Question

Please read the following before answering the next two questions :

It turns out that Jack’s data were 10.4, 10.1, 10. 9, 11.0.

 

A chemical instrument is calibrated so it has no bias in determining weights of chemical compounds. The manufacture claims that it has small random measurement error that follow a normal distribution with mean 0 and standard deviation σ = 0.2 grams. Suppose this claim is true.

Two students were assigned to measure the weight of a compound distilled in an experiment. The first student (Jack ) measured the compound 4 times and report the average reading = 10.6 grams while the second student (Jill ) measured the compound 9 times and report the average reading = 10.9. Denote the true weight of the compound by μ .

(hint : Jack’s estimator can be represented as ( X1 + ...+X4 )/4 and Jill’s estimator can be represented as
(X5 + ...+ X13)/9 )

8. Use Jack’s data to find a 95% confidence interval of μ . Use Jill’s data to find a 99% confidence interval of μ. Which interval turns out shorter ?

9. Find the MSE of Jack’s estimator. Find the MSE of Jill’s estimator.

10. Another student (Tom) was asked to combine the data obtained by Jack and Jill . He took a simple average = (10.6 + 10.9 )/2 = 10. 75 . Find the MSE of Tom’s estimator.

11. Can you find an estimator which has a smaller MSE than Tom’s ?

12. Find a 95% confidence interval based only on Jack’s data (ignoring the information about σ provided by the manufacture ).
13. Use Jack’s data to find a 95% confidence interval of σ .

Answer 1