Answer to Question #237633 in Statistics and Probability for martin

Question

                  Suppose the measures to ensure that all manufactured motor vehicles meet requirements 

with regards to carbon monoxide emission indicate that the standard deviation per unit 

volume of exhaust fumes must not be in excess of 0.03 percent carbon monoxide emission 

particles per unit volume. A random sample of ten newly manufactured motor vehicles are 

tested and the following carbon monoxide emissions are recorded: 

  

Answer 1

QUESTION

Suppose the measures to ensure that all manufactured motor vehicles meet requirements with regards to carbon monoxide emission indicate that the standard deviation per unit volume of exhaust fumes must not be in excess of 0.03 percent carbon monoxide emission particles per unit volume. A random sample of ten newly manufactured motor vehicles are tested and the following carbon monoxide emissions are recorded: 0.23, 0.27, 0.31, 0.25, 0.30, 0.29, 0.32, 0.26, 0.24, 0.22

A). Calculate the point estimate of the standard deviation of the carbon monoxide emissions.

B). What is the 99% confidence interval estimate for the population variance of the carbon monoxide emissions?

C). Provide an interpretation for the answer in b).

D). Suppose the level of confidence is reduced, what will happen to the answer in b)? Motivate. 

SOLUTIONS

A). Calculate the point estimate of the standard deviation of the carbon monoxide emissions.

Solution:

$s^2=\frac{1}{n-1}\lbrack \Sigma x {2 \atop i}-\frac{1}{n}\Sigma (x_n)^2 \rbrack$

$s^2=\frac{1}{10-1} \lbrack0.7345-\frac{1}{10}(2.69)^2\rbrack$

$s^2=0.0348$

$Answer: 0.0348$

B). What is the 99% confidence interval estimate for the population variance of the carbon monoxide emissions?

Solution:

$CI=\lbrack \frac{(n-1)s^2}{x^2}_{\alpha/2}, \frac{(n-1)s^2}{x^2}_{1-\alpha/2}\rbrack$

$CI=\lbrack\frac{(10-1)0.0012}{23.5894}, \frac{(10-1)0.0012}{1.7349}\rbrack$

$CI=\lbrack0.0005,0.0063\rbrack$

$Answer:\lbrack0.0005,0.0063\rbrack$

C). Provide an interpretation for the answer in b).

Solution:

It is 99% confident that population variance lies between 0.0005 and 0.0063

D). Suppose the level of confidence is reduced, what will happen to the answer in b)? Motivate. 

Solution:

The lower limit will be increased and the upper limit will be reduced when the confidence level is reduced.