Questions: 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 0.039 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 H0 and Ha. H0: μ ≥ 0.037 Ha: μ < 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 t0= (Use a comma to separate answers as needed. Round to two decimal places as needed.)

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 0.039 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 H0 and Ha.
H0: μ ≥ 0.037
Ha: μ < 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 t0=
(Use a comma to separate answers as needed. Round to two decimal places as needed.)

Transcript text: 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 0.039 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. (a) Identify the claim and state $\mathrm{H}_{0}$ and $\mathrm{H}_{3}$. \begin{tabular}{l|l|l|l} $\mathrm{H}_{0}:$ & $\mu$ & $\geq 0.037$ \\ $\mathrm{H}_{\mathrm{a}}:$ & $\mu$ & $<0.037$ \end{tabular} (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.)

Solution

Solution Steps

Step 1: State the Hypotheses

We are testing the claim that the mean amount of lead in the air in U.S. cities is less than $0.037$ microgram per cubic meter. The hypotheses are defined as follows:

Null Hypothesis ($H_0$): $ \mu \geq 0.037 $
Alternative Hypothesis ($H_a$): $ \mu < 0.037 $

Step 2: Calculate the Standard Error

The standard error ($SE$) is calculated using the formula:

\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{0.068}{\sqrt{56}} \approx 0.0091 \]

Step 3: Calculate the Test Statistic

The test statistic ($Z_{test}$) is calculated using the formula:

\[ Z_{test} = \frac{\bar{x} - \mu_0}{SE} = \frac{0.039 - 0.037}{0.0091} \approx 0.2201 \]

Step 4: Calculate the P-value

For a left-tailed test, the P-value is determined as follows:

\[ P = T(z) \approx 0.5871 \]

Step 5: Determine the Critical Value

For a left-tailed test at a significance level of $\alpha = 0.01$, the critical value ($t$) is found using the t-distribution with $n - 1 = 55$ degrees of freedom:

\[ \text{Critical Value} \approx -2.3961 \]

Step 6: Identify the Rejection Region

The rejection region for this left-tailed test is defined as:

\[ t < -2.3961 \]

Step 7: Make a Decision

We compare the test statistic to the critical value: