In Exercises 5–16, use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal.


Measured and Reported Weights Listed below are measured and reported weights (lb) of random female subjects (from Data Set 4 “Measured and Reported” in Appendix B).


a. Use a 0.05 significance level to test the claim that for females, the measured weights tend to be higher than the reported weights.

Verified step by step guidance

1. Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis states that the measured weights are not significantly higher than the reported weights (H₀: μ_d ≤ 0), while the alternative hypothesis states that the measured weights are significantly higher than the reported weights (H₁: μ_d > 0).
2. Step 2: Calculate the differences between the paired measured and reported weights for each subject. For each pair, subtract the reported weight from the measured weight (d = Measured - Reported).
3. Step 3: Compute the mean of the differences (μ_d) and the standard deviation of the differences (s_d). Use the formulas for sample mean and sample standard deviation: μ_d = (Σd) / n and s_d = sqrt((Σ(d - μ_d)²) / (n - 1)), where n is the number of pairs.
4. Step 4: Perform a t-test for paired samples. Calculate the test statistic using the formula: t = (μ_d - 0) / (s_d / sqrt(n)). Here, μ_d is the mean of the differences, s_d is the standard deviation of the differences, and n is the number of pairs.
5. Step 5: Compare the calculated t-value to the critical t-value from the t-distribution table at a significance level of 0.05 and degrees of freedom (df = n - 1). If the calculated t-value is greater than the critical t-value, reject the null hypothesis and conclude that the measured weights tend to be higher than the reported weights.

Key Concepts

Paired Sample Data

Paired sample data involves two related groups where each subject in one group is matched with a subject in the other group. This design is often used in studies to compare two measurements taken on the same subjects, such as measured and reported weights in this case. The analysis of paired samples helps to control for variability between subjects, allowing for a more accurate assessment of the differences.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. In this context, the null hypothesis (H0) would state that there is no difference between measured and reported weights, while the alternative hypothesis (H1) posits that measured weights are higher. The significance level (0.05) indicates the threshold for determining whether to reject the null hypothesis based on the calculated p-value.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. In this question, it is assumed that the differences between measured and reported weights follow an approximately normal distribution, which is crucial for applying certain statistical tests, such as the t-test, to analyze the paired sample data.