Hypothesis Testing Using Rejection Regions In Exercises 19–26, (a) identify the claim and state H0 and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic t, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the population is normally distributed.


Lead Levels As part of your work for an environmental awareness group, you want to test a claim that the mean amount of lead in the air in U.S. cities is less than 0.032 microgram per cubic meter. You find that the mean amount of lead in the air for a random sample of 56 U.S. cities is 0.021 microgram per cubic meter and the standard deviation is 0.034 microgram per cubic meter. At α=0.01, can you support the claim?

Verified step by step guidance

1. Step 1: Identify the claim and state the null and alternative hypotheses. The claim is that the mean amount of lead in the air in U.S. cities is less than 0.032 microgram per cubic meter. This translates to the alternative hypothesis (Ha): μ < 0.032. The null hypothesis (H0) is the opposite of the claim: μ = 0.032.
2. Step 2: Determine the critical value(s) and rejection region(s). Since this is a left-tailed test (based on the alternative hypothesis μ < 0.032), use the significance level α = 0.01 and a t-distribution table to find the critical t-value. The degrees of freedom (df) are calculated as n - 1, where n is the sample size. The rejection region will be t < critical value.
3. Step 3: Calculate the standardized test statistic t. Use the formula t = (x̄ - μ) / (s / √n), where x̄ is the sample mean (0.021), μ is the population mean under the null hypothesis (0.032), s is the sample standard deviation (0.034), and n is the sample size (56). Substitute the values into the formula to compute t.
4. Step 4: Compare the calculated t-value to the critical value. If the calculated t-value falls within the rejection region (t < critical value), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
5. Step 5: Interpret the decision in the context of the original claim. If the null hypothesis is rejected, it supports the claim that the mean amount of lead in the air in U.S. cities is less than 0.032 microgram per cubic meter. If the null hypothesis is not rejected, there is insufficient evidence to support the claim.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents a statement of no effect or no difference, and the alternative hypothesis (Ha), which represents the claim being tested. The goal is to determine whether there is enough evidence in the sample data to reject H0 in favor of Ha.

Rejection Region

The rejection region is a range of values for the test statistic that leads to the rejection of the null hypothesis. It is determined by the significance level (α), which defines the probability of making a Type I error (rejecting H0 when it is true). In this case, with α=0.01, the rejection region is based on the critical value(s) that correspond to the chosen significance level, indicating where the test statistic must fall to reject H0.

Standardized Test Statistic

The standardized test statistic, often denoted as t or z, measures how far the sample mean is from the population mean under the null hypothesis, expressed in terms of standard deviations. It is calculated using the sample mean, population mean (from H0), sample standard deviation, and sample size. This statistic is crucial for determining whether the observed sample data is significantly different from what is expected under the null hypothesis.