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The problem provides a table of values and their frequencies for two data sets, A and B, each consisting of 27 values. It then asks which of the given statements correctly compares the means of the two data sets.

Let's go through the steps to compare the means of data sets A and B.

### Step 1: Calculate the Mean of Data Set A

For Data Set A, we will use the formula for the mean:
\[
\text{Mean of A} = \frac{\sum (\text{Value} \times \text{Frequency})}{\text{Total Frequency}}
\]

Using the values and frequencies given:

\[
\text{Mean of A} = \frac{(30 \times 2) + (34 \times 4) + (38 \times 7) + (42 \times 8) + (46 \times 6)}{27}
\]

\[
\text{Mean of A} = \frac{60 + 136 + 266 + 336 + 276}{27} = \frac{1074}{27} \approx 39.78
\]

### Step 2: Calculate the Mean of Data Set B

Similarly, we calculate the mean for Data Set B:

\[
\text{Mean of B} = \frac{(30 \times 3) + (34 \times 9) + (38 \times 7) + (42 \times 4) + (46 \times 4)}{27}
\]

\[
\text{Mean of B} = \frac{90 + 306 + 266 + 168 + 184}{27} = \frac{1014}{27} \approx 37.56
\]

### Step 3: Compare the Means

From the calculations:

- Mean of Data Set A ≈ 39.78
- Mean of Data Set B ≈ 37.56

Clearly, the mean of Data Set A is greater than the mean of Data Set B.

### Conclusion:
The correct answer is:
**A) The mean of data set A is greater than the mean of data set B.**

Would you like further details or have any questions?

Here are some related questions:

1. How do frequency tables help in calculating the mean?
2. What is the difference between mean and median in data sets?
3. How can you determine if one data set is more spread out than another?
4. Why might the mean not fully represent the central tendency of a data set?
5. How would outliers affect the mean of a data set?
6. Can you compare the means of data sets without calculating them directly?
7. What are some ways to visually represent frequency data?
8. How does the concept of weighted mean relate to this problem?

**Tip:** When calculating the mean from a frequency table, always ensure that the total frequency is consistent with the number of data points in each set.

Explore how to compare the means of two data sets using frequency tables and statistical analysis. This problem involves calculating means from frequency data and determining which data set has a higher mean value.

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