Question

Hosmer Winery has a machine for dispensing their wine into bottles. When the machine is in statistical control, the amount dispensed has a mean of 752 ml. Four random observations taken yesterday were 745, 740, 750, 745 ml. Construct a 95% confidence interval for the mean amount dispensed.

Answer 1

Solution :

We are given a data of sample size n = 4

745, 740, 750, 745

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

=   

= (745 + 740.......+ 745)/4

= 745

Now ,

s=   

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

s = 4.08

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 = 3

    =    =  0.025,3 = 3.182

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

The margin of error is given by

E =  /2,d.f. * ( / n )

  = 3.182* ( 4.08/ 4 )

= 6.492

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

( - E ) <   <  ( + E)

( 745 - 6.492 )   <   <  ( 745 + 6.492 )

738.508 <   < 751.492

Required 95% confidence interval is ( 738.508 , 751.492 )