Question

test a claim that the mean amount of lead in the air in u.s. cities is less than 0.037 microgram per cubic meter. it was found that the mean amount of lead in the air for the random sample of 56 u.s. cities is 0.038 microgram per cubic meter and the standard deviation is 0.068 microgram per cubic meter. at α = 0.01, 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 0.037 )
( h_a: mu < 0.037 )
(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 type of test

This is a left - tailed \(t\) - test because the alternative hypothesis \(H_{a}:\mu < 0.037\). The degrees of freedom \(df=n - 1\), where \(n = 56\), so \(df=56-1 = 55\).

Step2: Find the critical value

Using a \(t\) - distribution table or a calculator with the significance level \(\alpha = 0.01\) and \(df = 55\).
The critical value \(t_{0}\) for a left - tailed test with \(\alpha=0.01\) and \(df = 55\) can be found. Using a calculator (e.g., TI - 84: invT(0.01,55)), we get \(t_{0}\approx - 2.40\)

Answer:

\(-2.40\)