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.


Is the Diet Practical? When 40 people used the Weight Watchers diet for one year, their mean weight loss was 3.0 lb and the standard deviation was 4.9 lb (based on data from “Comparison of the Atkins, Ornish, Weight Watchers, and Zone Diets for Weight Loss and Heart Disease Reduction,” by Dansinger et al., Journal of the American Medical Association, Vol. 293, No. 1). Use a 0.01 significance level to test the claim that the mean weight loss is greater than 0. Based on these results, does the diet appear to have statistical significance? Does the diet appear to have practical significance?

Verified step by step guidance

1. Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: μ = 0 (the mean weight loss is 0), and the alternative hypothesis is H₁: μ > 0 (the mean weight loss is greater than 0). This is a one-tailed test since the claim is about the mean being greater than 0.
2. Step 2: Identify the test statistic formula. Since the population standard deviation is unknown and the sample size is relatively small (n = 40), use the t-test statistic formula: 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 degrees of freedom (df) for the t-distribution. The degrees of freedom are given by df = n - 1. In this case, df = 40 - 1 = 39.
4. Step 4: Determine the critical value or P-value. Using a significance level of α = 0.01 and df = 39, find the critical t-value from a t-distribution table or calculate the P-value corresponding to the test statistic. For a one-tailed test, compare the test statistic to the critical value or compare the P-value to α to decide whether to reject H₀.
5. Step 5: Draw a conclusion. If the test statistic exceeds the critical value or if the P-value is less than α, reject the null hypothesis and conclude that the mean weight loss is greater than 0 (statistical significance). Then, assess practical significance by considering whether the mean weight loss of 3.0 lb is meaningful in the context of weight loss goals.

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 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., 0.01), the null hypothesis is rejected.

Statistical vs. Practical Significance

Statistical significance refers to the likelihood that a result or relationship is caused by something other than mere random chance, typically assessed through P-values. Practical significance, on the other hand, considers whether the size of the effect is large enough to be of real-world importance. A result can be statistically significant without being practically significant if the effect size is too small to have meaningful implications in practice.