Question

a scientist estimates that the mean nitrogen dioxide level in a city is greater than 30 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 7. at α = 0.11, 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 leq 30 )
( h_a: mu > 30 ) (claim)
b. ( h_0: mu leq 30 ) (claim)
( h_a: mu > 30 )
c. ( h_0: mu < 30 )
( h_a: mu geq 30 ) (claim)
d. ( h_0: mu geq 30 ) (claim)
( h_a: mu < 30 )
e. ( h_0: mu = 30 )
( h_a: mu > 30 ) (claim)
f. ( h_0: mu = 30 ) (claim)
( h_a: mu > 30 )
(b) find the critical value and identify the rejection region.
( z_0 = 1.23 ) (round to two decimal places as needed.)
rejection region: ( z > 1.23 )
(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

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

We know that \(\mu = 30\), \(\sigma=7\), \(n = 31\), and \(\bar{x}=30\)

Substitute these values into the formula:

$$ z=\frac{30 - 30}{\frac{7}{\sqrt{31}}}=0 $$

Answer:

\(z = 0.00\)