Question

A magazine published data on the best small firms in a certain year. These were firms that had been publicly traded for at least a year, have a stock price of at least $5 per share, and have reported annual revenue between $5 million and $1 billion. The table below shows the ages of the corporate CEOs for a random sample of these firms. 49 57 51 60 57 59 74 63 53 50 59 60 60 57 46 55 63 57 47 55 57 43 61 62 49 67 67 55 55 49 Use this sample data to construct a 90% confidence interval for the mean age of CEO's for these top small firms. Use the Student's t-distribution. (Round your answers to two decimal places.) ,

Answer 1

49, 57, 51, 60, 57, 59, 74, 63, 53, 50, 59, 60, 60, 57, 46, 55, 63, 57, 47, 55, 57, 43, 61, 62, 49, 67, 67, 55, 55, 49

Solution:

sample size = n = 30

x	x2
49	2401
57	3249
51	2601
60	3600
57	3249
59	3481
74	5476
63	3969
53	2809
50	2500
59	3481
60	3600
60	3600
57	3249
46	2116
55	3025
63	3969
57	3249
47	2209
55	3025
57	3249
43	1849
61	3721
62	3844
49	2401
67	4489
67	4489
55	3025
55	3025
49	2401
Sum = 1697	97351

Sample Mean =   = 1697/30= 56.566666666667

Now ,

Sample variance s² =

= [1/(30 - 1)][97351 - (1697²/30) ]

= 46.805747126437

Now ,

sample standard deviation s = variance = 46.805747126437= 6.8414725846441

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

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 = 30 - 1 = 29

    =    =  _0.05,29 = 1.699

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

The margin of error is given by

E =  /2,d.f. * ( / n)

= 1.699 * (6.8414725846441 / 30)

= 2.12

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

( - E ) <   <  ( + E)

(56.566666666667 - 2.12)   <   <  (56.566666666667 + 2.12)

54.45 <   < 58.69

Required 90% confidence interval for  the mean age of CEO's for these top small firms is

54.45 <   < 58.69

i.e.

(54.45 , 58.69)