Question

Use R commands and the information given in Exercise, that is, n-36, S-4.1, H0 : μ-1 vs Ha : μ > 1, and α-0.05, to complete the following. (a) Find the probability of type Il error when the true concentration is 2 ppm (b) OSHA would like the probability of type II error not to exceed 1% when the true concentration is 2 ppm, while keeping the level of significance at 0.05. What sample size should be used? Exercise Studies have shown that people who work with benzene longer than five years have 20 times the incidence of leukemia than the general population. As a result, OSHA (Occupational Safety and Health Administration) has set a time-weighted average permissible exposure limit of 1 ppm. A steel manufacturing plant, which exposes its workers to benzene daily, is under investigation for possible violations of the permissible exposure limit. Thirty-six measurements, taken over a period of 1.5 months, yielded 2.1 ppm, S 4.1 ppm. Complete the following parts to determine if there is sufficient evidence to conclude that the steel manufacturing plant is in violation of the OSHA exposure limit. (a) State the null and alternative hypotheses. (b) Carry out the test at a 0.05 and state your conclusion. What assumptions, if any, are needed for the validity of this test? 3. Textbook 8.4.4. Modify this problem in the following way. Assume ơ-41 (the value given for s). With ơ known you nnay now do a z test rather than at test. Based on this z test, you should be able to do parts a) and b) by hand, algebraically.

Answer 1

a)

> mu=1
> mu0=2
> sd=4.1
> alpha=0.05
> n=36
> (z=(mu-mu0)/sd*sqrt(n))
[1] -1.463415
> (Power=pnorm(abs(z)-qnorm(1-alpha)))
[1] 0.4280115

Power=1-Beta

Beta =1-Power =1-0.4280115 =0.5719885

b)

> mu=1
> mu0=2
> sd=4.1
> alpha=0.05
> beta=0.01
> (n=(sd*(qnorm(1-alpha)+qnorm(1-beta))/(mu-mu0))^2)
[1] 265.1011
> ceiling(n)
[1] 266