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 \\( \alpha = 0.01 \\), can the claim be supported? complete parts (a) through (e) below. assume the population is normally distributed.
(b) find the critical value(s) and identify the rejection region(s).
the critical value(s) is/are \\( t _ { 0 } = - 2.40 \\).
(use a comma to separate answers as needed. round to two decimal places as needed)
choose the graph which shows the rejection region.
a.
b.
c.
d.
\\( t < t _ { 0 } \\)
\\( t < - t _ { 0 }, t > t _ { 0 } \\)
\\( t > t _ { 0 } \\)
\\( - t _ { 0 } < t < t _ { 0 } \\)
(c) find the standardized test statistic, t.
the standardized test statistic is \\( t = 0.11 \\).
(round to two decimal places as needed.)
(d) decide whether to reject or fail to reject the null hypothesis.
\\( h _ { 0 } \\) because the standardized test statistic in the rejection region

Explanation:

Step1: Recall the decision rule for t - test

If the standardized test statistic \(t\) is in the rejection region, we reject \(H_0\). If not, we fail to reject \(H_0\). The rejection region is \(t < t_0=- 2.40\) (from part (b)).

Step2: Compare the test statistic and the critical value

We are given \(t = 0.11\) (from part (c)). Since \(0.11>-2.40\) (i.e., \(t\) is not in the rejection region \(t < - 2.40\)).

Answer:

Fail to reject \(H_0\) because the standardized test statistic is not in the rejection region.