Paper For Above instruction
Introduction
The analysis of cigarette tar content comparing unfiltered king-size cigarettes and filtered 100 mm cigarettes involves statistical hypothesis testing to determine whether there is a significant difference between their mean tar levels. Given the sample data and the assumption of normality, an independent two-sample t-test with unequal variances (Welch’s t-test) is appropriate for this comparison. The significance level is set at 0.05 to evaluate the claim that unfiltered king-size cigarettes have higher mean tar content than filtered cigarettes.
Methodology
The data comprises two independent samples: unfiltered king-size cigarettes with a sample mean of 21.1 mg and standard deviation of 3.2 mg, and filtered 100 mm cigarettes with a mean of 13.2 mg and standard deviation of 3.7 mg, each with a sample size of 25. The hypotheses are formulated as:
Null hypothesis (H
0 ): µ
unfiltered
≤ µ filtered
Alternative hypothesis (H
): µ unfiltered > µ
filtered
This is a one-tailed test comparing the means of two independent samples, without assuming equal variances.
Calculations
Step 1: Compute the test statistic
The formula for the t-statistic in Welch's t-test is:
\[ t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]
Where:
\(\bar{x}_1 = 21.1\) mg (mean of unfiltered)
\(s_1 = 3.2\) mg (standard deviation of unfiltered)
\(n_1 = 25\)
\(\bar{x}_2 = 13.2\) mg (mean of filtered)
\(s_2 = 3.7\) mg (standard deviation of filtered)
\(n_2 = 25\)
Calculating the numerator:
\[ 21.1 - 13.2 = 7.9 \]
Calculating the denominator:
\[ \sqrt{\frac{3.2^2}{25} + \frac{3.7^2}{25}} = \sqrt{\frac{10.24}{25} + \frac{13.69}{25}} = \sqrt{0.4096 + 0.5476} = \sqrt{0.9572} ≈ 0.9784 \]
Thus, the t-statistic is:
\[ t = \frac{7.9}{0.9784} ≈ 8.07 \]
Step 2: Degrees of freedom (df)
Using the Welch-Satterthwaite equation:
\[ df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{ \frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1} } \]
Calculating numerator:
\[ (0.4096 + 0.5476)^2 = (0.9572)^2 ≈ 0.915 \]
Calculating denominator:
For the first term:
\[ \frac{(0.4096)^2}{24} = \frac{0.1678}{24} ≈ 0.0070 \]
For the second term:
\[ \frac{(0.5476)^2}{24} = \frac{0.2999}{24} ≈ 0.0125 \]
Adding these:
\[ 0.0070 + 0.0125 = 0.0195 \]
Degrees of freedom:
\[ df ≈ \frac{0.915}{0.0195} ≈ 46.92 \]
<-p>We’ll approximate df ≈ 47 for practical purposes.
Results
Using a t-distribution table or calculator, with df ≈ 47 and significance level α=0.05 for a one-tailed test, the critical t-value is approximately 1.68.
The computed t-value of approximately 8.07 exceeds 1.68 substantially, indicating strong evidence against the null hypothesis.
Conclusion
At the 0.05 significance level, the data provide sufficient evidence to conclude that unfiltered king-size
cigarettes have a statistically higher mean tar content than filtered 100 mm cigarettes. The observed mean difference is substantial, and the calculated t-statistic confirms the significance of this difference.
Discussion
This analysis underscores the impact of filtering on tar content in cigarettes. The significant difference suggests that filtering reduces tar levels, which could have implications for public health and regulatory policies. However, even with statistically significant results, it's important to consider other factors such as potential confounders, sample representativeness, and measurement accuracy before drawing definitive policy conclusions.
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