Question

Sample Size = 500 Standard Deviation = 0.170758 Sample Mean = 0.03

a.) What are the end points of a 90% confidence interval for the population mean? (round to 3 digits after Decimal point)

b.) What are the end points of a 95% confidence interval fro the population mean? (round to 3 digits after Decimal point)

Answer 1

Solution:

Given that,

   = 0.03 ....... Sample mean

s = 0.170758   ......Sample standard deviation

n = 500 ....... Sample size

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

a)

Our aim is to construct 90% confidence interval.   

c = 0.90

= 1- c = 1- 0.90 = 0.10

  /2 = 0.10 2 = 0.05

Also, d.f = n - 1 = 500 - 1 = 499

    =    =  _0.05,6 = 1.648

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

The margin of error is given by

E =  /2,d.f. * ( / n)

= 1.648 * (0.170758 / 500 )

= 0.013

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

( - E ) <   <  ( + E)

(0.03 - 0.013)   <   <  (0.03 + 0.013)

0.017 <   < 0.043

Required 90% confidence interval is (0.017 , 0.043)

b)

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 = 500 - 1 = 499

    =    =  _0.025,6 = 1.965

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

The margin of error is given by

E =  /2,d.f. * ( / n )

=  1.965 * (0.170758  / 500 )

= 0.015

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

( - E ) <   <  ( + E)

(0.03 - 0.015)   <   <  (0.03 + 0.015)

0.015 <   < 0.045

Required 95% confidence interval is (0.015 , 0.045)