Question

1. 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 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 5 months, yielded x=2.1 ppm, s=4.1 ppm Use this information to answer the following questions.
   1. Define the parameter of interest in words and notation.
   2. State the null and alternative hypotheses in proper notation if the investigators are looking for evidence that the permissible exposure limit has been exceeded.
   3. Calculate the test statistic for these data. Provide this statistic in proper notation.
   4. Report the p-value for your calculated test statistic using the appropriate function in R.
   5. (5 points) Provide a conclusion for this study at the 1% significance level. In context of the problem. Be sure to include all parts of the conclusion as discussed in class.

Answer 1

Part a

The parameter of interest is given as population average permissible exposure limit.

Notation: µ = 1 ppm

Part b

The null and alternative hypotheses are given as below:

Null hypothesis: H₀: The permissible exposure limit is 1 ppm.

Alternative hypothesis: H_a: The permissible exposure limit is greater than 1 ppm.

H₀: µ = 1 versus H_a: µ > 1

This is an upper tailed test.

Part c

The test statistic formula is given as below:

t = (Xbar - µ)/[S/sqrt(n)]

We are given

Xbar = 2.1

S = 4.1

n = 36

df = n – 1 = 35

t = (2.1 – 1)/[4.1/sqrt(36)]

t = 1.6098

Test statistic = t = 1.6098

Part d

We have t = 1.6098

df = 35

So, p-value by using t-table is given as below:

P-value = 0.0582

Part e

We are given

P-value = 0.0582

α = 0.01

P-value > α = 0.01

So, we do not reject the null hypothesis

There is insufficient evidence to conclude that permissible exposure limit has been exceeded.