Suppose a sample of size n is to be drawn from a normal dist

Suppose a sample of size n is to be drawn from a normal distribution where standard deviation is know to be 14.3. How large does n have to be to guarentee that the length of the 95% interval for Mu will be less than 3.06?

Solution

If the length is less than 3.06, then the margin of error is less than 3.06/2 = 1.53.

Note that      
      
n = z(alpha/2)^2 s^2 / E^2      
      
where      
      
alpha/2 = (1 - confidence level)/2 =    0.025  
      
Using a table/technology,      
      
z(alpha/2) =    1.959963985  
      
Also,      
      
s = sample standard deviation =    14.3  
E = margin of error =    1.53  
      
Thus,      
      
n =    335.571752  
      
Rounding up,      
      
n =    336