Question

test a claim that the mean amount of lead in the air in u.s. cities is less than 0.036 microgram per cubic meter. it was found that the mean amount of lead in the air for the random sample of 54 u.s. cities is 0.039 microgram per cubic meter and the standard deviation is 0.068 microgram per cubic meter. at \\( \alpha = 0.05 \\), can the claim be supported? complete parts (a) through (e) below. assume the population is normally distributed.
(a) identify the claim and state \\( h _ { 0 } \\) and \\( h _ { a } \\).
\\( h _ { 0 } : \mu \geq 0.036 \\)
\\( h _ { a } : \mu < 0.036 \\)
(type integers or decimals. do not round.)
the claim is the alternative hypothesis.
(b) find the critical value(s) and identify the rejection region(s).
the critical value(s) is/are \\( t _ { 0 } = \square \\).
(use a comma to separate answers as needed. round to two decimal places as needed.)

Explanation:

Step1: Determine the degrees of freedom

The degrees of freedom \(df=n - 1\), where \(n = 54\). So \(df=54-1 = 53\).

Step2: Find the critical value

Since the test is left - tailed (\(H_{a}:\mu<0.036\)) and \(\alpha = 0.05\). Using a t - distribution table or a calculator with t - distribution functions (for example, in Excel: =T.INV(0.05,53)), we find the critical value.

Answer:

\(-1.67\)