Questions: Test a claim that the mean amount of lead in the air in U.S. cities is less than 0.035 microgram per cubic meter. It was found that the mean amount of lead in the air for the random sample of 58 U.S cities is 0.038 microgram per cubic meter and the standard deviation is 0.069 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 ≥ 0.035 Ha: μ < 0.035 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= .

Test a claim that the mean amount of lead in the air in U.S. cities is less than 0.035 microgram per cubic meter. It was found that the mean amount of lead in the air for the random sample of 58 U.S cities is 0.038 microgram per cubic meter and the standard deviation is 0.069 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 ≥ 0.035
Ha: μ < 0.035

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= .

Transcript text: Test a claim that the mean amount of lead in the air in U.S. cities is less than 0.035 microgram per cubic meter. It was found that the mean amount of lead in the air for the random sample of 58 U.S cities is 0.038 microgram per cubic meter and the standard deviation is 0.069 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 $\mathrm{H}_{a}$ \[ \begin{array}{ll} \mathrm{H}_{0}: \mu & \geq 0.035 \\ \mathrm{H}_{\mathrm{a}}: \mu & <0.035 \end{array} \] The claim is the alternative hypothesis. (b) Find the critical value(s) and identify the rejection region(s). The critical value(s) is/are $\mathrm{t}_{0}=$ $\square$ .

Solution

Solution Steps

Step 1: Identify the Claim and Hypotheses

The claim is that the mean amount of lead in the air in U.S. cities is less than $0.035$ microgram per cubic meter. We can express this in terms of hypotheses:

\[ \begin{align_} H_0: & \quad \mu \geq 0.035 \\ H_a: & \quad \mu < 0.035 \end{align_} \]

Step 2: Calculate the Standard Error

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

\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{0.069}{\sqrt{58}} \approx 0.0091 \]

Step 3: Calculate the Test Statistic

The test statistic ($Z_{test}$) is calculated as follows:

\[ Z_{test} = \frac{\bar{x} - \mu_0}{SE} = \frac{0.038 - 0.035}{0.0091} \approx 0.3311 \]

Step 4: Determine the Critical Value and Rejection Region

For a left-tailed test at $\alpha = 0.01$, the critical value is:

\[ t_0 \approx -2.3263 \]

The rejection region is defined as:

\[ t < -2.3263 \]

Step 5: Calculate the P-value

The P-value associated with the test statistic is:

\[ P \approx 0.6297 \]

Step 6: Conclusion

Since the test statistic $Z_{test} \approx 0.3311$ does not fall within the rejection region $t < -2.3263$ and the P-value $0.6297$ is greater than $\alpha = 0.01$, we fail to reject the null hypothesis $H_0$.