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.


Got a Minute? Students of the author estimated the length of one minute without reference to a watch or clock, and the times (seconds) are listed below. Use a 0.05 significance level to test the claim that these times are from a population with a mean equal to 60 seconds. Does it appear that students are reasonably good at estimating one minute?


69 81 39 65 42 21 60 63 66 48 64 70 96 91 95

Verified step by step guidance

1. Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: μ = 60 (the population mean is equal to 60 seconds). The alternative hypothesis is H₁: μ ≠ 60 (the population mean is not equal to 60 seconds). This is a two-tailed test.
2. Step 2: Calculate the sample mean (x̄) and the sample standard deviation (s) using the given data. Use the formulas: x̄ = (Σx) / n and s = sqrt((Σ(x - x̄)²) / (n - 1)), where n is the sample size.
3. Step 3: Compute the test statistic using the t-test formula for a single sample: t = (x̄ - μ) / (s / sqrt(n)), where μ is the hypothesized population mean (60 seconds), x̄ is the sample mean, s is the sample standard deviation, and n is the sample size.
4. Step 4: Determine the critical value(s) or the P-value. For the critical value method, use a t-distribution table with degrees of freedom (df = n - 1) and a significance level of 0.05 for a two-tailed test. For the P-value method, use the calculated t-value and a t-distribution table or statistical software to find the P-value.
5. Step 5: Compare the test statistic to the critical value(s) or compare the P-value to the significance level (α = 0.05). If the test statistic falls outside the critical value range 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 students are reasonably good at estimating one minute.

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 (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 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., 0.05), the null hypothesis is rejected.

Significance Level

The significance level, often 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. Commonly set at 0.05, this level indicates that there is a 5% risk of concluding that a difference exists when there is none. Choosing an appropriate significance level is crucial for the validity of hypothesis testing.