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 α = 0.01, can you support the scientist’s 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)

Explanation:

In hypothesis testing, the null hypothesis \(H_0\) is a statement of equality or no difference, and the alternative hypothesis \(H_a\) is the claim we are trying to find evidence for. The scientist's claim is that the mean nitrogen dioxide level \(\mu> 26\). The null hypothesis is the opposite of the claim for a one - tailed test (since we are testing if it is greater). So \(H_0:\mu\leq26\) and \(H_a:\mu > 26\) (where the claim is \(H_a\)).

Answer:

A. \(H_0:\mu = 26\), \(H_a:\mu>26\) (claim) is incorrect because the null hypothesis should be the non - claim in this case.
B. \(H_0:\mu\leq26\) (claim) is incorrect as the claim is \(H_a\).
C. \(H_0:\mu = 26\) (claim) is incorrect as the claim is \(H_a\).
D. \(H_0:\mu\leq26\), \(H_a:\mu>26\) (claim) is correct. The null hypothesis \(H_0\) is the statement of no effect (the mean is less than or equal to 26) and the alternative hypothesis \(H_a\) is the claim (the mean is greater than 26).
E. \(H_0:\mu\geq26\) (claim) is incorrect as the claim is \(H_a\).
F. \(H_0:\mu<26\) is incorrect as the null hypothesis should be a statement that includes equality for a one - tailed test when the claim is \(H_a:\mu>26\).

So the answer is D. \(H_0:\mu\leq26\), \(H_a:\mu>26\) (claim).