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 57 u.s. cities is 0.039 microgram per cubic meter and the standard deviation is 0.067 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.

(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.

(c) find the standardized test statistic, t.
the standardized test statistic is ( t = 0.22 )
(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 the null hypothesis \(H_0\). If it is not in the rejection region, we fail to reject \(H_0\).

Step2: Compare the test statistic and the critical value

We have the critical value \(t_0=- 2.40\) (left - tailed test since the claim is \(\mu<0.037\)) and the test statistic \(t = 0.22\).
Since \(t=0.22>-2.40\) (i.e., \(t\) is not in the rejection region \(t < t_0\)).

Answer:

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