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 0.069 microgram per cubic meter. at \\( \alpha = 0.10 \\), 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

Since the population standard deviation is unknown and we are testing a claim about the population mean, we use a t - test. The sample size \(n = 57\), so the degrees of freedom \(df=n - 1=57-1 = 56\).

Step2: Find the critical value

The significance level \(\alpha = 0.10\) and the test is left - tailed (because \(H_{a}:\mu<0.037\)). Using a t - distribution table or a calculator with a t - distribution function (such as in Excel: =T.INV(0.10,56)), we find the critical value.

Answer:

The critical value \(t_{0}\approx - 1.29\)