddof=1)
Vars = 0.2**2
n = 10
m = 8
dof = [[n-1 , m-1]] #자유도
trust = 90
trust = round((1- trust/100)/2,3)
sample_x = 17.5
stand_x = 3.1
sample_y = 21.2
stand_y = 2.8
STDS = math.sqrt(Vars)
MO_std = 0.3
for i in dof:
ax = sns.lineplot(X , scipy.stats.f(i[0] , i[1]).pdf(X))
X_r = scipy.stats.f(dof[0][0], dof[0][1]).ppf(0.95)
X_l = scipy.stats.f(dof[0][0], dof[0][1]).ppf(0.05)
# t_r = round( (x_0 - (0)) / (math.sqrt(33.463) * math.sqrt(1/16 + 1/16)), 3)
print(X_r)
ax.fill_between(X, scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X) , 0 , where = (X<=X_r) & (X>=X_l) , facecolor = 'orange')
ax.fill_between(X, scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X) , 0 , where = (X>=X_r) | (X<=X_l) , facecolor = 'skyblue') # x값 , y값 , 0 , X조건 인곳 , 색깔
ax.vlines(x = X_r ,ymin=0 , ymax= scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_r) , colors = 'black')
ax.vlines(x = X_l ,ymin=0 , ymax= scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_l) , colors = 'black')
plt.annotate('' , xy=(X_r , scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_r)/2), xytext=(X_r+2 ,.1) , arrowprops = dict(facecolor = 'black'))
plt.annotate('' , xy=(X_l , scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_l)/2), xytext=(X_l + .5 , scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_l)/2) , arrowprops = dict(facecolor = 'black'))
area = round(1- scipy.stats.f(dof[0][0] , dof[0][1]).cdf(X_r) ,4)
ax.text(X_r+1.5 , scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_r) + .08 , r'$P(F\leqq f_{%.3f,%2d,%2d})$' % (trust,dof[0][0] , dof[0][1]) + f'= {area}' , fontsize = 14)
ax.text(X_l + .8 , scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_l)/2 , r'$P(F\leqq f_{%.3f,%2d,%2d})$' % (1-trust,dof[0][0] , dof[0][1]) + f'= {area}' , fontsize = 14)
ax.text(X_l + 0.2 , scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_l)/6 , r'$P(f_{%.3f,%2d,%2d})$' % (1-trust,dof[0][0] , dof[0][1]) + f'= {round(X_l , 2)}' , fontsize = 14)
ax.text(X_r - 1.5, scipy.stats.f(dof[0][0] , dof[0][1]).pdf(X_r)/6, r'$P(f_{%.3f,%2d,%2d})$' % (trust,dof[0][0] , dof[0][1]) + f'= {round(X_r , 2)}' , fontsize = 14)
ax.text(6 , 0.6 , r'$\overline{x} = {%.3f} , \overline{y} = {%.3f}$' % (sample_x, sample_y) + f'\n' + r'$s_1 = {%.3f} , s_2 = {%.3f}$' %(stand_x, stand_y), fontsize = 15)
ax.text(6 , 0.4 , f'{ (1- (trust*2))*100}%신뢰도 에서의 신뢰구간\n' + r'$\left(\dfrac{s^2_1}{s^2_2}*\dfrac{1}{f_{0.05 , n-1 , m-1}} , \dfrac{s^2_1}{s^2_2}*\dfrac{1}{f_{0.95 , n-1 , m-1}}\right)$' + f'\n =' + r'$\left( {%.3f} , {%.3f} \right)$' % (round(stand_x**2 / stand_y**2 / X_r,4) , round(stand_x**2 / stand_y**2 / X_l,4)) , fontsize = 16)
# #=================================가설검정=====================================
#
ax.set_title('양측 검정' , fontsize = 17)
#
# X_L_1 = (n-1) * Vars / (MO_std**2) #검정값
# print(f'X_L_1 : {X_L_1}' )
# X_L_1 = abs(round(X_L_1,4))
#
# X_R_1 = round(float(scipy.stats.chi2(dof_2).ppf(1- scipy.stats.chi2(dof_2).cdf(X_L_1))),4)
# print(X_R_1)
# ax.fill_between(X, scipy.stats.chi2(dof_2).pdf(X) , 0 , where = (X>=X_r) | (X<=X_l) , facecolor = 'red') # x값 , y값 , 0 , X조건 인곳 , 색깔
#
#
# area = round(float(scipy.stats.chi2(dof_2).cdf(X_L_1) + 1 - (scipy.stats.chi2(dof_2).cdf(X_R_1))),4)
#
#
# ax.vlines(x= X_L_1, ymin= 0 , ymax= stats.chi2(dof_2).pdf(X_L_1) , color = 'green' , linestyle ='solid' , label ='{}'.format(2))
# ax.vlines(x= X_R_1, ymin= 0 , ymax= stats.chi2(dof_2).pdf(X_R_1) , color = 'green' , linestyle ='solid' , label ='{}'.format(2))
# #
# annotate_len = stats.chi2(dof_2).pdf(X_R_1) /2
# plt.annotate('' , xy=(X_L_1, annotate_len), xytext=((X_R_1-X_L_1)/2 , annotate_len) , arrowprops = dict(facecolor = 'black'))
# plt.annotate('' , xy=(X_R_1, annotate_len), xytext=((X_R_1-X_L_1)/2 , annotate_len) , arrowprops = dict(facecolor = 'black'))
# ax.text( (X_R_1-X_L_1)/2 + 5, annotate_len+0.005 , f'P-value : \nP(X<={X_L_1}) + P(X>={X_R_1}) \n = {area}',fontsize=15)
신뢰구간 : (0.095 , 3.959)
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