10. Hypothesis Testing for Two Samples
Two Proportions
2:30 minutesProblem 9.4.10b
Textbook Question
Second-Hand Smoke Samples from Data Set 15 “Passive and Active Smoke” include cotinine levels measured in a group of smokers ( n = 40, x_bar = 172.48 ng/mL, 119.50 ng/mL ) and a group of nonsmokers not exposed to tobacco smoke ( n = 40, x_bar = 16.35 ng/mL, 62.53 ng/mL ). Cotinine is a metabolite of nicotine, meaning that when nicotine is absorbed by the body, cotinine is produced.
b. The 40 cotinine measurements from the nonsmoking group consist of these values (all in ng/mL): 1, 1, 90, 244, 309, and 35 other values that are all 0. Does this sample appear to be from a normally distributed population? If not, how are the results from part (a) affected?
Verified step by step guidance1
Step 1: Understand the problem. The goal is to determine whether the sample of cotinine measurements from the nonsmoking group appears to come from a normally distributed population. If it does not, we need to consider how this affects the results from part (a).
Step 2: Analyze the data distribution. The nonsmoking group has 40 measurements, with 35 values being 0, and the remaining values are 1, 1, 90, 244, and 309. A normal distribution typically has a symmetric, bell-shaped curve, so we need to assess whether this data aligns with that shape.
Step 3: Use a histogram or boxplot to visualize the data. Plot the data to observe its shape. If the data is heavily skewed or has extreme outliers (e.g., the values 244 and 309), it is unlikely to be normally distributed.
Step 4: Perform a normality test. Use a statistical test such as the Shapiro-Wilk test or Anderson-Darling test to formally assess normality. These tests will provide a p-value, and if the p-value is below a significance level (e.g., 0.05), we reject the null hypothesis that the data is normally distributed.
Step 5: Interpret the results. If the data is not normally distributed, the results from part (a) may be affected because many statistical methods (e.g., t-tests) assume normality. In such cases, alternative methods like nonparametric tests or data transformations may be needed to ensure valid conclusions.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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. It is characterized by its bell-shaped curve, defined by its mean and standard deviation. Understanding whether a dataset follows a normal distribution is crucial for applying many statistical tests, which often assume normality.
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Cotinine Levels
Cotinine is a metabolite of nicotine, commonly used as a biomarker to measure exposure to tobacco smoke. In the context of the provided data, cotinine levels are compared between smokers and nonsmokers to assess the impact of second-hand smoke. Analyzing these levels helps in understanding the health implications of tobacco exposure and the effectiveness of smoking cessation efforts.
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Statistical Tests for Normality
Statistical tests for normality, such as the Shapiro-Wilk test or the Kolmogorov-Smirnov test, are used to determine if a dataset follows a normal distribution. If the sample does not appear to be normally distributed, it may affect the validity of parametric tests, which rely on this assumption. Non-normal data may require the use of non-parametric tests or data transformation techniques to ensure accurate analysis.
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