Chapter 9: Inferences Concerning Variances¶

Example, Page-269¶

In :
# Variable declaration
from scipy import *
from pylab import *

val = 0
Mine1 = array([8260,8130,8350,8070,8340])

# Calculation

Mean1 = sum(Mine1)/len(Mine1)

for each in Mine1:
val = val + (Mean1-each)**2

var = (val)/(len(Mine1)-1)
std_dev = sqrt(var)

d2 = 2.326   # for n=5

est_std_dev = (max(Mine1)-min(Mine1))/d2

# Result
print "Estimated standard deviation: ",round(est_std_dev,1)
print "Actual standard deviation: ",round(std_dev,1)
Estimated standard deviation:  120.4
Actual standard deviation:  125.5

Example, page-270¶

In :
# Variable declaration
v = 19                              # degree of freedom
Variance = 1.2* pow(10,-4)          # variance

# Calculation
from scipy import *
from pylab import *

# alpha = 0.05
chi_square1 = 8.907                 # chi square value at alpha/2 from table-5
chi_square2 = 32.852                # chi square value at 1-(alpha/2) from table-5

# we need to find limits of sigma

y1 = (v*Variance) / chi_square2
y2 = (v*Variance) / chi_square1

y1 = round( sqrt(y1), 4)            #lower limit
y2 = round( sqrt(y2), 4)            #upper limit

# Result
print "95% confidence interval: (",y1," , ",y2,")"
95% confidence interval: ( 0.0083  ,  0.016 )

Example, page-271¶

In :
# Variable declaration
alpha = 0.05      # level of significance
v = 14            # degree of freedom
std_dev = 0.64    # standard deviation(in mil)
sigma = 0.5

# Calculation
from scipy import *
from pylab import *

# null hypothesis: if sigma = 0.5 , Alternative hypothesis if sigma > 0.5

chi_sq_thr = 23.685                                      # theoritical value of chi square at alpha = 0.05 with v = 14
chi_sq_prt = ( v*pow(std_dev,2) ) /pow(sigma,2)          # calculated value of chi square
chi_sq_prt = round(chi_sq_prt,2)

# Result
if(chi_sq_thr > chi_sq_prt):
print "null hypothesis can not be rejected"
print "we can not conclude that lapping process is unsatisfatory"
else:
print "null hypothesis can be rejected"
print "we can conclude that lapping process is unsatisfatory"
null hypothesis can not be rejected
we can not conclude that lapping process is unsatisfatory

Example, page-273¶

In :
# Variable declaration
v = 11            # degree of freedom
s1 = 0.035        # company-1 work yield
s2 = 0.062        # company-2 work yield
alpha = 0.05      # level of significance

# Calculation
from scipy import *
from pylab import *

# null hypothesis: if square(sigma1) = square(sigma2) , Alternative hypothesis if square(sigma1) < square(sigma2)

f_thr = 2.82                                   # theoritical value of F at alpha = 0.05 with v = 11
f_prt = square(s2) / square(s1)                # calculated value of F
f_prt = round(f_prt,2)

# Result
if(f_thr > f_prt):
print "null hypothesis can not be rejected"
print "data does not support contention"
else:
print "null hypothesis must be rejected"
print "data supports contention"
null hypothesis must be rejected
data supports contention

Example, Page-273¶

In :
# Variable declaration
alpha = 0.02      # level of significance
val1 = 0
val2 = 0

Mine1 = array([8260,8130,8350,8070,8340])
Mine2 = array([7950,7890,7900,8140,7920,7840])

# Calculation
from scipy import *
from pylab import *

Mean1 = sum(Mine1)/len(Mine1)
Mean2 = sum(Mine2)/len(Mine2)

for each in Mine1:
val1 = val1 + (Mean1-each)**2

for each in Mine2:
val2 = val2 + (Mean2-each)**2

# null hypothesis: if square(sigma1) = square(sigma2) , Alternative hypothesis if square(sigma1) < square(sigma2)
s1_square = val1/(len(Mine1)-1)
s2_square = val2/(len(Mine2)-1)

f_thr = 11.4                   # theoritical value of F at alpha = 0.02 for 4 & 5 DOF
f_prt = s1_square / float(s2_square)            # calculated value of F
f_prt = round(f_prt,2)

# Result
print "Practical F value: ",f_prt
if(f_thr > f_prt):
print "null hypothesis can not be rejected"
print "It can be assumed that variances of two populations sampled are equal."
else:
print "null hypothesis must be rejected"
print "It can not be assumed that variances of two populations sampled are equal."
Practical F value:  1.44
null hypothesis can not be rejected
It can be assumed that variances of two populations sampled are equal.

Example, page-275¶

In :
# Variable declaration
v1 = 8                    # degree of freedom of sample-1
v2 = 8                    # degree of freedom of sample-2
alpha = 0.02              # level of significance
s1_square = 0.4548        # for sample-1
s2_square = 0.1089        # for sample-2

# Calculation
from scipy import *
from pylab import *

# 98% confidence interval

f1 = 6.03                                    # f value at 0.01
f2 = 1 / 6.03                                # f value at 0.99

y1 = (f2)*(s2_square / s1_square)            # lower limit
y2 = (f1)*(s2_square / s1_square)            # upper limit

y1 = round(y1,2)
y2 = round(y2,2)

# Result
print "98% confidence interval: (",y1," , ",y2,")"
98% confidence interval: ( 0.04  ,  1.44 )