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. (c) find the standardized test statistic, t. the standardized test statistic is \\( t = 0.32 \\). (round to two decimal places as needed.) (d) decide whether to reject or fail to reject the null hypothesis. fail to reject \\( h _ { 0 } \\) because the standardized test statistic is not in the rejection region. (e) interpret the decision in the context of the original claim. there enough evidence at the \\( \\% \\) level of significance to the claim that the mean amount of lead in the air in u.s. cities is microgram per cubic meter. (type integers or decimals. do not round.)

Explanation:

Step1: Determine the null and alternative hypotheses

The claim is that the mean amount of lead in the air in U.S. cities is less than \(0.036\) microgram per cubic meter. So, \(H_0:\mu\geq0.036\) and \(H_1:\mu < 0.036\) (left - tailed test).

Step2: Recall the formula for the t - test statistic

The formula for the t - test statistic is \(t=\frac{\bar{x}-\mu}{s/\sqrt{n}}\), where \(\bar{x} = 0.039\), \(\mu=0.036\), \(s = 0.068\), and \(n = 54\).

Step3: Calculate the t - test statistic

Substitute the values into the formula:

$$
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$$

Step4: Find the critical value

For a left - tailed test with \(\alpha = 0.05\) and \(n-1=54 - 1=53\) degrees of freedom (using a t - table or calculator approximation), \(t_{0}\approx - 1.674\)

Step5: Make a decision

Since the calculated \(t = 0.32\) and the critical value \(t_{0}=-1.674\), and \(0.32>-1.674\) (the test statistic is not in the rejection region \(t < t_{0}\)), we fail to reject \(H_0\)

Step6: Interpret the decision

There is not enough evidence at the \(5\%\) level of significance to support the claim that the mean amount of lead in the air in U.S. cities is less than \(0.036\) microgram per cubic meter.

Answer:

There is not enough evidence at the \(5\%\) level of significance to support the claim that the mean amount of lead in the air in U.S. cities is less than \(0.036\) microgram per cubic meter.