Confidence Interval
Assignment # 2

Briefly describe following with the help of
examples:
Point Estimation of Parameters
 Confidence Intervals
 Students’ t Distribution
 Chi Square Distribution
 Testing of Hypothesis



Hand Written assignments be submitted by 20
Dec 2012.
Copying and Late Submissions will have
appropriate penalty.
Estimation Process
Population
Mean, , is
unknown
Sample
Random Sample
Mean
X = 50
I am 95%
confident that
 is between 40
& 60.
Confidence Interval provides more
information
about
a
population
characteristic than does a point estimate.
It provides a confidence level for the
estimate.

Lower
Confidence
Limit
Upper
Point Estimate
Width of
Confidence Interval
Confidence
Limit
Interval Estimates

Provides range of values





Takes into consideration variation in
sample statistics from sample to sample
Is based on observation from one sample
Gives information about closeness to
unknown population parameters
Is stated in terms of level of confidence
Never 100% certain
Confidence Interval Estimates
Confidence
Intervals
Variance
Mean
2 Known
2Unknown
Confidence Interval for µ
( б2 Known)
Assumptions
 Population standard deviation is known
 Population is normally distributed
 If population is not normal, use large
sample
 Confidence interval estimate



Conf  ( x  k    x  k ).. where .. k 
c
n
Steps to Determine
Confidence Interval




Choose a Confidence Level γ (95%, 90%, etc)
Determine the corresponding c [ Z(D) ]
γ
0.90
0.95
0.99
0.999
C
1.645
1.960
2.576
3.291
Compute the mean x of the sample x1, x2,…, xn
Confidence interval estimate


Conf  ( x  k    x  k ).. where .. k 
c
n
Problem 1
Find a 95% Confidence Interval for
the mean µ of a normal population
with standard deviation 4.00 from the
sample 30, 42, 40, 34, 48, 50
Confidence Interval for µ
( б2 Unknown)
Choose a Confidence Level γ (95%, 90%, etc)
 Determine the solution C of the equation
F(C) = ½ ( 1 + γ)
[Table: t distribution with n-1 degrees of freedom]
 Compute the mean x and variance s2 of the
sample x1, x2,…, xn
 Confidence interval estimate


cs
Conf  ( x  k    x  k ).. where .. k 
n

Student’s t Distribution
Standard
Normal
Bell-Shaped
Symmetric
‘Fatter’
Tails
t (df = 13)
t (df = 5)
0
Z
t
Degrees of Freedom (df )



Number of observations that are free to
vary after sample mean has been calculated
df = n - 1
Example: Mean of 3 numbers is 2
degrees of freedom
= n -1
= 3 -1
=2
Student’s t Distribution
Let X1, X2,…, Xn
be independent random
variables with the same mean µ and the same
variance б2. Then, the random variable

T 
x 
S /
n
has a t distribution with n-1 degrees of freedom
S 
2
1

n
(X

n 1
j 1
j
X)
2
Student’s t Table
Upper Tail Area
df
.25
.10
.05
Let: n = 3
df = n - 1 = 2
 = .10
/2 =.05
1 1.000 3.078 6.314
2 0.817 1.886 2.920
 / 2 = .05
3 0.765 1.638 2.353
t Values
0 2.920
t
Problem 1
Find a 95% Confidence Interval for
the mean µ of a normal population
with standard deviation 4.00 from the
sample 30, 42, 40, 34, 48, 50
Problem 9
Find a 99% Confidence Interval for
the mean of a normal population. The
length of 20 bolts with Sample Mean
20.2 cm and Sample Variance 0.04
cm2.
Confidence Interval for the
Variance
Choose a Confidence Level γ (95%, 90%, etc)
 Determine the solutions C1 and C2 of the equation
F(C1) = ½ ( 1 - γ) and F(C2) = ½ ( 1 + γ)
[Table A10: Chi Square distribution with n-1
degrees of freedom]
 Compute (n-1)S2, where S2 is the variance of the
sample x1, x2,…, xn

Confidence Interval for the
Variance


Compute
k1 = (n-1)S2 / C1
k2 = (n-1)S2 / C2
Confidence Interval Estimate
Conf  ( k 2  
2
 k1 )
Chi Square Distribution
Let X1, X2,…, Xn
be independent random
variables with the same mean µ and the same
variance б2. Then, the random variable
  ( n  1)
With
S 
2
1
j 1

2

n
(X

n 1
S
2
j
X)
2
has a Chi Square
distribution with
n-1 degrees of
freedom
Chi Square Distribution
CDF
PDF
Problem 15
Find a 95% Confidence Interval for
the variance of a normal population.
The
Sample
has
variance 0.0007.
30
values
with
Problem 2
Does the Interval in Problem 1 get
longer or shorter, if we take γ = 0.99
instead of 0.95? By what factor?
Problem 3
By what factor does the length of
the Interval in Problem 1 change, if we
double the Sample Size?
Problem 5
What Sample Size would be needed
for
obtaining
a
95%
Confidence
Interval (3) of length 2 б? Of length б?
Problem 7
What Sample Size is needed to
obtain a 99% Confidence Interval of
length 2.0 for the mean of a normal
population with variance 25?
Problem 9
Find a 99% Confidence Interval for
the mean of a normal population. The
length of 20 bolts with Sample Mean
20.2 cm and Sample Variance 0.04
cm2.
Problem 11
Find a 99% Confidence Interval for
the mean of a normal population. The
Copper Content (%) of brass is
66, 66, 65, 64, 66, 67, 64, 65, 63, 64
Problem 13
Find a 95% Confidence Interval for
the percentage of cars on a certain
highway that have poorly adjusted
brakes, using a random sample of 500
cars stopped at a road block on a
highway, 87 of which had poorly
adjusted brakes.
Problem 17
Find a 95% Confidence Interval for
the variance of a normal population.
The Sample is the Copper Content (%)
of brass:
66, 66, 65, 64, 66, 67, 64, 65, 63, 64
Problem 19
Find a 95% Confidence Interval for
the variance of a normal population.
The Sample has Mean Energy (keV) of
delayed neutron group (Group 3, half
life 6.2 sec) for uranium U235 fission:
435, 451, 430, 444, 438
Problem 21
If X is normal with mean 27 and
variance 16, what distribution do –X,
3X and 5X-2 have?
Problem 23
A machine fills boxes weighing Y lb
with X lb of salt, where X and Y are
normal with mean 100 lb and 5 lb and
standard deviation 1 lb and 0.5 lb,
respectively. What percent of filled
boxes weighing between 104 lb and
106 lb are to be expected?
Confidence Interval for Variance
Choose a Confidence Level γ (95%, 90%, etc)
 Determine the solution C1 and C2 of the
equation
F(C1) = ½ ( 1 - γ) and F(C2) = ½ ( 1 + γ)
[Table: Chi Square distribution with n-1 degrees
of freedom]
 Compute (n-1) s2, where s2 is variance of the
sample x1, x2,…, xn

Confidence Interval for Variance

Compute k1 = (n-1) s2/C1
and k2 = (n-1) s2/C2

Confidence interval estimate
Conf  ( k 2    k 1 )
2
Elements of
Confidence Interval Estimation

Level of confidence


Precision (range)


Confidence in which the interval will contain
the unknown population parameter
Closeness to the unknown parameter
Cost

Cost required to obtain a sample of size n
Level of Confidence


Denoted by 1 0 0  1    %
A relative frequency interpretation
 In the long run, 1 0 0  1    % of all the confidence
intervals that can be constructed will contain the
unknown parameter

A specific interval will either contain or not
contain the parameter

No probability involved in a specific interval
Interval and Level of Confidence
Sampling Distribution of the
_ Mean
X
  Z  / 2 X
Intervals
extend from
X  Z X
to
X  Z X
 /2
1
X  
  Z  / 2 X
 /2
X
1    1 0 0 %
of intervals
constructed
contain  ;
1 0 0 %
do
not.
Confidence IntervalsChap 7-Confidence
Intervals-37
Factors Affecting
Interval Width (Precision)

Data variation


Sample size


Measured by 
X 
Intervals Extend from
X - Z
x
to X + Z 
x

n
Level of confidence
 1 0 0 1    %
© 1984-1994 T/Maker Co.
Chap 7-Confidence Intervals-38
Determining Sample Size (Cost)
Too Big:
Too small:
• Requires
too many
resources
• Won’t do
the job
Chap 7-Confidence Intervals-39
Determining Sample
Size for Mean
What sample size is needed to be 90% confident of
being correct within ± 5? A pilot study suggested that
the standard deviation is 45.
Z 
2
n
2
Error
2

1.645
2
5
 45 
2
2
 219.2  220
Round Up
Chap 7-Confidence Intervals-40
Determining Sample
Size for Mean in PHStat


PHStat | sample size | determination for the
mean …
Example in excel spreadsheet
Chap 7-Confidence Intervals-41
Confidence Interval for 
(  Unknown)

Assumptions





Population standard deviation is unknown
Population is normally distributed
If population is not normal, use large sample
Use student’s t distribution
Confidence interval estimate

S
S
X  t / 2 , n  1
   X  t / 2 , n  1
n
n
Chap 7-Confidence Intervals-42
Example
A random sam ple of n  25 has X  50 and S  8.
Set up a 95% confidence interval estim at e for 
X  t / 2 , n  1
50  2.0639
S
n
8
   X  t / 2 , n  1
   50  2.0639
25
S
n
8
25
46.69    53.30
Chap 7-Confidence Intervals-43
Confidence Interval
Estimate for Proportion p

Assumptions




Two outcomes (0;1)
Number of ‘1’ in n trials follows B(n,p)
Normal approximation can be used if
np  5 and
n 1  p   5
Confidence interval estimate

pS  Z / 2
p S 1  p S
n

 p  pS  Z / 2
p S 1  p S

n
Chap 7-Confidence Intervals-44
Example
A random sample of 400 voters showed 32
preferred candidate A. Set up a 95% confidence
interval estimate for p.
ps  Z / 2
.08  1.96
p s 1  p s 
n
.08  1  .08 
 p  ps  Z / 2
 p  .08  1.96
400
p s 1  p s 
n
.08 1  .08 
400
.053  p  .107
Normal Table
Chap 7-Confidence Intervals-45
Confidence Interval Estimate for
Proportion in PHStat


PHStat | confidence interval | estimate for the
proportion …
Example in excel spreadsheet
Chap 7-Confidence Intervals-46
Determining Sample Size for
Proportion
Out of a population of 1,000, we randomly
selected 100, of which 30 were defective. What
sample size is needed to be within ± 5% with
90% confidence?
Z p 1  p 
2
n
E rror
2

1.645
2
 0.3   0.7 
0.05
2
 227.3  228
Round Up Chap 7-Confidence Intervals-47
Determining Sample Size for
Proportion in PHStat


PHStat | sample size | determination for the
proportion …
Example in excel spreadsheet
Chap 7-Confidence Intervals-48
Confidence Interval for
Population Total Amount

Point estimate


NX
Confidence interval estimate

N X  N  t / 2 , n  1 
S
n
N  n
 N  1
Chap 7-Confidence Intervals-49
Confidence Interval for
Population Total: Example
An auditor is faced with a
population of 1000 vouchers
and wants to estimate the
total value of the population.
A sample of 50 vouchers is
selected with average
voucher amount of
$1076.39, standard deviation
of $273.62. Set up the 95%
confidence interval estimate
of the total amount for the
population of vouchers.
Chap 7-Confidence Intervals-50
Example Solution
N  1000
n  50
N X  N  t / 2 , n  1 
S
n
X  $1076.39
S  $273.62
N  n
 N  1
  1000   1076.39    1000   2.0096 
273.62
1000  50
100
1000  1
 1, 076, 390  75, 830.85
The 95% confidence interval for the population total
amount of the vouchers is between 1,000,559.15, and
1,152,220.85
Chap 7-Confidence Intervals-51
Confidence Interval for Total
Difference in the Population

Point estimate


n
Where D 
difference
ND
D
i
is the sample average
i 1
n
Confidence interval estimate

N D  N  t / 2 , n  1 
SD
n
n

Where
SD 
 D
i
N  n
 N  1
 D
2
i 1
n 1
Chap 7-Confidence Intervals-52
Estimation for Finite Population

Samples are selected without replacement

Confidence interval for the mean (  unknown)


X  t / 2 , n  1
S
n
N  n
 N  1
Confidence interval for proportion

pS  Z / 2
p S 1  p S
n
 N  n
 N  1
Chap 7-Confidence Intervals-53
Sample Size Determination
for Finite Population

Samples are selected without replacement

When estimating the mean
Z  / 2
2


n0 
e
2
2
When estimating the proportion
Z  / 2 p 1  p 
2

n0 
e
2
Chap 7-Confidence Intervals-54
Ethical Considerations




Report confidence interval (reflect sampling
error) along with the point estimate
Report the level of confidence
Report the sample size
Provide an interpretation
of the confidence interval estimate
Chap 7-Confidence Intervals-55
Chapter Summary






Illustrated estimation process
Discussed point estimates
Addressed interval estimates
Discussed confidence interval estimation
for the mean (  known)
Addressed determining sample size
Discussed confidence interval estimation
for the mean (  unknown)
Chap 7-Confidence Intervals-56
Chapter Summary





(continued)
Discussed confidence interval estimation for
the proportion
Addressed confidence interval estimation for
population total
Discussed confidence interval estimation for
total difference in the population
Addressed estimation and sample size
determination for finite population
Addressed confidence interval estimation and
ethical issues
Chap 7-Confidence Intervals-57