You're correct that the sum of the relative frequencies should be 100%, and I appreciate the clarification. Let's go through the calculation again using the correct method.

Step 1: Calculate Total Frequencies

• Nonfiltered Cigarettes:
  • Total Frequency = $2 + 0 + 14 + 8 + 1 = 25$
• Filtered Cigarettes:
  • Total Frequency = $2 + 1 + 7 + 15 = 25$

Step 2: Calculate Relative Frequencies

To calculate the percentage frequency for each class, we'll use the formula:

$\text{Percentage Frequency} = \left(\frac{\text{Class Frequency}}{\text{Total Frequency}}\right) \times 100\%$

Nonfiltered Cigarettes:

• 17-22 mg: $\frac{2}{25} \times 100 = 8\%$
• 23-28 mg: $\frac{0}{25} \times 100 = 0\%$
• 29-34 mg: $\frac{14}{25} \times 100 = 56\%$
• 35-40 mg: $\frac{8}{25} \times 100 = 32\%$
• 41-46 mg: $\frac{1}{25} \times 100 = 4\%$

These frequencies indeed sum to 100%.

Filtered Cigarettes:

• 5-10 mg: $\frac{2}{25} \times 100 = 8\%$
• 11-16 mg: $\frac{1}{25} \times 100 = 4\%$
• 17-22 mg: $\frac{7}{25} \times 100 = 28\%$
• 23-28 mg: $\frac{15}{25} \times 100 = 60\%$

Again, these frequencies sum to 100%.

Step 3: Construct the Relative Frequency Table

Here’s the corrected table:

Tar (mg)	Relative Frequency (Nonfiltered)	Relative Frequency (Filtered)
5-10	0%	8%
11-16	0%	4%
17-22	8%	28%
23-28	0%	60%
29-34	56%	0%
35-40	32%	0%
41-46	4%	0%

Step 4: Analyze Effectiveness of Filters

• Nonfiltered Cigarettes: The highest relative frequencies are in the 29-34 mg and 35-40 mg ranges.
• Filtered Cigarettes: The highest relative frequencies are in the 17-22 mg and 23-28 mg ranges, which are significantly lower than the tar content in nonfiltered cigarettes.

Conclusion: The filters appear to be effective as they shift the relative frequencies to lower tar ranges.

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Would you like further clarification, or do you have any additional questions?

Here are some related questions:

1. How would the results change if there were a greater total frequency for filtered cigarettes?
2. What is the impact of rounding errors in calculating percentage frequencies?
3. How can you interpret relative frequencies in terms of probability?
4. What other statistical measures could be used to analyze this data?
5. How does the shape of the distribution affect the interpretation of these frequencies?

Tip: Always re-check your calculations to ensure all relative frequencies add up to 100%.