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.


Diastolic Blood Pressure Diastolic blood pressure levels of 60 mm Hg or lower are considered to be too low. For the 300 diastolic blood pressure levels listed in Data Set 1 “Body Data” from Appendix B, the mean is 70.75333 mm Hg and the standard deviation is 11.61618 mm Hg. Use a 0.01 significance level to test the claim that the sample is from a population with a mean greater than 60 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 to be tested, and the alternative hypothesis represents the claim we are trying to support. Here, H₀: μ ≤ 60 mm Hg (the population mean is less than or equal to 60 mm Hg), and H₁: μ > 60 mm Hg (the population mean is greater than 60 mm Hg). This is a one-tailed test.
2. Step 2: Identify the test statistic to be used. Since the population standard deviation is unknown and the sample size is large (n = 300), we use the t-test for the sample mean. The formula for the test statistic is t = (x̄ - μ₀) / (s / √n), where x̄ is the sample mean, μ₀ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.
3. Step 3: Calculate the test statistic using the given values. Substitute x̄ = 70.75333, μ₀ = 60, s = 11.61618, and n = 300 into the formula t = (x̄ - μ₀) / (s / √n). Simplify the expression to find the value of the test statistic.
4. Step 4: Determine the critical value or P-value. For a one-tailed test with a significance level of α = 0.01, use a t-distribution table or statistical software to find the critical t-value corresponding to df = n - 1 = 299. Alternatively, calculate the P-value by finding the probability of obtaining a test statistic as extreme as the one calculated in Step 3.
5. Step 5: Compare the test statistic to the critical value or compare the P-value to the significance level. If the test statistic exceeds the critical value or if the P-value is less than α = 0.01, 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 population mean diastolic blood pressure is greater than 60 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 the P-value is less than the significance level (e.g., 0.01), the null hypothesis is rejected.

Significance Level

The significance level, 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. 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.