Page 399 - DMTH404_STATISTICS

Unit 29: Estimation of Parameters: Criteria for Estimates

The methods of construction of confidence intervals in various situations are explained through Notes
the following examples.
Confidence Interval for Population Mean

Example 5: Construct 95% and 99% confidence intervals for mean of a normal population.
Solution.

Let X , X , ...... X be a random sample of size n from a normal population with mean m and
1 2 n
standard deviation s.

We know that sampling distribution of X is normal with mean m and standard error .
n
X 
-
Therefore, z = will be a standard normal variate.
 / n
From the tables of areas under standard normal curve, we can write

X 
-
P[- 1.96  z  1.96] = 0.95 or P[- 1.96   1.96] = 0.95 .... (1)
 / n
X 
-
The inequality - 1.96  can be written as
 / n
 

-
- 1.96  X  or  X + 1.96 .... (2)
n n
X 
-
Similarly, from the inequality  1.96, we can write
 / n


³ X - 1.96 .... (3)
n
Combining (2) and (3), we get

 
X - 1.96   X + 1.96

n n
Thus, we can write equation (1) as

æ   ö

P ç X - 1.96   X + 1.96 ÷ = 0.95
è n n ø

This gives us a 95% confidence interval for the parameter m. The lower limit of  is X - 1.96
n

and the upper limit is X + 1.96 . The probability of m lying between these limits is 0.95 and
n
therefore, this interval is also termed as 95% confidence interval for .
In a similar way, we can construct a 99% confidence interval for m as

æ   ö

P ç X - 2.58   X + 2.58 ÷ = 0.99
è n n ø

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