Chapter No.12
Estimation
ESTIMATION
Short Questions
Q.1
What is meant by statistical inference?
Ans: It is the conclusion made about the unknown value of the parameter of the
population using information contained in an observed sample taken from
population at random.
Q.2
Define estimation.
Ans: The statistical estimation is a procedure of making judgment about the unknown
value of a population parameter by using sample observation.
Q.3
Differentiate between estimator and estimate.
Ans: Estimator is the rule or method that used to estimate a parameter, whereas an
estimate is the numerical value obtained by sample data.
Q.4
Explain estimator and estimate by example.
Ans:
If X1 ,X 2 ,.......,X n is a random sample of size"n"taken from population with mean μ,
then X=
Q.5
 X is an estimator of μ and the value X is an estimate of μ.
n
What are the different types of estimation?
Ans: Two types of estimation are: (1) Point Estimation (2) Interval Estimation.
Q.6
Define point estimation.
Ans: The method of finding the single unknown value of parameter from the sample
values by using an estimator is called point estimation.
Q.7
Define point estimate.
Ans: A single numerical value calculated from sample data by using an estimator is
called point estimate.
Q.8
Define Interval estimation.
Ans: The method of finding an interval, on the belief that it will include the parameter
 with a known probability is called interval estimation.
Q.9
Define interval estimate.
Ans: An interval, calculated on the belief that it will include the parameter  with a
known probability is called interval estimate.
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Chapter No.12
Estimation
Q.10 What is meant by interval?
Ans: The range of values is called interval. Example  a, b
Q.11 What is meant by confidence interval?
Ans: An interval to which 100(1-  )% probability is associated that it will include the
parameter is called confidence interval. Example P(L<  < U) = 1 - 
Q.12 What is confidence coefficient?
Ans: 1 -  or 100(1-  )% probability associated with an interval that it will contain the
parameter is called confidence coefficient. Confidence coefficient is also called
level of confidence.
Q.13 Estimator is a statistic, explain?
Ans: Estimator is sample statistic used to estimate true value of population parameter,
hence it is always statistic which is both function and random variable with
probability function.
Q.14 What are the properties of good point estimator?
Ans: Properties of good point estimator are:
(1) unbiased ness (2) efficiency ( 3) consistency (4) sufficiency
Q.15 Explain unbiasedness.
Ans: An estimator is defined to be unbiased when an estimator has its expected value
equal to the true value of population parameter.
Suppose  be an estimator and  is a parameter then,
E(  ) =  , then  is an unbiased estimator of 
E(  )   , then  is a biased estimator of 
E(  )>  , then  is a Positive biased estimator of 
E(  )<  , then  is a Negative biased estimator of 
Q.16 Write the formula of bias.
Ans: Bias = E(  ) - 
Q.17 Is an estimator a random variable?
Ans: An estimator is a random variable having its own probability distribution.
Q.18 Why interval estimation is useful?
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Chapter No.12
Estimation
Ans: Interval estimation is useful, because interval estimation provide precision of the
estimate.
Q.19 What are advantages of interval estimation?
Ans:

It provides an interval of values that is lively to include the unknown true
value of the parameter.

Interval estimation provides precision of the estimate.
Q.20 Describe methods for increasing precision.
Ans:
 By decreasing standard error to estimate.
 By decreasing the confidence coefficient.
Q.21 Define lower and upper confidence limits.
Ans: The end point “L” and “U” that bound the confidence interval are called lower and
upper confidence limit for parameter  . Example P(L<  < U) = 1 - 
Q.22 What is the effect on confidence interval for  , if “n” decreases?
Ans: The length of confidence interval for “  ” increases by “n” decreasing.
Q.23 When finite population correction factor can be ignored for estimating  ?
Ans: The finite population correction factor can be ignored when sample size “n” is less
than 5% of population size “N”.
Q.24 What are large sample and small sample?
Ans: When n  30 samples is called large size otherwise small sample.
Q.25 Define t – statistic.
Ans: Let a small sample of size n<30 is drawn from a normal population with unknown
variance, then the sampling distribution of the statistic
t=
X-μ
with "v" d.f
s
n
Q.26 Describe the term degree of freedom.
Ans: Degree of freedom represents the number of independent random variable that
expresses the sampling distribution.
Q.27 How the width of confidence interval can be decreased?
Ans: The width of confidence interval can be decreased by:
(i)
Increasing the sample size.
(ii)
Decreasing confidence coefficient (level of confident)
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Chapter No.12
Estimation
Q.28 What is the best estimator of population proportion?
Ans:
x
n
The best estimator of population proportion “  ”is P =
Q.29 What is the pooled estimator of population mean and population proportion?
Ans:
The pooled estimator of population mean is X p 
n1 X1  n2 X 2
n1  n2
The pooled estimator of population proportion is  
n1 P1  n2 P2
n1  n2
Q.30 Give pool estimator of population variance.
Ans:
The pooled estimator of variance is s 2p 
n1S12  n2 S22
n1  n2  2
IMPORTANT FORMULAE
Confidence interval for population mean
(1)
(2)
(3)
When  is known: X  Z

n
2
   X  Z

2
n
When  is unknown and sample size large: X  Z
When  is unknown and sample size small: X  t
S
S
   X  Z
2
n
n
2
s
(v)
n
2
   X  t
S
(v)
2
n
Confidence interval for difference between population mean ( 1  2 )
(1) When Variances are known:
 12
 22

 1  2  ( X 1  X 2 )  Z
2
n1 n2
(2) When Variances are unknown for large samples:
( X 1  X 2 )  Z
2
 12
n1
 22

n2
S12 S22
S12 S22

 1  2  ( X 1  X 2 )  Z

2
2
n1 n2
n1 n2
(3) When Variances are unknown for small samples:
( X 1  X 2 )  Z
( X1  X 2 )  t
(v) S p
1

1
 1  2  ( X1  X 2 )  t
n1 n2
Confidence interval for paired observation:
d t
(v)

2
2
sd
   d  t
d
n
2
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(v)
2
(v) S p
1

1
n1 n2
sd
n
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Chapter No.12
Estimation
Confidence interval for Population proportion:
pq
pq
p  z
   p  z
2
2
n
n
Confidence interval for difference between Population proportions:
( p1  p2 )  z
2
p1q1 p2 q2

   ( p1  p2 )  z
2
n1
n2
p1q1 p2 q2

n1
n2
Multiple Choice Questions
Each question has four possible answers. Select correct answer.
1
The statistical inferences has types:
(a) two
2
(b) data
(c) variable
(d) population
(b) three
(c) four
(d) five
(b) number
(c) data
(d) value
(b) value
(c) statistic
(d) parameter
(b) statistic
(c) estimate
(d) parameter
(b) estimate
(c) observed value
(d) observation
A specific value calculated from sample is called
(a) error
11
(d) values
An point estimate is a specific value of an
(a) estimator
10
(c) parameters
Point estimator is used to estimate the unknown true value of population
(a) data
9
(b) observation
A point estimator is a sample
(a)estimate
8
(d) unknown
The object of point estimation is to obtain a single
(a) digit
7
(c) universe
The statistical estimation of population has types
(a) two
6
(b) sample data
Statistical estimation examine the entire
(a) sample
5
(d)five
Statistical estimation procedures provide estimates of population
(a) data
4
(c) four
Population parameters are estimated from
(a) random variable
3
(b) three
(b) statistic
(c) parameter
(d) bias
A function that used to estimate a parameter is called
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Chapter No.12
(a) bias
12
Estimation
(b) error
(c) estimator
(d) estimation
(c) biased estimator
(d) unbiased
If E(  ) =  , then  is called
(a) negatively biased
(b) positively biased
estimator
13
A statistic  is said to be positively biased if
(a) E( 
14
)> 

(b) sample proportion
(c) sample median
(d) all of these
(c) P
(d)
(b) estimator
(c) estimation
(d) error
(b) level of confidence
(c) power test
(d) none of these
X
X
(b) 90%
(c) 95%
(d) 99%
(b)

(c) 1
-
(d)

(b) confidence interval
(c) size of AR
(d) size of CR
(b) 40
(c) 1000
(d) all of these
(c) 30
(d) all of these
(c) n is large
(d) none of these
A small sample contains less than values
(b) 25
In applying t –test when  is
(a) known
25
(d) none of these
A large sample contains more than values
(a) 5
24
)=
Level of significance is also called
(a) 30
23
(c) E( 
Level of significance is denoted by
(a) power test
22
)<
Which of the following is level of confidence, if level of significance is 0.05
(a) 1-
21
(d) none of these
1 -  is called
(a) 5%
20
)=
E(T) -  is equal to
(a) level of significance
19
(b) E( 
(b)
(a) bias
18
(c) E( 
The point estimator of “u” is
(a) X
17
)<
These statistic’s are an unbiased
(a) sample mean
16
(b) E( 
A statistic  is said to be negatively biased if
(a) E( 
15
)> 
(b) unknown
t – distribution is used when
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Chapter No.12
Estimation
(a) population is normal
26
(b) three
(b) biased
(b) different
(d) none of these
(b) English letter
(c) Latin letters
(d) Roman letters
(b) decreasing
(c) decumulative
(d) none of these
(b) increase
(c) no effect
(b) no effect
(d) none of these
(c) fixed
(d) decrease
(c) > 
(d) none of these
(c) variance
(d) proportion
X is an unbiased estimator if E( X ) is

(b) <

“P” is an unbiased estimator of
(a) mean
(b) median
n
s2 
(X
i 1
i
 X )2
n 1
n
S2 
(c) three values
The width of confidence interval decrease if the confidence co-efficient is
(a) =
35
(d) b & c
By decreasing X the length of confidence interval for 
(a) increase
34
(d) none of these
a&b
Bias is
(a) decrease
33
(c)
Population parameters are denoted by
(a) cumulative
32
(d) ten
(c) opposite
(b) range of value
(a) Greek letters
31
(c) five
In interval estimation, we always get
(a) single value
30
(d) all of these
Interval estimation and confidence interval are
(a) same
29
(c) n is small
If E(s2) =  2 then s2 is estimator
(a) unbiased
28
 is unknown
The precision can be increased by methods
(a) two
27
(b)
(X
i 1
i
is an unbiased estimator of  2
 X )2
n
is a biased estimator of  2
AHMAD NAWAZ NASIR
SSS (STATISTICS)
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