Question

A chemical plant is required to maintain sulfur dioxide levels in the working environment atmosphere at an average level of no more than 0.125 parts per million (ppm). Safety engineers measure the levels at a randomly chosen 10 intervals each week. If the sample mean sulfur dioxide level is more than 0.15 ppm, the safety protocols say that the plant will be evacuated while the air is scrubbed and machines are adjusted. The standard deviation of the measurements is known to be 0.04 ppm, and the testing scenario tests the hypotheses H_0:\mu=0.125H 0 ​ :μ=0.125 vs H_a:\mu>0.125H a ​ :μ>0.125

Using the decision rule in the prompt above, what is the chance the safety engineers will make a Type I error?
Using the same decision rule, what is the power of this test if the mean pollution level is truly 0.14 ppm?

Answer 1

Answer:

1. Type I error = 0.0241

2. Power of test = 0.2146

Step-by-step explanation:

1.  

Test statistic:

z = (x- µ)/(σ/√n)

  = (0.15 - 0.125)/(0.04/√10)

  = 0.025/0.0126

 = 1.9764

p-value = 1- NORM.S.DIST(1.9764, 1)

            = 0.0241

Type I error = 0.0241

2.

If true µ₁ = 0.14,

then Power, (1 - β) is given as;

1 - β = P(X ≥ 0.15 | µ₁ = 0.14)

      = P(Z ≥ (0.15 - 0.14)/(0.04/√10) )

      = P(Z ≥ 0.79)

      = 1 - P(Z < 0.79)

Using excel function :  

  = 1 - NORM.S.DIST(0.79, 1)

  = 1 - 0.7854

  = 0.2146

Power of test = 0.2146