Question

How much money do winners go home with from the television quiz show Jeopardy? To determine an answer, a random sample of winners was drawn and the amount of money each won was recorded and listed below. Estimate with 92% confidence the mean winning's for all the show's players.

34135 29640 26440 19111 34635 24903 20591 34012 33338 23721 19518 32867 21627 17659 28450 34135 19111 20591 23721 21627 29640 34635 34012 19518 17659 26440 24903 33338 32867 28450

Upper confidence limit =

Lower confidence limit =

Answer 1

Solution:

We are given a data of sample size n = 30

34135,29640 ,26440 ,19111 ,34635 ,24903, 20591 ,34012 ,33338 ,23721, 19518 ,32867, 21627 ,17659 ,28450, 34135, 19111 ,20591, 23721, 21627, 29640 ,34635, 34012, 19518 ,17659, 26440, 24903 ,33338 ,32867 ,28450

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

=   

= (34135+ 29640+ .......+ 28450)/30

= 26709.8

Now ,

s=   

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

s = 6055.1977

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

Our aim is to construct 92% confidence interval.   

c = 0.92

= 1- c = 1- 0.92 = 0.08

  /2 = 0.08 2 = 0.04

Also, d.f = n - 1 = 30 - 1= 29

    =    =  _0.04,29 = 1.814

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

The margin of error is given by

E =  /2,d.f. * ( / n)

=    1.814 * (6055.1977 / 30 )

= 2005.4183

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

( - E ) <   <  ( + E)

(26709.8 - 2005.4183)   <   <  (26709.8 + 2005.4183)

24704.3817 <   <  28715.2183

Required interval is (24704.3817 , 28715.2183)

Upper confidence limit = 28715.2183 nearly 28715

Lower confidence limit = 24704.3817 nearly 24704