Answer:
99% confidence interval for the given specimen is [3.4125 , 3.4155].
Step-by-step explanation:
We are given that a laboratory scale is known to have a standard deviation (sigma) or 0.001 g in repeated weighing. Scale readings in repeated weighing are Normally distributed with mean equal to the true weight of the specimen. 
Three weighing of a specimen on this scale give 3.412, 3.416, and 3.414 g.
Firstly, the pivotal quantity for 99% confidence interval for the true mean specimen is given by;
         P.Q. =  ~ N(0,1)
 ~ N(0,1)
where,  = sample mean weighing of specimen =
 = sample mean weighing of specimen =  = 3.414 g
 = 3.414 g
              = population standard deviation = 0.001 g
 = population standard deviation = 0.001 g
             n = sample of specimen = 3
              = population mean
 = population mean
<em>Here for constructing 99% confidence interval we have used z statistics because we know about population standard deviation (sigma).</em>
So, 99% confidence interval for the population mean,  is ;
 is ;
P(-2.5758 < N(0,1) < 2.5758) = 0.99  {As the critical value of z at 0.5% level 
                                                             of significance are -2.5758 & 2.5758}
P(-2.5758 <  < 2.5758) = 0.99
 < 2.5758) = 0.99
P(  <
 <  <
 <  ) = 0.99
 ) = 0.99
P(  <
 <  <
 <  ) = 0.99
 ) = 0.99
<u>99% confidence interval for</u>  = [
 = [  ,
 ,  ]
 ]
                                              = [  ,
 ,  ]
 ]
                                              = [3.4125 , 3.4155]
Therefore, 99% confidence interval for this specimen is [3.4125 , 3.4155].