Question

government regulations restrict the amount of pollutants that can be released to the atmosphere through industrial smokestacks. to demonstrate that their smokestacks are releasing pollutants below the mandated limit of 7 parts per billion pollutants, rem industries collects a random sample of 24 readings. the mean pollutant level for the sample is 6.55 parts per billion with a population standard deviation of 0.9 parts per billion. do the data support the claim that the average pollutants produced by rem industries are below the mandated level at a 0.05 significance level? assume the population of readings is approximately normally distributed. step 3 of 3: draw a conclusion and interpret the decision. answer we fail to reject the null hypothesis and conclude that there is sufficient evidence at a 0.05 level of significance to support the claim that the average pollutants produced by rem industries are less than 7 parts per billion pollutants. we reject the null hypothesis and conclude that there is insufficient evidence at a 0.05 level of significance to support the claim that the average pollutants produced by rem industries are less than 7 parts per billion pollutants. we reject the null hypothesis and conclude that there is sufficient evidence at a 0.05 level of significance to support the claim that the average pollutants produced by rem industries are less than 7 parts per billion pollutants. we fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.05 level of significance to support the claim that the average pollutants produced by rem industries are less than 7 parts per billion pollutants.

Explanation:

Step1: Set up hypotheses

$H_0:\mu\geq7$, $H_1:\mu < 7$

Step2: Calculate test - statistic

The test - statistic for a one - sample z - test (since population standard deviation $\sigma = 0.9$ is known) is $z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$, where $\bar{x}=6.55$, $\mu = 7$, $\sigma=0.9$, and $n = 24$.
$z=\frac{6.55 - 7}{\frac{0.9}{\sqrt{24}}}=\frac{- 0.45}{\frac{0.9}{4.899}}\approx\frac{-0.45}{0.184}\approx - 2.45$

Step3: Find critical value

For a one - tailed test with $\alpha = 0.05$, the critical value $z_{\alpha}=-1.645$.
Since $z=-2.45<-1.645$, we reject the null hypothesis.

Answer:

We reject the null hypothesis and conclude that there is sufficient evidence at a 0.05 level of significance to support the claim that the average pollutants produced by REM Industries are less than 7 parts per billion pollutants.