Questions: The gas mileages (in miles per gallon) for 27 cars are shown in the frequency distribution. Approximate the mean of the frequency distribution. Gas Mileage (in miles per gallon) Frequency --- --- 31-35 8 36-40 13 41-45 2 46-50 4 The approximate mean of the frequency distribution is □ (Round to one decimal place as needed.)

Solution Steps

To approximate the mean of a frequency distribution, we first need to find the midpoint of each class interval. The midpoint is calculated as the average of the lower and upper bounds of each interval.

For the class intervals:

• \(31-35\), the midpoint is \(\frac{31 + 35}{2} = 33.0\)
• \(36-40\), the midpoint is \(\frac{36 + 40}{2} = 38.0\)
• \(41-45\), the midpoint is \(\frac{41 + 45}{2} = 43.0\)
• \(46-50\), the midpoint is \(\frac{46 + 50}{2} = 48.0\)

Next, we multiply each midpoint by the frequency of its class interval to find the sum of the products.

\[ \begin{align_} 33.0 \times 8 &= 264.0 \\ 38.0 \times 13 &= 494.0 \\ 43.0 \times 2 &= 86.0 \\ 48.0 \times 4 &= 192.0 \\ \end{align_} \]

Add the products obtained in the previous step to get the total sum.

\[ 264.0 + 494.0 + 86.0 + 192.0 = 1036.0 \]

The total number of data points is the sum of all frequencies.

\[ 8 + 13 + 2 + 4 = 27 \]

Finally, divide the sum of the products by the total number of data points to find the approximate mean.

\[ \text{Approximate Mean} = \frac{1036.0}{27} \approx 38.3704 \]

Final Answer