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.
a. \\( h _ { 0 } : \mu = 26 \\)
\\( h _ { a } : \mu > 26 \\) (claim)
b. \\( h _ { 0 } : \mu \leq 26 \\) (claim)
\\( h _ { a } : \mu > 26 \\)
c. \\( h _ { 0 } : \mu = 26 \\) (claim)
\\( h _ { a } : \mu > 26 \\)
d. \\( h _ { 0 } : \mu \leq 26 \\)
\\( h _ { a } : \mu > 26 \\) (claim)
e. \\( h _ { 0 } : \mu \geq 26 \\) (claim)
\\( h _ { a } : \mu < 26 \\)
f. \\( h _ { 0 } : \mu < 26 \\)
\\( h _ { a } : \mu \geq 26 \\) (claim)
(b) find the critical value and identify the rejection region.
\\( z _ { 0 } = \square \\) (round to two decimal places as needed )

Explanation:

Step1: Identify the claim and hypotheses

The scientist's claim is that the mean nitrogen dioxide level is greater than 26. In hypothesis - testing, the null hypothesis \(H_0\) is a statement of equality or non - effect, and the alternative hypothesis \(H_a\) is the claim we are trying to find evidence for. So, \(H_0:\mu\leq26\) and \(H_a:\mu > 26\) (where the claim is \(H_a\)).

Step2: Find the critical value

Since the test is a right - tailed test (\(H_a:\mu>26\)) with \(\alpha = 0.01\).
We look up the \(z\) - value in the standard normal distribution table. The critical value \(z_0\) for a right - tailed test with \(\alpha=0.01\) is the \(z\) - value such that \(P(Z>z_0)=0.01\), or \(P(Z\leq z_0)=1 - 0.01=0.99\).
Looking up in the standard normal table (or using a calculator with a normal - distribution function, e.g., in Excel: =NORM.S.INV(0.99)), we get \(z_0 = 2.33\).
The rejection region is \(z>2.33\).

Answer:

(a) D. \(H_0:\mu\leq26\), \(H_a:\mu > 26\) (claim)
(b) \(z_0 = 2.33\), rejection region \(z>2.33\)