Math Problem Statement

A cigarette manufacturer claims that the tar content of a new brand of cigarettes is 17 milligrams. A random sample of 30 cigarettes is selected and the tar content measured. The results are shown below in milligrams: 16.9, 16.6, 17.3, 17.5, 17.0, 17.2, 16.1, 16.4, 17.3, 15.9, 17.7, 18.3, 19, 18.5, 15, 15.6, 16.8, 17.1, 17.2, 16.4, 18.1, 17.4, 16.7, 16.9, 16.0, 16.5, 17.8, 18, 17 Is there evidence that the median tar content of this new brand is different from 17 miligrams?

Solution

To assess whether the median tar content of this cigarette brand is statistically different from 17 milligrams, we use a non-parametric test such as the Wilcoxon signed-rank test, which is suitable for comparing a sample median to a hypothesized value. This test does not assume normality and is often used for testing the median of paired or single-sample data.

Steps to Analyze:

  1. Null Hypothesis (H0H_0): The median tar content is 1717 milligrams. H0:Median=17H_0: \text{Median} = 17

  2. Alternative Hypothesis (HaH_a): The median tar content is different from 1717 milligrams. Ha:Median17H_a: \text{Median} \neq 17

  3. Test Statistic: Compute the Wilcoxon signed-rank test statistic by:

    • Calculating the differences between the observed values and the hypothesized median (1717).
    • Ranking the absolute differences and assigning positive/negative signs based on whether the observed value is above or below 1717.
    • Summing the signed ranks to compute the test statistic.
  4. Significance Level (α\alpha): Usually set to 0.050.05.

  5. Decision Rule: Compare the p-value of the test to α\alpha. If pαp \leq \alpha, reject the null hypothesis.

Let’s compute these values step-by-step. numpy as np from scipy.stats import wilcoxon

Data: Tar content in milligrams

tar_content = np.array([16.9, 16.6, 17.3, 17.5, 17.0, 17.2, 16.1, 16.4, 17.3, 15.9, 17.7, 18.3, 19, 18.5, 15, 15.6, 16.8, 17.1, 17.2, 16.4, 18.1, 17.4, 16.7, 16.9, 16.0, 16.5, 17.8, 18, 17])

Hypothesized median value

hypothesized_median = 17

Perform Wilcoxon signed-rank test

stat, p_value = wilcoxon(tar_content - hypothesized_median)

stat, p_valueThe results of the Wilcoxon signed-rank test are as follows:

  • Test Statistic: W=178.0W = 178.0
  • p-value: p=0.791p = 0.791

Interpretation:

At a significance level of α=0.05\alpha = 0.05:

  • Since p=0.791>0.05p = 0.791 > 0.05, we fail to reject the null hypothesis.

Conclusion:

There is insufficient evidence to suggest that the median tar content of the cigarettes is different from 17 milligrams.


Would you like to explore further details, such as the ranking process or alternative methods for verification?

Related Questions:

  1. What is the difference between the Wilcoxon signed-rank test and a one-sample t-test?
  2. How does the sample size affect the power of the Wilcoxon test?
  3. Can this test be used for non-numeric categorical data?
  4. What assumptions does the Wilcoxon signed-rank test make about the data?
  5. How would results differ if we assumed a normal distribution for the data?

Tip:

Use non-parametric tests like Wilcoxon when the data distribution is unknown or violates normality assumptions.

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Math Problem Analysis

Mathematical Concepts

Statistics
Non-parametric Testing
Hypothesis Testing

Formulas

Wilcoxon Signed-Rank Test Statistic
p-value Calculation

Theorems

Wilcoxon Signed-Rank Test Theorem
Null Hypothesis and Alternative Hypothesis Framework

Suitable Grade Level

Undergraduate Statistics