To calculate the approximate mean of a frequency distribution, we use the formula:

$\text{Mean} = \frac{\sum (f \cdot x)}{N}$

Where:

• $f$ is the frequency of each class.
• $x$ is the midpoint of each class.
• $N$ is the total number of observations (sum of frequencies).

Step-by-Step Calculation:

1. Identify midpoints for each class interval:

   • The midpoint of the interval $28-31$ is $\frac{28+31}{2} = 29.5$
   • The midpoint of the interval $32-35$ is $\frac{32+35}{2} = 33.5$
   • The midpoint of the interval $36-39$ is $\frac{36+39}{2} = 37.5$
   • The midpoint of the interval $40-43$ is $\frac{40+43}{2} = 41.5$

2. Multiply each midpoint by its corresponding frequency:

   • For the first interval, $29.5 \times 9 = 265.5$
   • For the second interval, $33.5 \times 12 = 402$
   • For the third interval, $37.5 \times 3 = 112.5$
   • For the fourth interval, $41.5 \times 6 = 249$

3. Calculate the total of $f \cdot x$: $265.5 + 402 + 112.5 + 249 = 1029$

4. Find the total number of observations (N): $N = 9 + 12 + 3 + 6 = 30$

5. Calculate the mean: $\text{Mean} = \frac{1029}{30} = 34.3$

Conclusion:

The approximate mean of the gas mileage distribution is 34.3 miles per gallon.

Would you like more details or have any questions? Here's a related list of questions to explore:

1. How do you interpret the frequency distribution in a practical context?
2. What would happen if the intervals were not equal?
3. How can we calculate the standard deviation of a frequency distribution?
4. What is the importance of using midpoints when calculating the mean of a frequency distribution?
5. How would this mean change if the highest class interval were extended?

Tip: When working with frequency distributions, always check if the intervals are equally spaced to ensure accurate midpoint calculations!