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.


Systolic Blood Pressure Systolic blood pressure levels above 120 mm Hg are considered to be high. For the 300 systolic blood pressure levels listed in Data Set 1 “Body Data” from Appendix B, the mean is 122.96000 mm Hg and the standard deviation is 15.85169 mm Hg. Use a 0.01 significance level to test the claim that the sample is from a population with a mean greater than 120 mm Hg.

Verified step by step guidance

1. Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis represents the claim that the population mean is equal to 120 mm Hg, while the alternative hypothesis represents the claim that the population mean is greater than 120 mm Hg. Mathematically, H₀: μ = 120 and H₁: μ > 120.
2. Step 2: Identify the significance level (α) and the test type. The significance level is given as 0.01, and since the alternative hypothesis is testing for a mean greater than 120, this is a one-tailed test.
3. Step 3: Calculate the test statistic using the formula for a one-sample z-test: z = (x̄ - μ₀) / (σ / √n), where x̄ is the sample mean (122.96), μ₀ is the hypothesized population mean (120), σ is the population standard deviation (15.85169), and n is the sample size (300).
4. Step 4: Determine the critical value or P-value. For the critical value method, find the z-critical value corresponding to a significance level of 0.01 for a one-tailed test. For the P-value method, calculate the P-value using the z-test statistic obtained in Step 3 and compare it to the significance level.
5. Step 5: Make a decision and state the conclusion. If the test statistic exceeds the critical value or if the P-value is less than the significance level, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Based on this decision, conclude whether there is sufficient evidence to support the claim that the population mean is greater than 120 mm Hg.

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 it is less than the predetermined significance level (e.g., 0.01), the null hypothesis is rejected.

Significance Level

The significance level, denoted as alpha (α), is the threshold used to decide whether to reject the null hypothesis. It represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. Common significance levels are 0.05, 0.01, and 0.10. In this context, a significance level of 0.01 means that there is a 1% risk of concluding that a difference exists when there is none.