Testing Hypotheses
In Exercises 13–24, assume that a simple random sample has been selected and test the given claim. Unless specified by your instructor, use either the P-value method or the critical value method for testing hypotheses. Identify the null and alternative hypotheses, test statistic, P-value (or range of P-values), or critical value(s), and state the final conclusion that addresses the original claim.


Lead in Medicine Listed below are the lead concentrations (in ) measured in different Ayurveda medicines. Ayurveda is a traditional medical system commonly used in India. The lead concentrations listed here are from medicines manufactured in the United States (based on data from “Lead, Mercury, and Arsenic in US and Indian Manufactured Ayurvedic Medicines Sold via the Internet,” by Saper et al., Journal of the American Medical Association, Vol. 300, No. 8). Use a 0.05 significance level to test the claim that the mean lead concentration for all such medicines is less than 14 μg/g.


3.0 6.5 6.0 5.5 20.5 7.5 12 20.5 11.5 17.5

Verified step by step guidance

1. Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: μ ≥ 14 μg/g, which states that the mean lead concentration is greater than or equal to 14 μg/g. The alternative hypothesis is H₁: μ < 14 μg/g, which states that the mean lead concentration is less than 14 μg/g.
2. Step 2: Calculate the sample mean (x̄) and the sample standard deviation (s) using the given data: 3.0, 6.5, 6.0, 5.5, 20.5, 7.5, 12, 20.5, 11.5, 17.5. Use the formulas for the sample mean and standard deviation: x̄ = (Σx) / n and s = sqrt((Σ(x - x̄)²) / (n - 1)), where n is the sample size.
3. Step 3: Determine the test statistic. Since the population standard deviation is not provided, use the t-test formula: t = (x̄ - μ₀) / (s / sqrt(n)), where μ₀ is the hypothesized population mean (14 μg/g), x̄ is the sample mean, s is the sample standard deviation, and n is the sample size.
4. Step 4: Find the critical value or P-value. For a one-tailed t-test with a significance level of α = 0.05 and degrees of freedom df = n - 1, use a t-distribution table or statistical software to find the critical t-value or calculate the P-value corresponding to the test statistic.
5. Step 5: Compare the test statistic to the critical value or compare the P-value to the significance level. If the test statistic is less than the critical value or if the P-value is less than 0.05, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. State the conclusion in the context of the problem: whether there is sufficient evidence to support the claim that the mean lead concentration is less than 14 μg/g.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters 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 (H1), which indicates the presence of an effect or difference. The goal is to determine whether there is enough evidence in the sample data to reject the null hypothesis in favor of the alternative.

P-value

The P-value is a measure that helps determine the strength of the evidence against the null hypothesis. It represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. A smaller P-value indicates stronger evidence against the null hypothesis, and if the P-value is less than the predetermined significance level (e.g., 0.05), the null hypothesis is rejected.

Significance Level

The significance level, often denoted as alpha (α), is a threshold set by the researcher before conducting a hypothesis test. It defines the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. Commonly used significance levels are 0.05 and 0.01. In this context, a significance level of 0.05 means that there is a 5% risk of concluding that a difference exists when there is none.