Question

5. Confidence Intervals Suppose you have the following X: (6, 5, 4, 1, 1). What is the mean of X? What is the Standard deviation of X? What is the Standard Error of X? Calculate the 95% confidence interval for X. Calculate the 99% confidence interval for X.

Answer 1

Solution :

We are given a data of sample size n = 5

6,5,4,1,1

Using this, first we find sample mean() and sample standard deviation(s).

=   

= ( 6 + 5 +.......+ 1 ) / 5

= 3.4

Now ,

s=   

Using given data, find Xi - for each term.Take square for each.Then we can easily find s.

s = 2.302

The Standard Error =  ( / n ) = 2.302 / 5 = 1.0294

Note that, Population standard deviation() is unknown. So we use t distribution.

Our aim is to construct 95% confidence interval.

c = 0.95

= 1- c = 1- 0.95 = 0.05

  /2 = 0.05 2 = 0.025

Also, d.f = n - 1 = 4

    =    =  0.025,4 = 2.776

( use t table or t calculator to find this value..)

The margin of error is given by

E =  /2,d.f. * ( / n )

= 2.776 * (2.302 / 5 )

= 2.858

Now , confidence interval for mean() is given by:

( - E ) <   <  ( + E)

( 3.4 - 2.858 )   <   <  ( 3.4 + 2.858)

0.542 <   < 6.258

Required 95% confidence interval is ( 0.542 , 6.258 )

Solution:

We are given a data of sample size n = 5

6,5,4,1,1

Using this, first we find sample mean() and sample standard deviation(s).

=   

= ( 6 + 5.......+ 1 ) / 5

= 3.4

Now ,

s=   

Using given data, find Xi - for each term.Take square for each.Then we can easily find s.

s = 2.302

The Standard Error =  ( / n ) = 2.302 / 5 = 1.0294

Note that, Population standard deviation() is unknown..So we use t distribution.

Our aim is to construct 99% confidence interval.

   c = 0.99

= 1 - c = 1 - 0.99 = 0.01

  /2 = 0.01 2 = 0.005

Also, d.f = n - 1 = 4  

    =    =  0.005,4 = 4.604

( use t table or t calculator to find this value..)

The margin of error is given by

E =  /2,d.f. * ( / n )

= 4.604 * (2.302 / 5 )

= 4.74

Now , confidence interval for mean() is given by:

( - E ) <   <  ( + E)

( 3.4 - 4.74 )   <   <  ( 3.4 + 4.74 )

-1.34 <   < 8.14

Required 99% confidence interval is ( -1.34 , 8.14 )