Question

a scientist estimates that the mean nitrogen dioxide level in a city is greater than 31 parts per billion. to test this estimate, you determine the nitrogen dioxide levels for 31 randomly selected days. the results (in parts per billion) are listed to the right. assume that the population standard deviation is 11. at \\( \alpha=0.12 \\), can you support the scientists estimate? complete parts (a) through (e).
(a) write the claim mathematically and identify \\( h_{0} \\) and \\( h_{a} \\). choose from the following
\\( \bigcirc \\) a. \\( h_{0}: \mu=31 \\)
\\( h_{a}: \mu>31 \\)
(claim)
\\( \bigcirc \\) b. \\( h_{0}: \mu \geq 31 \\)
(claim)
\\( h_{a}: \mu<31 \\)
\\( \bigcirc \\) c. \\( h_{0}: \mu<31 \\)
\\( h_{a}: \mu \geq 31 \\)
(claim)
\\( \bigcirc \\) d. \\( h_{0}: \mu=31 \\)
(claim)
\\( h_{a}: \mu>31 \\)
\\( \bigcirc \\) e. \\( h_{0}: \mu \leq 31 \\)
(claim)
\\( h_{a}: \mu>31 \\)
\\( \bigcirc \\) f. \\( h_{0}: \mu \leq 31 \\)
\\( h_{a}: \mu>31 \\)
(claim)
(b) find the critical value and identify the rejection region.
\\( z_{0}=1.17 \\) (round to two decimal places as needed.)
rejection region \\( z \\)

Explanation:

Step1: Determine the hypotheses

The claim is that the mean nitrogen dioxide level is greater than 31. The null hypothesis \(H_0\) is the statement of no - effect or equality. The alternative hypothesis \(H_a\) is the claim. So \(H_0:\mu\leq31\) and \(H_a:\mu > 31\) (claim).

Step2: Find the critical value

Since the significance level \(\alpha = 0.12\) and the test is right - tailed (because \(H_a:\mu>31\)), we look up the \(z\) - value in the standard normal distribution table. The critical value \(z_0\) is the \(z\) - value such that \(P(Z>z_0)=\alpha\). Using a standard normal table or a calculator, \(z_0\approx1.17\)

Step3: Identify the rejection region

For a right - tailed \(z\) - test, the rejection region is \(z>z_0\).

Answer:

(a) The correct option is F: \(H_0:\mu\leq31\), \(H_a:\mu > 31\) (claim)
(b) Critical value \(z_0 = 1.17\), Rejection region \(z>1.17\)