Use the following grouped data set (and your calculator if you want to and know how) to find the quantity requested below.
\begin{tabular}{|c|c|}
\hline Classes & Frequency \\
\hline $10-< 20$ & 23 \\
$20-< 30$ & 21 \\
$30-< 40$ & 24 \\
$40-< 50$ & 5 \\
$50-< 60$ & 25 \\
$60-< 70$ & 17 \\
\hline & 115 \\
\hline
\end{tabular}
Calculate the sample mean for the data set. Enter your answer using 2 decimal places.
Note: $\sum x^{2}=205,275$

Step 1 ：Given the following grouped data set: \begin{tabular}{|c|c|} \hline Classes & Frequency \\ \hline $10-<20$ & 23 \\ $20-<30$ & 21 \\ $30-<40$ & 24 \\ $40-<50$ & 5 \\ $50-<60$ & 25 \\ $60-<70$ & 17 \\ \hline & 115 \\ \hline \end{tabular}

Step 2 ：We can calculate the sample mean for the data set using the formula: \[\bar{x} = \frac{\sum f_i x_i}{\sum f_i}\] where $f_i$ is the frequency of each class, $x_i$ is the midpoint of each class, and the sum is taken over all classes.

Step 3 ：The midpoint of each class can be calculated as the average of the lower and upper bounds of the class. The midpoints for the given classes are: [15.0, 25.0, 35.0, 45.0, 55.0, 65.0].

Step 4 ：Multiplying each midpoint by its corresponding frequency and summing these products gives us $\sum f_i x_i = 4415.0$.

Step 5 ：The sum of the frequencies, $\sum f_i$, is 115.

Step 6 ：Substituting these values into the formula for the sample mean gives us: \[\bar{x} = \frac{4415.0}{115} = 38.391304347826086\]

Step 7 ：Rounding to two decimal places, we find that the sample mean for the data set is \(\boxed{38.39}\).