Question

A laboratory procedure is used to measure the level of cadmium in soil. It is
applied to a specimen that has a controlled level of 1.5 μg/g of cadmium. Three measurements
are obtained by splitting the specimen into three parts of equal weight and applying the
procedure to each part. Here are the results: 1.64, 1.85, and 1.94.
(a) Does a 95% confidence interval give an indication that the laboratory procedure may be
biased?
b. What are your assumptions that you are making to produce the confidence interval?

second part of question: For 95% confidence interval of (1.38, 1.92) is given for the mean of a population
based on the normal distribution. Re-express this as a 90% confidence interval

Answer 1

a)

sample std dev ,    s = √(Σ(X- x̅ )²/(n-1) ) =   0.1539
Sample Size ,   n =    3
Sample Mean,    x̅ = ΣX/n =    1.8100

Level of Significance ,    α =    0.05          
degree of freedom=   DF=n-1=   2          
't value='   tα/2=   4.303   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   0.1539   / √   3   =   0.0889
margin of error , E=t*SE =   4.3027   *   0.0889   =   0.3824
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    1.81   -   0.382428   =   1.4276
Interval Upper Limit = x̅ + E =    1.81   -   0.382428   =   2.1924
95%   confidence interval is (   1.4276   < µ <   2.1924   )

b)

assumption : population from which samples are taken is normally distributed

sample are random and independent