Question

a scientist estimates that the mean nitrogen dioxide level in a city is greater than 26 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 = 0.01 \\), 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 = 26 \\)
\\( h _ { a } : \mu > 26 \\) (claim)
\\( \bigcirc \\) b. \\( h _ { 0 } : \mu \leq 26 \\) (claim)
\\( h _ { a } : \mu > 26 \\)
\\( \bigcirc \\) c. \\( h _ { 0 } : \mu = 26 \\) (claim)
\\( h _ { a } : \mu > 26 \\)
\\( \bigcirc \\) d. \\( h _ { 0 } : \mu \leq 26 \\)
\\( h _ { a } : \mu > 26 \\) (claim)
\\( \bigcirc \\) e. \\( h _ { 0 } : \mu \geq 26 \\) (claim)
\\( h _ { a } : \mu < 26 \\)
\\( \bigcirc \\) f. \\( h _ { 0 } : \mu < 26 \\)
\\( h _ { a } : \mu \geq 26 \\) (claim)
(b) find the critical value and identify the rejection region.
\\( z _ { 0 } = 2.33 \\) (round to two decimal places as needed)
rejection region: \\( z > 2.33 \\)
(c) find the standardized test statistic.
\\( z = \square \\) (round to two decimal places as needed.)

Explanation:

Step1: Calculate the sample mean

First, sum up all the data values:

$$
LATEXBLOCK0
$$

The sample size \(n = 31\). The sample mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}=\frac{930}{31}=30\)

Step2: Calculate the standardized test statistic \(z\)

The formula for the \(z\) - test statistic in a one - sample z - test (when population standard deviation \(\sigma\) is known) is \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\)

We know that \(\mu = 26\), \(\sigma=11\), \(n = 31\), and \(\bar{x}=30\)

Substitute the values into the formula:

$$
LATEXBLOCK1
$$

Answer:

\(z\approx2.02\)