Question

In a calibration test, 50 measurements were taken of a laboratory gas sample that is known to have a CO concentration of 70 parts per million (ppm). A measurement is considered to be satisfactory if it is within 5 ppm of the true concentration. Of the 50 measurements, 37 were satisfactory. [5 + 10 points]
(a) Suppose it is desirable to estimate the population proportion of satisfactory measurement within ±0.03, how big a sample of measurements should be?
(b) Without using the result from the pilot study of observed 50 observations, determine how big a sample of measurements should be if it is desirable to estimate the population proportion of satisfactory measurement within ±0.03.

Answer 1

a)

sample proportion ,   p̂ =x/n = 37/50 = 0.74  
sampling error ,    E =   0.03  
Confidence Level ,   CL=   95%  
          
alpha =   1-CL =   5%  
Z value =    Zα/2 =    1.9600   [excel formula =normsinv(α/2)]
          
Sample Size,n =    (Z / E)²*p̂*(1-p̂)=   821.2185  
          
          
so,Sample Size required=       822  

b)

without using the result from the pilot study of observed 50 observations,

let sample proportion ,   p̂ =    0.5  
sampling error ,    E =   0.03  
Confidence Level ,   CL=   95%  
          
alpha =   1-CL =   5%  
Z value =    Zα/2 =    1.9600   [excel formula =normsinv(α/2)]
          
Sample Size,n =    (Z / E)²*p̂*(1-p̂)=   1067.0719  
          
          
so,Sample Size required=       1068