Chapter 7: Inferences Concerning a Mean

Example, page-204

In [1]:
# Variable declaration
l = [2.4,2.9,2.7,2.6,2.9,2.0,2.8,2.2,2.4,2.4,2.0,2.5]            # list of lead concentration
x = [1,1,1,1,1.1,1,1,1,1.1,1.2,1.1,1]
n = 12                                                           # sample size

# Calculation
from scipy import *
from pylab import *
%matplotlib inline

scatter(l,x)
title("DOT DIAGRAM")
xlabel("$ Lead(micro gram/l$")

Mean = sum(l)/len(l)                                             # Mean of sample
Mean = round(Mean,3)
Variance = (sum(square(l)) - pow(sum(l),2)/ n) / (n-1)           # Variance of sample
Variance = round(Variance,5)

# Result
std_err = round(sqrt(Variance / n),3)                            # Estimated standard error
print "Estimated standard error: ",std_err

Estimated standard error:  0.09

Example, Page-207

In [2]:
# Variable declaration
n = 150
std_dev = 6.2                              # Standard deviation

# as alpha = 0.01  , z(alpha/2) = 2.575
Z = 2.575                                  # Z value

# Calculation
from scipy import *
from pylab import *

E = (Z*std_dev)/sqrt(n)
E = round(E,2)

# Result
print "Maximum Error: ",E

Maximum Error:  1.3

Example, page-208

In [3]:
# Variable declaration
E = 0.50                                   # Atmost error
std_dev = 1.6                              # Standard deviation

# as alpha = 0.05  , z(alpha/2) = 1.96
Z = 1.96                                   # Z value

# Calculation
from scipy import *
from pylab import *

n = pow( (Z*std_dev) / E,2 )               # Sample size
n = round(n,-1)

# Result
print "Sample size: ",int(n)

Sample size:  40

Example, page-209

In [4]:
# Variable declaration
n = 6                                               # number of observations
std_dev = 0.14                                      # standard deviation(in degree)
# as alpha = 0.02  , t(alpha/2) = 3.365 for v = 5

# Calculation
from scipy import *
from pylab import *

t = 3.365                                           # i.e. t(0.01) for v=5
std_err = (t*std_dev) / sqrt(n)                     # estimated standard error
std_err = round(std_err,2)

# Result
print "Maximum error: ",std_err,"degree"

Maximum error:  0.19 degree

Example, Page-210

In [5]:
# Variable declaration
n = 100                                # sample size
Mean = 21.6                           # sample mean
std_dev = 5.1                        # standard deviation

# Calculation
from scipy import *
from pylab import *

# as alpha = 0.05  , z(alpha/2) = 1.96
Z = 1.96

y1 = Mean - Z*(std_dev / sqrt(n))           # lower limit of range
y2 = Mean + Z*(std_dev / sqrt(n))           # upper limit of range
y1 = round(y1,1)
y2 = round(y2,1)

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

95% confidence interval: ( 20.6 , 22.6 )

Example, page-210

In [6]:
# Variable declaration
n = 50                                # sample size
Mean = 305.58                         # sample mean(in nm)
std_dev = 36.97                       # standard deviation(in nm)

# Calculation
from scipy import *
from pylab import *

# as alpha = 0.01  , z(alpha/2) = 2.575
Z = 2.575

y1 = Mean - Z*(std_dev / sqrt(n))           # lower limit of range
y2 = Mean + Z*(std_dev / sqrt(n))           # upper limit of range
y1 = round(y1,2)
y2 = round(y2,2)

# Result
print "99% confidence interval(in nm): (",y1,",",y2,")"

99% confidence interval(in nm): ( 292.12 , 319.04 )

Example, Page-211

In [7]:
# Variable declaration
n = 16                      # sample size
Mean = 3.42                 # sample mean
std_dev = 0.68              # standard deviation

# Calculation
from scipy import *
from pylab import *

# t(0.05) = 2.947
t = 2.947

y1 = Mean - t*(std_dev / sqrt(n))           # lower limit of range
y2 = Mean + t*(std_dev / sqrt(n))           # upper limit of range
y1 = round(y1,2)
y2 = round(y2,2)

# Result
print "99% confidence interval(in grams): (",y1,",",y2,")"

99% confidence interval(in grams): ( 2.92 , 3.92 )

Example, page-219

In [8]:
# Variable declaration
count = [7,3,1,2,4,1,2,3,1,2]                                               # count list for 10 days

# Calculation
from scipy import *
from pylab import *

lembda = float(sum(count)) / len(count)                                     # maximaum likelihood estimate of lembda
lembda = round(lembda,2)

# we need to find P(x=0 or x=1) using poisson distribution
p = exp(-lembda) + (exp(-lembda) * lembda) / math.factorial(1)              # maximum estimated probability
p = round(p,3)

# Result
print "Maximum estimated probability: ",p

Maximum estimated probability:  0.267

Example, Page-220

In [10]:
#Variable declaration
l = array([5.57,5.76,4.18,4.64,7.02,6.62,6.33,7.24,5.57,7.89,4.67,7.24,6.43,5.59,5.39])
          
# calculation
Mean = mean(l)
var = 0

for each in l:
    var = var + (each-Mean)**2

var = var/len(l)

coff = sqrt(var)/Mean

# Results
print "Maximum likelihood estimates of Mean:",round(Mean,3),"  Variance:",round(var,3)
print "Cofficient of variation:",round(coff,3)          

Maximum likelihood estimates of Mean: 6.009   Variance: 1.084
Cofficient of variation: 0.173

Example, page-231

In [11]:
# Variable declaration
Mean = 4.5                 # mean of normal distribution
std_dev = 1.5              # standard deviation of normal distribution
n = 25                     # number of vinyl specimens
x = 3.9

# Calculation
from scipy import *
from pylab import *

# corresponding to x = 3.9 ,  Z = (x-Mean) / (std_dev/sqrt(n))
Z = (x-Mean) / (std_dev/sqrt(n))           # Z value
Z = round(Z,0)

# from Normal Table P(Z>2.00) = 0.0228 which is same as P(Z<-2.00)
p = 0.0228                                                              #  probability P(z<-2.00)
p_val = 2*p                                                             # Required P-Value

# Result
print "P-Value: ",p_val

P-Value:  0.0456

Example, page-232

In [12]:
# Variable declaration
alpha = 0.01         # level of significance
x = 27463            # in miles
Mean = 28000         # Mean( in miles)
std_dev = 1348       # standard deviation(in miles)
n = 40               # sample size

# Calculation
from scipy import *
from pylab import *

# null hypothesis is accepted if Z< -z(alpha)  and rejected if Z> -z(alpha), z(0.01) = 2.33
Z = (x-Mean) / (std_dev / sqrt(n))          # Z value corresponding to x
Z = round(Z,2)

# Result
if(Z<-2.33):
    print "Null hypothesis rejected"
else:
    print "Null hypothesis accepted"

Null hypothesis rejected

Example, page-233

In [13]:
# Variable declaration

alpha = 0.01         # level of significance
Mean = 180           # Mean( in pound)
n = 5                # sample size
std_dev = 5.7        # standard deviation(in pound)
x = 169.5            # in pound

# Calculation
from scipy import *
from pylab import *

# consider (1) null hypothesis if value=180 punds (2) alternative hypothesis if value < 180 pounds

t = (x-Mean) / (std_dev / sqrt(n))          # t value corresponding to x
t = round(t,2)

# Result
if(t < -3.747):
    print "Null hypothesis rejected"
else:
    print "Null hypothesis accepted"

print "The breaking strength is below specifications"

Null hypothesis rejected
The breaking strength is below specifications

Example, Page-236

In [14]:
# Variable declaration
n = 16                      # sample size
Mean = 3.42                 # sample mean
std_dev = 0.68              # standard deviation

# Calculation
from scipy import *
from pylab import *

# t(0.025) = 2.131
t = 2.131

y1 = Mean - t*(std_dev / sqrt(n))           # lower limit of range
y2 = Mean + t*(std_dev / sqrt(n))           # upper limit of range
y1 = round(y1,2)
y2 = round(y2,2)

# Result
print "95% confidence interval(in grams): (",y1,",",y2,")"

95% confidence interval(in grams): ( 3.06 , 3.78 )

Example, Page-239

In [15]:
# Variable declaration
n = 15                     # sample size
Mean1 = 75.20              # sample mean
Mean2 = 77                 # sample mean
std_dev = 3.6              # standard deviation

# Calculation
from scipy import *
from pylab import *

# z(0.05) = 1.645
Z = 1.645

y1 = Z + sqrt(n)*((Mean1-Mean2) / std_dev)           # lower limit of range
y1 = round(y1,3)

# probability corresponding to Z>y1 is 0.614
p = 0.614   # P(Z > -0.219) = 0.614

prob = 1 - p

# Result
print "Type-2 error probability: ",prob

Type-2 error probability:  0.386

Example, Page-240

In [16]:
# Variable declaration
n = 30                     # sample size
Mean1 = 2.000              # sample mean
Mean2 = 2.010                 # sample mean
std_dev = 0.050              # standard deviation

# Calculation
from scipy import *
from pylab import *

# z(0.025) = 1.96
Z = 1.96

y1 = -Z + sqrt(n)*((Mean1-Mean2) / std_dev)           
y2 = Z + sqrt(n)*((Mean1-Mean2) / std_dev)           


# probability corresponding to Z>y1 is 0.614
p1 = 0.0015   # P(Z < Y1) = 0.001
p2 = 0.1945   # P(Z > Y2) = 0.194
p = p1 + p2

prob = 1 - p

# Result
print "Type-2 error probability: ",prob

Type-2 error probability:  0.804

Example, page-241

In [17]:
# Variable declaration

alpha = 0.05
beta = 0.10
Mean1 = 20          # mean corresponding to alpha
Mean2 = 21          # mean corresponding to beta
std_dev = 2.4       # standard deviation

# Calculation
from scipy import *
from pylab import *

Z1 = 1.645          # Z value corresponding to alpha=0.05
Z2 = 1.280          # Z value corresponding to beta=0.10

Size = pow( (std_dev*(Z1 + Z2))/(Mean1 - Mean2) ,2)               # Minimum sample size
Size = round(Size,-1)

# Result
print "Required sample size: ",int(Size)

Required sample size:  50