To assess the accuracy of laboratory scale a standard weight

To assess the accuracy of laboratory scale, a standard weight known to weigh 10 grams is weighed repeatedly. The weight is weighed 40 times. The mean result is 10.230 grams. The standard deviation of the scale readings is 0.020 gram.

(a) Give a 98% confidence interval for the mean of repeated measurements of the weight. Round your answers to three decimal places.

(b) How many measurements must be averaged to get a margin of error of + or - 0.001 with 98% confidence?

Solution

a)

Note that              
Margin of Error E = z(alpha/2) * s / sqrt(n)              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.01          
X = sample mean =    10.23          
z(alpha/2) = critical z for the confidence interval =    2.326347874          
s = sample standard deviation =    0.02          
n = sample size =    40          
              
Thus,              
Margin of Error E =    0.007356558          
Lower bound =    10.22264344          
Upper bound =    10.23735656          
              
Thus, the confidence interval is              
              
(   10.22264344   ,   10.23735656   ) [ANSWER]

**********************

b)

Note that      
      
n = z(alpha/2)^2 s^2 / E^2      
      
where      
      
alpha/2 = (1 - confidence level)/2 =    0.01  
      
Using a table/technology,      
      
z(alpha/2) =    2.326347874  
      
Also,      
      
s = sample standard deviation =    0.02  
E = margin of error =    0.001  
      
Thus,      
      
n =    2164.757772  
      
Rounding up,      
      
n =    2165   [ANSWER]