Question

A sample survey of 62 discount brokers showed that the mean price charged for a trade of 100 shares at $50 per share was $31.22. The survey is conducted annually. With the historical data available, assume a known population standard deviation of $19. (a) Using the sample data, what is the margin of error in dollars associated with a 95% confidence interval? (Round your answer to the nearest cant) $ (b) Develop a 95% confidence interval for the mean price in dollars charged by discount brokers for a trade of 100 shares at $50 per share. (Round your answers to the nearest cent.) to $ $

Answer 1

Solution:

Given,

$url=http://latex.codecogs.com/gif.latex?%5Cbar%20x&t=1596160577&ymreqid=50870fb5-0ab3-84d7-1c0d-d8004d010800&sig=DBgw3vWEKcbxLmTfiNIc5g--~D$ = 31.22

$\sigma$ = 19

n = 62

Note that, Population standard deviation($\sigma$) is known..So we use z distribution.

Our aim is to construct 95% confidence interval.

$\therefore$ c = 0.95

$\therefore$$\alpha$ = 1- c = 1- 0.95 = 0.05

$\therefore$  $\alpha$/2 = 0.05 $\slash$ 2 = 0.025 and 1- $\alpha$ /2 = 0.975

Search the probability 0.975 in the Z table and see corresponding z value

$\therefore$ = 1.96   

2012

a)

The margin of error is given by

E =  $url=https://latex.codecogs.com/gif.latex?Z&t=1596160577&ymreqid=50870fb5-0ab3-84d7-1c0d-d8004d010800&sig=94x0tHA5bCaTnWU1ssP01g--~D$$\alpha$/2 * ($\sigma$ / $\sqrt{}$ n )

= 1.96 * ( 19 / $\sqrt{}$ 62 )

= 4.73

Answer : Margin of error (E) = 4.73

b)

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

($url=http://latex.codecogs.com/gif.latex?%5Cbar%20x&t=1596160577&ymreqid=50870fb5-0ab3-84d7-1c0d-d8004d010800&sig=DBgw3vWEKcbxLmTfiNIc5g--~D$ - E ) <  $\mu$ <  ($url=http://latex.codecogs.com/gif.latex?%5Cbar%20x&t=1596160577&ymreqid=50870fb5-0ab3-84d7-1c0d-d8004d010800&sig=DBgw3vWEKcbxLmTfiNIc5g--~D$ + E)

( 31.22 - 4.73 )   <  $\mu$ <  ( 31.22 + 4.73)

26.49 <  $\mu$ < 35.95

Answer : $ 26.49 to $ 35.95