Question

Government standards limit the airborne amount of a known cancer causing gas. A company using such a substance in its production process limits the exposure of its employees to the gas by shutting down production when the amount of the gas reaches 3 ppm(parts per million) with a standard deviation of 0.5 ppm.

(2) If you were the plant manager and you are told management wants as little down time as possible, would you choose a larger or a smaller value of alpha to test whether the amount of gas is dangerous? Explain.

(2) If you were an employee in the plant, would you want the choice of alpha to be larger or smaller?     Explain.

(5) A random sample of 50 air specimens from various sensors in the plant yielded an average of 3.1 ppm. Will this sample data indicate the company should shut down production given their criteria described above? Use alpha = 0.01.

(2) Describe both Type I and Type II errors in the context of this problem.

(5) Calculate the probability of Type II error if in fact the true average plant level is 3.1 ppm. Use alpha = 0.01.

(1) What is the value of the Power of the test in this case?

(5) Calculate the probability of Type II error if in fact the true mean plant level is 3.1 ppm when alpha = 0.05.

(1) What is the Power of the test now?

(2) What do the findings of parts e, f , h , and i confirm about the relationships between alpha, beta and the Power of the test?

Answer 1

(2)

Plant manager would choose smaller value of alpha. It means probability to get null hypothesis true would be very less in order to get the null hypothesis get rejected.

3)

If you were an employee alpha choosen would be larger so that production can be stopped even with larger probability of null being true.

4)

Ho :   µ =   3                  
Ha :   µ >   3       (Right tail test)          
                          
Level of Significance ,    α =    0.01                  
population std dev ,    σ =    0.5000                  
Sample Size ,   n =    50                  
Sample Mean,    x̅ =   3.1000                  
                          
'   '   '                  
                          
Standard Error , SE = σ/√n =   0.5000   / √    50   =   0.0707      
Z-test statistic= (x̅ - µ )/SE = (   3.100   -   3   ) /    0.0707   =   1.41
                          
critical z value, z* =       2.3263   [Excel formula =NORMSINV(α/no. of tails) ]              
                          
p-Value   =   0.0786   [ Excel formula =NORMSDIST(z) ]              
Decision:   p-value>α, Do not reject null hypothesis   

sample data indicate the company should not shut down production                    

5)

Type 1 error: When we stop the production but infact we should not.

Type 2 error: When we do not stop the production but infact we should .

THANKS

revert back for doubt

please upvote