The Mean Tail Content Of A Simple Random Sample Of 25 Unfilt

The Mean Tar Content Of A Simple Random Sample Of 25 Unfiltered King S

The mean tar content of a simple random sample of 25 unfiltered king-size cigarettes is 21.1 mg, with a standard deviation of 3.2 mg. The mean tar content of a simple random sample of 25 filtered 100 mm cigarettes is 13.2 mg with a standard deviation of 3.7 mg. Assume that the two samples are independent, simple, random samples, selected from normally distributed populations. Do not assume that the population standard deviations are equal. Use a 0.05 significance level to test the claim that unfiltered king-size cigarettes have mean tar content greater than that of filtered 100 mm cigarettes.

Paper For Above instruction

The objective of this analysis is to evaluate whether unfiltered king-size cigarettes have a higher average tar content than filtered 100 mm cigarettes, based on a statistical hypothesis test at the 0.05 significance level. Given the data collected from two independent, randomly-selected samples, we will perform a two-sample t-test considering unequal variances (Welch’s t-test), which is appropriate given the differences in sample standard deviations and sample sizes.

The salient data points are as follows: the unfiltered king-size cigarette sample has a mean tar content of 21.1 mg with a standard deviation of 3.2 mg, based on 25 samples. Concurrently, the filtered 100 mm cigarette sample has a mean tar content of 13.2 mg with a standard deviation of 3.7 mg, also based on 25 samples. Since the samples are independent and drawn from normally distributed populations, the assumptions for the t-test hold reasonably.

Hypotheses Formulation

The null hypothesis (H0): There is no difference in mean tar content, or the mean tar content of unfiltered king-size cigarettes is less than or equal to that of filtered cigarettes. Mathematically, H0: μ1 ≤ μ2, where μ1 is the mean tar content of unfiltered cigarettes, and μ2 is that of filtered cigarettes.

The alternative hypothesis (H1): Unfiltered king-size cigarettes have a greater mean tar content than filtered cigarettes. Mathematically, H1: μ1 > μ2. This constitutes a one-tailed test.

Statistical Calculation

To perform the test, we use Welch’s t-test formula for independent samples with unequal variances:

t = (X̄1 - X̄2) / √(s1²/n1 + s2²/n2)

Where:

- X̄1 = 21.1 mg (mean of unfiltered cigarettes)

- X̄2 = 13.2 mg (mean of filtered cigarettes)

- s1 = 3.2 mg (standard deviation of unfiltered cigarettes)

- s2 = 3.7 mg (standard deviation of filtered cigarettes)

- n1 = n2 = 25 (sample sizes)

Calculating the numerator: 21.1 - 13.2 = 7.9 mg

Calculating the denominator: √((3.2²/25) + (3.7²/25)) = √((10.24/25) + (13.69/25)) = √(0.4096 + 0.5476) = √0.9572 ≈ 0.9784 mg

So, t ≈ 7.9 / 0.9784 ≈ 8.07

Degrees of Freedom and Critical Value

Using Welch’s approximation for degrees of freedom:

df ≈ [(s1²/n1 + s2²/n2)²] / [ (s1⁴) / (n1² (n1 - 1)) + (s2⁴) / (n2² (n2 - 1)) ]

Calculations:

- Numerator: (0.4096 + 0.5476)² = (0.9572)² ≈ 0.916

- Denominator:

- (s1⁴)/(n1²(n1-1)) = (10.24²) / (25²24) = (104.86) / (625*24) ≈ 104.86 / 15000 ≈ 0.007

- (s2⁴)/(n2²(n2-1)) = (13.69²) / (25²24) = (187.57) / 15000 ≈ 0.0125

- Sum: 0.007 + 0.0125 = 0.0195

- Degrees of freedom: 0.916 / 0.0195 ≈ 46.9

With approximately 47 degrees of freedom, the critical t-value for a one-tailed test at α=0.05 is about 1.68 (from t-distribution tables).

Decision and Conclusion

The calculated t-value of approximately 8.07 is significantly greater than the critical value of 1.68. Therefore, we reject the null hypothesis and conclude that there is strong statistical evidence to support the claim that unfiltered king-size cigarettes have a higher mean tar content than filtered 100 mm cigarettes at the 5% significance level.

This result has health implications, emphasizing the higher tar exposure associated with unfiltered cigarettes. Public health policies could leverage this evidence to inform consumers about the relative risks of various cigarette types, encouraging reductions in tar intake and smoking-related diseases.

Additionally, this analysis underscores the importance of proper statistical testing methodologies when comparing groups with unequal variances and small sample sizes—highlighting the robustness of Welch’s t-test in such contexts.

Limitations and Recommendations

While the findings are statistically significant, it is important to recognize limitations, such as the assumption of normality and the representativeness of the samples. Further research could involve larger, more diverse samples to confirm these results. Continuous monitoring of tobacco product contents alongside health studies will strengthen the evidence base for regulatory actions.

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